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Speak in Old English manner similar to how Shakespeare writes his sonnets.", "logging": true, "query_route": "" } INITIALIZATION Knowledgebase: ki-dev-large Base Query: in 5 sentences summarize the context Model: gpt-4o-mini-2024-07-18 Use Curl?: None ================================================== **Elapsed Time: 0.00 seconds** ================================================== ROUTING Query type: summary ================================================== **Elapsed Time: 2.49 seconds** ================================================== RAG PARAMETERS Max Context To Include: 100 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.13 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: True Use Similarity search?: False ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 6.52 seconds** ================================================== PRIMER Primer: You are Simon, a highly intelligent personal assistant in a system called KIOS. You are a chatbot that can read knowledgebases through the "CONTEXT" that is included in the user's chat message. Your role is to act as an expert at summarization and analysis. In your responses to enterprise users, prioritize clarity, trustworthiness, and appropriate formality. Be honest by admitting when a topic falls outside your scope of knowledge, and suggest alternative avenues for obtaining information when necessary. Make effective use of chat history to avoid redundancy and enhance response relevance, continuously adapting to integrate all necessary details in your interactions. Use as much tokens as possible to provide a detailed response. ###################### Summarize in 5 sentences the contents of your context. Speak in Old English manner similar to how Shakespeare writes his sonnets. ================================================== **Elapsed Time: 0.18 seconds** ================================================== FINAL QUERY Final Query: CONTEXT: ########## File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 1 Context: # A Cool Brisk Walk Through Discrete Mathematics **Version 2.2.1** **Stephen Davies, Ph.D.** ## Table of Contents 1. Introduction 2. Concepts - Sets - Functions - Relations 3. Applications 4. Conclusion ## Introduction In this document, we explore various topics under discrete mathematics. ## Concepts ### Sets - Definition - Notation - Operations ### Functions - Definition - Types of functions - Examples ### Relations - Definition - Properties | Concept | Description | |-----------|-----------------------------------| | Set | A collection of distinct objects | | Function | A relation from a set to itself | | Relation | A connection between two sets | ## Applications The concepts of discrete mathematics can be applied in various fields, including computer science, information technology, and combinatorial design. ## Conclusion Discrete mathematics provides fundamental knowledge required for numerous applications in modern science and technology. Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution: - **Image Identification and Numbering:** - This is a single image and is referred to as **Image 1**. #### 2. Object Detection and Classification: - **Image 1:** - The image depicts a forest scene with several birch trees. - **Key Features of Detected Objects:** - **Trees:** The trees are characterized by their tall, slender trunks with distinctive white bark and dark horizontal markings typical of birches. - **Forest Ground:** There are grassy elements and what appears to be a forest floor with various green and brown patches. #### 3. Scene and Activity Analysis: - **Image 1:** - **Scene Description:** - The image is an artistic painting capturing a serene forest with several birch trees. - There is no human activity depicted; the scene is calm, evoking a sense of peace and natural beauty. #### 4. Text Analysis: - **Image 1:** - **Detected Text:** - Top: None - Bottom: - "A Cool Brisk Walk" (Title) - "Through Discrete Mathematics" (Subtitle) - "Version 2.2.1" (Version Information) - "Stephen Davies, Ph.D." (Author Name) - **Text Significance:** - The text suggests that this image is probably the cover of a book titled "A Cool Brisk Walk Through Discrete Mathematics" by Stephen Davies, Ph.D. The version number indicates it is version 2.2.1, signifying a possibly updated or revised edition. #### 8. Color Analysis: - **Image 1:** - **Color Composition:** - The dominant colors are various shades of green (foliage), white, and light brown (birch trunks), and a soft blue (background). - **Color Impact:** - The colors create a cool, calming effect, reflecting the serene mood of a walk through a quiet forest. #### 9. Perspective and Composition: - **Image 1:** - **Perspective:** - The image appears to be from a standing eye-level perspective, giving a natural view as one would see while walking through the forest. - **Composition:** - The birch trees are placed prominently in the foreground, with the background filled with dense foliage. The positioning of the text at the bottom complements the overall composition without distracting from the artwork. #### 10. Contextual Significance: - **Image 1:** - **Overall Document Context:** - As a book cover, the image serves the purpose of drawing the reader's attention. The serene and inviting forest scene combined with the title implies a gentle, inviting approach to the subject of discrete mathematics. The image successfully combines art with educational content, aiming to make the subject of discrete mathematics more appealing and accessible to the reader. The natural, calming scenery juxtaposed with an academic theme may suggest that the book intends to provide a refreshing perspective on a typically challenging subject. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 3 Context: # Contents at a glance ## Contents at a glance - **Contents at a glance** …… i - **Preface** …… iii - **Acknowledgements** …… v 1. **Meetup at the trailhead** …… 1 2. **Sets** …… 7 3. **Relations** …… 35 4. **Probability** …… 59 5. **Structures** …… 85 6. **Counting** …… 141 7. **Numbers** …… 165 8. **Logic** …… 197 9. **Proof** …… 223 Also be sure to check out the forever-free-and-open-source instructional videos that accompany this series at [www.allthemath.org](http://www.allthemath.org)! #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 5 Context: # Preface Discrete math is a popular book topic — start Googling around and you’ll find a zillion different textbooks about it. Take a closer look, and you’ll discover that most of these are pretty thick, dense volumes packed with lots of equations and proofs. They’re principled approaches, written by mathematicians and (seemingly) to mathematicians. I speak with complete frankness when I say I’m comforted to know that the human race is well covered in this area. We need smart people who can derive complex expressions and prove theorems from scratch, and I’m glad we have them. Your average computer science practitioner, however, might be better served by a different approach. There are elements to the discrete math mindset that a budding software developer needs experience with. This is why discrete math is (properly, I believe) part of the mandatory curriculum for most computer science undergraduate programs. But for future programmers and engineers, the emphasis should be different than it is for mathematicians and researchers in computing theory. A practical computer scientist mostly needs to be able to use these tools, not to derive them. She needs familiarity, and practice, with the fundamental concepts and the thought processes they involve. The number of times the average software developer will need to construct a proof in graph theory is probably near zero. But the times she'll find it useful to reason about probability, logic, or the principles of collections are frequent. I believe the majority of computer science students benefit most from simply gaining an appreciation for the richness and rigor of #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 8 Context: I'm unable to view images. Please provide the text in Markdown format for me to assist you. Image Analysis: ### Image Analysis: #### 1. **Localization and Attribution:** - **Image Number**: Image 1 #### 2. **Object Detection and Classification:** - **Detected Objects**: The image includes a cup of coffee on a saucer, three coffee beans, and some sparkles implying aroma or steam. - **Object Classification**: - **Cup of Coffee**: A central object, likely made of ceramic and filled with coffee. - **Coffee Beans**: Close to the cup, classifiable as roasted beans. - **Sparkles/Steam**: Above the cup, representing the aroma of the hot coffee. #### 3. **Scene and Activity Analysis:** - **Scene Description**: This image displays a cup of coffee on a saucer with coffee beans around it. Decorative sparkles or steam lines are depicted rising above the coffee, suggesting freshness or warmth. - **Activities**: No human activities observed; the focus is on the presentation of the coffee. #### 4. **Text Analysis:** - No text detected in this image. #### 5. **Diagram and Chart Analysis:** - Not applicable; no diagrams or charts are present in the image. #### 6. **Product Analysis:** - **Product**: A cup of coffee. - **Main Features**: - A white/neutral-colored ceramic cup and saucer filled with coffee. - Roasted coffee beans placed nearby, suggesting the quality or type of coffee used. - **Materials**: Ceramic cup and saucer, roasted coffee beans. - **Colors**: Predominantly brown (coffee and beans) and white (cup and saucer). #### 7. **Anomaly Detection:** - No anomalies detected; the elements are consistent with a typical coffee presentation. #### 8. **Color Analysis**: - **Dominant Colors**: Brown and white. - **Brown**: Evokes warmth and richness, associated with coffee and coffee beans. - **White**: Clean and neutral, ensures focus on the coffee. #### 9. **Perspective and Composition**: - **Perspective**: Bird’s eye view. - **Composition**: - Centralized with the cup of coffee as the main focus. - Balanced by the placement of coffee beans around the cup. - Decorative elements like sparkles/steam give a visual cue towards the coffee's freshness or aroma. #### 10. **Contextual Significance**: - The image can be used in contexts such as coffee advertisements, cafe menus, or articles about coffee. It emphasizes the quality and appeal of the coffee through its visual presentation. #### 11. **Metadata Analysis**: - No metadata available for analysis. #### 12. **Graph and Trend Analysis**: - Not applicable; no graphs or trends are present in the image. #### 13. **Graph Numbers**: - Not applicable; no data points or graph numbers are present in the image. ### Additional Aspects: #### **Ablaufprozesse (Process Flows)**: - Not applicable; no process flows depicted. #### **Prozessbeschreibungen (Process Descriptions)**: - Not applicable; no processes described. #### **Typen Bezeichnung (Type Designations)**: - Not applicable; no type designations specified. #### **Trend and Interpretation**: - Not applicable; no trends depicted. #### **Tables**: - Not applicable; no tables included. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 11 Context: # Understanding Integration and Differentiation People in your family, there will never be 5.3 of them (although there could be 6 someday). The last couple of entries on this list are worth a brief comment. They are math symbols, some of which you may be familiar with. On the right side — in the continuous realm — are \( J \) and \( \frac{d}{dx} \), which you’ll remember if you’ve taken calculus. They stand for the two fundamental operations of integration and differentiation. Integration, which can be thought of as finding “the area under a curve,” is basically a way of adding up a whole infinite bunch of numbers over some range. When you “integrate the function \( x^2 \) from 3 to 5,” you’re really adding up all the tiny, tiny little vertical strips that comprise the area from \( x = 3 \) on the left to \( x = 5 \) on the right. Its corresponding entry in the left-column of the table is \( \Sigma \), which is just a short-hand for “sum up a bunch of things.” Integration and summation are equivalent operations; it’s just that when you integrate, you’re adding up all the (infinitely many) slivers across the real-line continuum. When you sum, you’re adding up a fixed sequence of entries, one at a time, like in a loop. \( \Sigma \) is just the discrete “version” of \( J \). The same sort of relationship holds between ordinary subtraction (\(-\)) and differentiation (\(\frac{d}{dx}\)). If you’ve plotted a bunch of discrete points on \( x \)-\( y \) axes, and you want to find the slope between two of them, you just subtract their \( y \) values and divide by the \( x \) distance between them. If you have a smooth continuous function, on the other hand, you use differentiation to find the slope at a point: this is essentially subtracting the tiny tiny difference between supremely close points and then dividing by the distance between them. Don’t worry; you don’t need to have fully understood any of the integration or differentiation stuff I just talked about, or even to have taken calculus yet. I’m just trying to give you some feel for what “discrete” means, and how the dichotomy between discrete and continuous really runs through all of math and computer science. In this book, we will mostly be focusing on discrete values and structures, which turn out to be of more use in computer science. That’s partially because, as you probably know, computers #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 13 Context: # 1.1 EXERCISES 1. **What’s the opposite of concrete?** Abstract. 2. **What’s the opposite of discrete?** Continuous. 3. **Consider a quantity of water in a glass. Would you call it abstract, or concrete? Discrete, or continuous?** Concrete, since it’s a real entity you can experience with the senses. Continuous, since it could be any number of ounces (or liters, or tablespoons, or whatever). The amount of water certainly doesn’t have to be an integer. (Food for thought: since all matter is ultimately comprised of atoms, are even substances like water discrete?) 4. **Consider the number 27. Would you call it abstract, or concrete? Discrete, or continuous?** Abstract, since you can’t see or touch or smell “twenty-seven.” Probably discrete, since it’s an integer, and when we think of whole numbers we think “discrete.” (Food for thought: in real life, how would you know whether I meant the integer “27” or the decimal number “27.00”? And does it matter?) Clearly it’s discrete. Abstract vs. concrete, though, is a little tricky. If we’re talking about the actual transistor and capacitor that’s physically present in the hardware, holding a thing in charge in some little chip, then it’s concrete. But if we’re talking about the value “1” that is conceptually part of the computer’s currently executing state, then it’s really abstract just like 27 was. In this book, we’ll always be talking about bits in this second, abstract sense. 5. **Consider a bit in a computer’s memory. Would you call it abstract, or concrete? Discrete, or continuous?** Any kind of abstract object that has properties we can reason about. 6. **If math isn’t just about numbers, what else is it about?** Any kind of abstract object that has properties we can reason about. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 14 Context: I'm unable to assist with that. Image Analysis: ### Analysis of Attached Visual Content #### Image Identification and Localization - **Image 1**: Single image provided in the content. #### Object Detection and Classification - **Image 1**: - **Objects Detected**: - Female figurine resembling a prehistoric representation. - The figurine appears to be crafted from a terracotta or similar clay-like material. - **Classification**: - The object is classified under 'Artifacts' and 'Historical Objects'. - **Key Features**: - The sculpture has exaggerated female attributes including a prominent chest and belly, which are indicative of fertility symbols. - The face does not have detailed features, implying focus on bodily form rather than facial details. - The object seems ancient, slightly weathered, suggesting it to be an archaeological artifact. #### Scene and Activity Analysis - **Image 1**: - **Scene Description**: - The image shows a close-up of a single artifact against a neutral background, possibly for the purpose of highlighting the artifact itself. - **Activities**: - No dynamic activity; the object is displayed presumably for appreciation or study. #### Perspective and Composition - **Image 1**: - **Perspective**: - The image is taken from a straight-on, eye-level view, ensuring the object is the primary focus. - Close-up perspective to capture detailed features. - **Composition**: - The object is centrally placed, drawing immediate attention. - The background is plain and undistracting, enhancing the focus on the artifact itself. #### Contextual Significance - **Image 1**: - **Overall Contribution**: - The artifact could be used in educational, historical, or museum contexts to study prehistoric cultures, their art, and societal values. - As a fertility symbol, it contributes to understanding sociocultural aspects of ancient civilizations. #### Color Analysis - **Image 1**: - **Dominant Colors**: - Shades of brown and beige are dominant, corresponding to the natural materials like terracotta. - **Impact on Perception**: - The earthy tones evoke a sense of antiquity and authenticity, reinforcing the perception of it being an ancient artifact. ### Conclusion The provided image is of a terracotta or similar material female figurine, likely of prehistoric origin. It is depicted with a neutral background to focus on its significant features, in particular, the enhanced bodily features indicative of fertility representations. The composition and color tones effectively highlight its historical and cultural importance. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 15 Context: # Chapter 2 ## Sets The place from which we'll start our walk is a body of mathematics called "set theory." Set theory has an amazing property: it's so simple and applicable that almost all the rest of mathematics can be based on it! This is all the more remarkable because set theory itself came along pretty late in the game (as things go) — it was singlehandedly invented by one brilliant man, Georg Cantor, in the 1870s. That may seem like a long time ago, but consider that by the time Cantor was born, mankind had already accumulated an immense wealth of mathematical knowledge: everything from geometry to algebra to calculus to prime numbers. Set theory was so elegant and universal, though, that after it was invented, nearly everything in math was redefined from the ground up to be couched in the language of sets. It turns out that this simple tool is an amazingly powerful way to reason about mathematical concepts of all flavors. Everything else in this book stands on set theory as a foundation. Cantor, by the way, went insane as he tried to extend set theory to fully encompass the concept of infinity. Don't let that happen to you! #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 19 Context: # 2.2. DEFINING SETS Note that two sets with different intensions might nevertheless have the same extension. Suppose \( O \) is "the set of all people over 25 years old" and \( R \) is "the set of all people who wear wedding rings." If our \( \Omega \) is the Davies family, then \( O \) and \( R \) have the same extension (namely, Mom and Dad). They have different intensions, though; conceptually speaking, they're describing different things. One could imagine a world in which older people don't all wear wedding rings, or one in which some younger people do. Within the domain of discourse of the Davies family, however, the extensions happen to coincide. **Fact:** We say two sets are equal if they have the same extension. This might seem unfair to intentionality, but that’s the way it is. So it is totally legit to write: \[ O = R \] since by the definition of set equality, they are in fact equal. I thought this was weird at first, but it’s really no weirder than saying "the number of years the Civil War lasted = Brett Favre’s jersey number when he played for the Packers." The things on the left and right side of that equals sign refer conceptually to two very different things, but that doesn’t stop them from both having the value 4, and thus being equal. By the way, we sometimes use the curly brace notation in combination with a colon to define a set intentionally. Consider this: \[ M = \{ k : k \text{ is between 1 and 20, and a multiple of } 3 \} \] When you reach a colon, pronounce it as “such that.” So this says "M is the set of all numbers \( k \) such that \( k \) is between 1 and 20, and a multiple of 3." (There's nothing special about \( k \), here; I could have picked any letter.) This is an intensional definition since we haven’t listed the specific numbers in the set, but rather given a rule for finding them. Another way to specify this set would be to write: \[ M = \{ 3, 6, 9, 12, 15, 18 \} \] which is an extensional definition of the same set. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 23 Context: # 2.6 Sets of Sets Sets are heterogeneous — a single set can contain four universities, seven integers, and an image — and so it might occur to you that they can contain other sets as well. This is indeed true, but let me issue a stern warning: you can get into deep water very quickly when you start thinking about “sets of sets.” In 1901, in fact, the philosopher Bertrand Russell pointed out that this idea can lead to unsolvable contradictions unless you put some constraints on it. What became known as “Russell’s Paradox” famously goes as follows: consider the set \( R \) of all sets that do not have themselves as a member. For this reason, we’ll be very careful to use curly braces to denote sets, and parentheses to denote ordered pairs. By the way, although the word “coordinate” is often used to describe the elements of an ordered pair, that’s really a geometry-centric word that implies a visual plot of some kind. Normally we won’t be plotting elements like that, but we will still have to deal with ordered pairs. I’ll just call the constituent parts “elements” to make it more general. Three-dimensional points need ordered triples \( (x, y, z) \), and it doesn’t take a rocket scientist to deduce that we could extend this to any number of elements. The question is what to call them, and you do sort of sound like a rocket scientist (or other generic nerd) when you say tuple. Some people rhyme this word with “Drupal,” and others with “couple,” by the way, and there seems to be no consensus. If you have an ordered pair-type thing with 5 elements, therefore, it’s a 5-tuple (or a quintuple). If it has 117 elements, it’s a 117-tuple, and there’s really nothing else to call it. The general term (if we don’t know or want to specify how many elements) is n-tuple. In any case, it’s an ordered sequence of elements that may contain duplicates, so it’s very different than a set. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 36 Context: # CHAPTER 2: SETS might be double-majoring in both. Hence, this group of subsets is neither mutually exclusive nor collectively exhaustive. It’s interesting to think about gender and partitions: when I grew up, I was taught that males and females were a partition of the human race. But now I’ve come to realize that there are non-binary persons who do not identify with either of those genders, and so it’s not a partition after all. ## Question: Is the number of students \( |S| \) equal to the number of off-campus students plus the number of on-campus students? Obviously yes. But why? The answer: because the off-campus and on-campus students form a partition. If we added up the number of freshmen, sophomores, juniors, and seniors, we would also get \( |S| \). But adding up the number of computer science majors and English majors would almost certainly not be equal to \( |S| \), because some students would be double-counted and others counted not at all. This is an example of the kind of beautiful simplicity that partitions provide. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 42 Context: I'm unable to view images or attachments. Please provide the text directly, and I'll help you format it in Markdown. Image Analysis: ### Image Analysis #### 1. Localization and Attribution: - **Image**: There is only one image provided. #### 2. Object Detection and Classification: - **Objects**: - Circular objects on a surface - Rectangular object in the middle - Grid or lines on the surface - Text in the center of the image #### 3. Scene and Activity Analysis: - **Scene Description**: - The scene appears to be a diagram or an illustrative layout on a surface. It depicts connections or pathways between various nodes or points. - **Activities**: - It resembles a chart or a flow diagram used for visualizing a process, hierarchy, or series of steps. #### 4. Text Analysis: - **Extracted Text**: - The text in the center reads: "Prototyping". - **Analysis**: - The term "Prototyping" indicates that the diagram is likely illustrating the steps, components, or elements involved in the process of creating prototypes. This could encompass anything from design thinking steps to development phases. #### 5. Diagram and Chart Analysis: - **Diagram Description**: - The diagram is a visually organized depiction with lines connecting different points. - These points could represent steps, features, or components in a process. - There is no additional text around the objects, making it a high-level illustration. - **Key Insights**: - The primary focus is on the process of "Prototyping." The connections indicate interactions or dependencies between different elements. #### 8. Color Analysis: - **Color Composition**: - The background is white. - The circular objects and lines are black. - The central rectangle has black text and a white background. - **Impact**: - The monochromatic color scheme with white background and black lines/objects makes it straightforward and easy to focus on the structure and relationships within the diagram. #### 9. Perspective and Composition: - **Perspective**: - The image appears to be taken from a bird’s eye view directly above the diagram. - **Composition**: - The central text "Prototyping" is the focal point. - Lines emanate from or lead to this central point, creating a balanced and symmetrical layout. ### Conclusion: The image is a visually minimalist and symmetrical diagram focusing on the concept of "Prototyping". It uses a central text element with connected nodes or points to represent a process or series of interactions. The monochrome color palette ensures clarity and directs attention towards the structure of the illustration without any distractions. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 43 Context: # Chapter 3 ## Relations Sets are fundamental to discrete math, both for what they represent in themselves and for how they can be combined to produce other sets. In this chapter, we’re going to learn a new way of combining sets, called relations. ### 3.1 The Idea of a Relation A relation between a set \( X \) and \( Y \) is a subset of the Cartesian product. That one sentence packs in a whole heck of a lot, so spend a moment thinking deeply about it. Recall that \( X \times Y \) yields a set of ordered pairs, one for each combination of an element from \( X \) and an element from \( Y \). If \( X \) has 5 elements and \( Y \) has 4, then \( X \times Y \) is a set of 20 ordered pairs. To make it concrete, if \( X \) is the set \{ Harry, Ron, Hermione \} and \( Y \) is the set \{ Dr. Pepper, Mt. Dew \}, then \( X \times Y \) is: - \( (Harry, Dr. Pepper) \) - \( (Harry, Mt. Dew) \) - \( (Ron, Dr. Pepper) \) - \( (Ron, Mt. Dew) \) - \( (Hermione, Dr. Pepper) \) - \( (Hermione, Mt. Dew) \) Convince yourself that every possible combination is in there. I listed them out methodically to make sure I didn’t miss any (all the Harry’s first, with each drink in order, then all the Ron’s, etc.), but of course there’s no order to the members of a set, so I could have listed them in any order. Now if I define a relation between \( X \) and \( Y \), I’m simply specifying that certain of these ordered pairs are in the relation, and certain #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 52 Context: # CHAPTER 3. RELATIONS I contend that in this toy example, “isAtLeastAsToughAs” is a partial order, and \( A \) along with \( isAtLeastAsToughAs \) together form a poset. I reason as follows. It’s reflexive, since we started off by adding every dog with itself. It’s antisymmetric, since when we need add both \( (x, y) \) and \( (y, x) \) to the relation. And it’s transitive, because if Rex is tougher than Fido, and Fido is tougher than Cuddles, this means that if Rex and Cuddles were met, Rex would quickly establish dominance. (I’m no zoologist, and am not sure if the last condition truly applies with real dogs. But let’s pretend it does.) It’s called a “partial order” because it establishes a partial, but incomplete, hierarchy among dogs. If we ask, “is dog X tougher than dog Y?” the answer is never ambiguous. We’re never going to say, “well, dog X was superior to dog A, who was superior to dog Y . . .” but then again, dog Y was superior to dog B, who was superior to dog X, so there’s no telling which of X and Y is truly toughest.” No, a partial order, because of its transitivity and antisymmetry, guarantees we never have such an unreconcilable conflict. However, we could have a lack of information. Suppose Rex has never met Killer, and nobody Rex has met has ever met anyone Killer has met. There’s no chain between them. They’re in two separate universes as far as we’re concerned, and we’d have no way of knowing which was toughest. It doesn’t have to be that extreme, though. Suppose Rex established dominance over Cuddles, and Killer also established dominance over Cuddles, but those are the only ordered pairs in the relation. Again, there’s no way to tell whether Rex or Killer is the tougher dog. They’d either too encounter a common opponent that only one of them can beat, or else get together for a throw-down. So a partial order gives us some semblance of structure — the relation establishes a directionality, and we’re guaranteed not to get wrapped up in contradictions — but it doesn’t completely order all the elements. If it does, it’s called a total order. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 68 Context: # 4.1 Outcomes and events Since life is uncertain, we don’t know for sure what is going to happen. But let’s start by assuming we know what things might happen. Something that might happen is called an outcome. You can think of this as the result of an experiment if you want to, although normally we won’t be talking about outcomes that we have explicitly manipulated and measured via scientific means. It’s more like we’re just curious how some particular happening is going to turn out, and we’ve identified the different ways it can turn out and called them outcomes. Now we’ve been using the symbol \( \Omega \) to refer to “the domain of discourse” or “the universal set” or “all the stuff we’re talking about.” We’re going to give it yet another name now: the sample space. \( \Omega \), the sample space, is simply the *set of all possible outcomes*. Any particular outcome — call it \( O \) — is an element of this set, just like in Chapter 1 every conceivable element was a member of the domain of discourse. If a woman is about to have a baby, we might define \( \Omega \) as \{ boy, girl \}. Any particular outcome is either boy or girl (not both), but both outcomes are in the sample space, because both are possible. If we roll a die, we’d define \( \Omega \) as \{ 1, 2, 3, 4, 5, 6 \}. If we’re interested in motor vehicle safety, we might define \( \Omega \) for a particular road trip as \{ safe, accident \}. The outcomes don’t have to be equally likely; an important point we’ll return to soon. In probability, we define an event as a *subset of the sample space*. In other words, an event is a group of related outcomes (though an event might contain just one outcome, or even zero). I always thought this was a funny definition for the word “event”: it’s not the first thing that word brings to mind. But it turns out to be a useful concept, because sometimes we’re not interested in any particular outcome necessarily, but rather in whether the outcome — whatever it is — has a certain property. For instance, suppose at the start of some game, my opponent and I each roll the die, agreeing that the highest roller gets to go first. Suppose he rolls a 2. Now it’s my turn. The \( \Omega \) for my die roll is of course \{ 1, 2, 3 \}. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 77 Context: # 4.4. CONDITIONAL PROBABILITY wait: suppose we had some additional information. Just before the outcome is announced, news is leaked through a Rupert Murdoch news source that the winner is a woman! If we believe this reporter, does that change our expectation about how old the winner is likely to be? Indeed it does. Knowing that the winner is female eliminates Dave from consideration. Looking back at Figure 4.1, we can see that once we know Dave is out of the running, the remaining pool consists of just F, which includes Kelly, Fantasia, and Carrie. The question is, how do we update our probability from 1 to reflect the fact that only these three ladies are left? In this case, F is the background knowledge: we know that the event F has occurred. And we want to know how likely U is to also have occurred. This is found easily: \[ Pr(U|F) = \frac{Pr(U \cap F)}{Pr(F)} = \frac{Pr\{Kelly,Fantasia\}}{Pr\{Kelly,Fantasia,Carrie\}} = \frac{4}{5} = 0.8. \] Our estimated chance of an underage winner doubled once we found out she was female (even though we don’t yet know which female). If you stare at the equation and diagram, you’ll see the rationale for this formula. Kelly and Fantasia originally had only 4 of the entire probability between them. But once David was asked, the question became: “what percentage of the remaining probability do Kelly and Fantasia have?” The answer was no longer 4 out of 1, but 4 out of 5, since only 5 of the whole was left post-David. This is why we divided by Pr(F): that’s what we know remains given our background fact. Now in this case, the conditional probability was higher than the original probability. Could it ever be lower? Easily. Consider the probability of a rock-star winner, Pr(R). A priori, it’s .7. But #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 81 Context: # 4.5 TOTAL PROBABILITY Pr(A) = Pr(A ∩ C₁) + Pr(A ∩ C₂) + ... + Pr(A ∩ C_N) = Pr(A|C₁)Pr(C₁) + Pr(A|C₂)Pr(C₂) + ... + Pr(A|C_N)Pr(C_N) = ∑ᵏ₌₁ Pr(A|Cₖ)Pr(Cₖ) is the formula.¹ Let's take an example of this approach. Suppose that as part of a promotion for Muvico Cinemas movie theatre, we're planning to give a door prize to the 1000th customer this Saturday afternoon. We want to know, though, the probability that this person will be a minor. Figuring out how many patrons overall will be under 18 might be difficult. But suppose we're showing these three films on Saturday: *Spider-Man: No Way Home*, *Here Before*, and *Sonic the Hedgehog 2*. We can estimate the fraction of each movie's viewers that will be minors: .6, .01, and .95, respectively. We can also predict how many tickets will be sold for each film: 2,000 for *Spider-Man*, 500 for *Here Before*, and 1,000 for *Sonic*. Applying frequentist principles, we can compute the probability that a particular visitor will be seeing each of the movies: - \(Pr(Spider-Man) = \frac{2000}{2000 + 500 + 1000} = 0.571\) - \(Pr(Here \ Before) = \frac{500}{2000 + 500 + 1000} = 0.143\) - \(Pr(Sonic) = \frac{1500}{2000 + 500 + 1000} = 0.286\) ¹ If you're not familiar with the notation in that last line, realize that \(Σ\) (a capital Greek "sigma") just represents a sort of loop with a counter. The \(k\) in the sign means that the counter is \(k\) and starts at 1. The "N" above the line means the counter goes up to \(N\), which is its last value. And what does the loop do? It adds up a cumulative sum. The thing being added to the total each time through the loop is the expression to the right of the sign. The last line with the \(Σ\) is just a more compact way of expressing the preceding line. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 85 Context: # 4.6. BAYES' THEOREM See how that works? If I do have the disease (and there's a 1 in 1,000 chance of that), there's a .99 probability of me testing positive. On the other hand, if I don't have the disease (a 999 in 1,000 chance of that), there's a .01 probability of me testing positive anyway. The sum of those two mutually exclusive probabilities is 0.01098. Now we can use our Bayes’ Theorem formula to deduce: \[ P(D|T) = \frac{P(T|D) P(D)}{P(T)} \] \[ P(D|T) = \frac{.99 \cdot \frac{1}{1000}}{0.01098} \approx .0902 \] Wow. We tested positive on a 99% accurate medical exam, yet we only have about a 9% chance of actually having the disease! Great news for the patient, but a head-scratcher for the math student. How can we understand this? Well, the key is to look back at that Total Probability calculation in equation 4.1. Remember that there were two ways to test positive: one where you had the disease, and one where you didn't. Look at the contribution to the whole that each of those two probabilities produced. The first was 0.00099, and the second was 0.9999, over ten times higher. Why? Simply because the disease is so rare. Think about it: the test fails once every hundred times, but a random person only has the disease once every thousand times. If you test positive, it’s far more likely that the test screwed up than that you actually have the disease, which is rarer than blue moons. Anyway, all of this about diseases and tests is a side note. The main point is that Bayes' Theorem allows us to recast a search for \(P(X|Y)\) into a search for \(P(Y|X)\), which is often easier to find numbers for. One of many computer science applications of Bayes' Theorem is in text mining. In this field, we computationally analyze the words in documents in order to automatically classify them or form summaries or conclusions about their contents. One goal might be to identify the true author of a document, given samples of the writing of various suspected authors. Consider the Federalist Papers, #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 88 Context: # CHAPTER 4. PROBABILITY Pr(L|F). Considering only the women, then, you compute Pr(L|F) = \(\frac{22}{198} \approx .111\) from the data in the table. Wait a minute. That’s exactly what we had before. Learning that we had chosen a woman told us nothing useful about her handedness. That’s what we mean by saying that the L and F events are independent of each other. The shrewd reader may object that this was a startling coincidence: the numbers worked out exactly perfectly to produce this result. The proportion of left-handed females was precisely the same as that of left-handed males, down to the penny. Is this really likely to occur in practice? And if not, isn’t independence theoretical as to be irrelevant? There are two ways of answering that question. The first is to admit that in real life, of course, we’re bound to get some noise in our data, just because the sample is finite and there are random fluctuations in who we happen to survey. For the same reason, if we flipped an ordinary coin 1,000 times, we aren’t likely to get exactly 500 heads. But that doesn’t mean we should rush to the conclusion that the coin is biased. Statisticians have sophisticated ways of answering this question by computing how much the experimental data needs to deviate from what we’d expect before we raise a red flag. Suffice to say here that even if the contingency table we collect isn’t picture perfect, we may still conclude that two events are independent if they’re “close enough” to independence. The other response, though, is that yes, the burden of proof is indeed on independence, rather than on non-independence. In other words, we shouldn’t start by cavalierly assuming all the events we’re considering are in fact independent, and only changing our mind if we see unexpected correlations between them. Instead, we should always be suspicious that two events will affect each other in some way, and only conclude they’re independent if the data we collect works out more or less “evenly” as in the example above. To say that Pr(A|B) is the same as Pr(A) is an aggressive statement, outside the norm, and we shouldn’t assume it without strong evidence. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 97 Context: # 5.1. GRAPHS ## Figure 5.2: A directed graph. ``` Sonny Liston / \ Muhammad Ali George Foreman \ / Joe Frazier ``` ## Figure 5.3: A weighted (and undirected) graph. ``` Washington, DC | 50 | Fredericksburg 150 | 60 | Virginia Beach | 100 | Richmond ``` **adjacent.** If two vertices have an edge between them, they are said to be adjacent. **connected.** The word connected has two meanings: it applies both to pairs of vertices and to entire graphs. We say that two vertices are connected if there is at least one path between them. Each vertex is therefore "reachable" from the other. In Figure 5.1, President and actor are connected, but Ford's Theatre and Civil War are not. "Connected" is also used to describe entire graphs, if every node can be reached from all others. It's easy to see that Fig. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 99 Context: # 5.1. GRAPHS A graph of dependencies like this must be both **directed** and **acyclic**, or it wouldn’t make sense. Directed, of course, means that task X can require task Y to be completed before it, without the reverse also being true. If they both depended on each other, we’d have an infinite loop, and no brownies could ever get baked! Acyclic means that no kind of cycle can exist in the graph, even one that goes through multiple vertices. Such a cycle would again result in an infinite loop, making the project hopeless. Imagine if there were an arrow from **bake for 30 mins** back to **grease pan** in Figure 5.4. Then, we’d have to grease the pan before pouring the goop into it, and we’d also have to bake before greasing the pan! We’d be stuck right off the bat: there’d be no way to complete any of those tasks since they all indirectly depend on each other. A graph that is both directed and acyclic (and therefore free of these problems) is sometimes called a **DAG** for short. ``` pour brown stuff in bowl ├── crack two eggs ├── measure 2 tbsp oil ├── preheat oven ├── grease pan └── mix ingredients └── pour into pan ├── bake for 30 mins └── cool └── enjoy! ``` *Figure 5.4: A DAG.* #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 102 Context: # CHAPTER 5. STRUCTURES ## Figure 5.7: A graph depicting a symmetric endorelationship. ``` (Harry, Ron) (Ron, Harry) (Ron, Hermione) (Hermione, Ron) (Harry, Harry) (Neville, Neville) ``` ``` Harry | | Ron / \ / \ Hermione Neville ``` Notice how the loops (edges from a node back to itself) in these diagrams represent ordered pairs in which both elements are the same. Another connection between graphs and sets has to do with partitions. Figure 5.7 was not a connected graph: Neville couldn't be reached from any of the other nodes. Now consider: isn't a graph like this similar in some ways to a partition of \( A \) — namely, this one? - \( \{ Harry, Ron, Hermione \} \) - \( \{ Neville \} \) We've simply partitioned the elements of \( A \) into the groups that are connected. If you remove the edge between Harry and Ron in that graph, you have: - \( \{ Harry \} \) - \( \{ Ron, Hermione \} \) - \( \{ Neville \} \) Then add one between Hermione and Neville, and you now have: - \( \{ Harry \} \) - \( \{ Ron, Hermione, Neville \} \) #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 106 Context: # CHAPTER 5. STRUCTURES Let's run this algorithm in action on a set of Facebook users. Figure 5.9 depicts eleven users and the friendships between them. First, we choose Greg as the starting node (not for any particular reason, just that we have to start somewhere). We mark him (in grey on the diagram) and put him in the queue (the queue contents are listed at the bottom of each frame, with the front of the queue on the left). Then, we begin our loop. When we take Greg off the queue, we visit him (which means we "do whatever we need to do to Greg") and then mark and enqueue his adjacent nodes Chuck and Lizzy. It does not matter which order we put them into the queue, just as it did not matter what node we started with. In pane 3, Chuck has been dequeued, visited, and his adjacent nodes put on the queue. Only one node gets enqueued here — Adrian — because obviously Greg has already been marked (and even visited, no less) and this marking allows us to be smart and not re-enqueue him. It’s at this point that the “breadth-first” feature becomes apparent. We've just finished with Chuck, but instead of proceeding Adrian next, we resume with Lizzy. This is because she has been waiting patiently on the queue, and her turn has come up. So we lay Adrian aside (in the queue, of course) and visit Lizzy, enqueueing her neighbor Elaine in the process. Then, we go back to Adrian. The process continues, in “one step on the top path, one step on the bottom path” fashion, until our two exploration paths actually meet each other on the back end. Visiting Lizzy causes us to enqueue Brittany, and then when we take Kim off the queue, we do not re-enqueue Brittany because she has been marked and we know she's already being taken care of. For space considerations, Figure 5.9 leaves off at this point, but our course would continue visiting nodes in the queue until the queue was empty. As you can see, Hank and Daniel will not have visited at all in this process: this is apparently because nobody they know knows anybody in the Greg crowd, and so there's no way to reach them from Greg. This is what I meant earlier by saying that as a side effect, the BFT algorithm tells us whether the graph is connected or not. All we have to do is start somewhere, run BFT, and then see whether any nodes have not been marked and visited. If there are any, we can continue with another starting point, and #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 107 Context: # 5.1. GRAPHS then repeat the process. | Step | Queue | Current Node | Visited Nodes | |------|---------------|--------------|---------------| | 1 | Greg | Jerry | | | 2 | Clark | Adrian | | | 3 | Judy | Kim | | | 4 | Adrian | Ethel | | | 5 | Ethel | Judy | | | 6 | Kim | Balthus | | | 7 | Balthus | | | Figure 5.9: The stages of breadth-first traversal. Marked nodes are grey, and visited nodes are black. The order of visitation is: G, C, I, A, E, J, K, F, B. ## Depth-first traversal (DFT) With depth-first traversal, we explore the graph boldly and recklessly. We choose the first direction we see, and plunge down it all the way to its depths, before reluctantly backing out and trying the other paths from the start. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 108 Context: # CHAPTER 5. STRUCTURES Start with an empty stack: Push a triangle, and we have: Push a star, and we have: Push a heart, and we have: Pop the heart, and we have: Push a club, and we have: Pop the club, and we have: Pop the star, and we have: Pop the triangle. We’re empty again: Figure 5.10: A stack in action. The horizontal bar marks the bottom of the stack, and the elements are pushed and popped from the top. The algorithm is almost identical to BFT, except that instead of a queue, we use a stack. A stack is the same as a queue except that instead of putting elements on one end and taking them off the other, you add and remove to the same end. This “end” is called the top of the stack. When we add an element to this end, we say we **push** it on the stack, and when we remove the top element, we say we **pop** it off. You can think of a stack as...well, a stack, whether of books or cafeteria trays or anything else. You can’t get anything out of the middle of a stack, but you can take items off and put more items on. The first item pushed is always the last one to be popped, and the most recent one pushed is always ready to be popped back off, and so a stack is also sometimes called a “LIFO” (last-in-first-out). The depth-first traversal algorithm itself looks like déjà vu all over again. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 113 Context: 5.1. GRAPHIS ================================= We've marked each node with a diamond containing the tentative shortest distance to it from Bordeaux. This is 0 for Bordeaux itself (since it’s 0 kilometers away from itself, duh), and infinity for all the others, since we haven't explored anything yet, and we want to start off as sensitively as possible. We'll update these distances to lower values as we find paths to them. We start with Bordeaux as the “current node,” marked in grey. In frame 2, we update the best-possible-path and the distance-of-that-path for each of Bordeaux’s neighbors. Nantes, we discover, is no longer “infinity away,” but a mere 150 km away, since there is a direct path to it from Bordeaux. Vicky and Toulouse are similarly updated. Note the heavy arrowed lines on the diagram, showing the best path (so far) to each of these cities from Bordeaux. Step 3 tells us to choose the node with the lowest tentative distance as the next current node. So for frame 3, Nantes fits the bill with a (tentative) distance of 150 km. It has only one unmarked neighbor, Paris, which we update with 450 km. Why 450? Because it took 150 to get from the start to Nantes, and another 300 from Nantes to Paris. After updating Paris, Nantes is now its own— we know we'll never encounter a better route to it than from Bordeaux directly. Frame 4 is our first time encountering a node that already has a non-infinite tentative distance. In this case, we don’t further update it, because our new opportunity (Bordeaux-to-Toulouse-to-Vichy) is 500 km, which is longer than going from Bordeaux to Toulouse direct. Lyon and Marseille are updated as normal. We now have two unmarked nodes that tie for shortest tentative distance: Paris and Vicky (450 km each). In this case, it doesn’t matter which one we choose. We’ll pick Vicky for no particular reason. Frame 5 then shows some interesting activity. We do not update the path to Paris; since it would be 800 km through Vichy, whereas Paris already had a much better 450 km path. Lyon is updated from infinity to 850 km, since we found our first path to it. But Lyon is the really interesting case. It already had a path — Bordeaux-to-Toulouse-to-Lyon — but that path was 800 km, and we have just found a better path: Bordeaux-to-Vichy-to-Lyon, which only #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 115 Context: # 5.1. GRAPHS How can we get all the cities connected to each other so we can safely deliver supplies between any of them, using the least possible amount of road? This isn’t just a military problem. The same issue came up in ancient Rome when aqueducts had to reach multiple cities. More recently, supplying neighborhoods and homes with power, or networking multiple computers with Ethernet cable, involves the same question. In all these cases, we’re not after the shortest route between two points. Instead, we’re sort of after the shortest route “between all the points.” We don’t care how each pair of nodes is connected, provided that they are connected. It’s the total length of the required connections that we want to minimize. To find this, we’ll use **Prim’s algorithm**, a technique named for the somewhat obscure computer scientist Robert Prim who developed it in 1957, although it had already been discovered much earlier (1930, by the Czech mathematician Vojtěch Jarník). Prim’s algorithm turns out to be much easier to carry out than Dijkstra’s algorithm, which I find surprising, since it seems to be solving a problem that’s just as hard. But here’s all you do: ## Prim’s minimal connecting edge set algorithm 1. Choose a node, any node. 2. While not all the nodes are connected, do these steps: 1. Identify the node closest to the already-connected nodes, and connect it to those nodes via the shortest edge. That’s it. Prim’s algorithm is an example of a **greedy algorithm**, which means that it always chooses the immediately obvious short-term best choice available. Non-greedy algorithms can say, “although node X would give the highest short-term satisfaction, I can look ahead and see that choosing Y instead will lead to a better overall result in the long run.” Greedy algorithms, by contrast, always gobble up what seems best at the time. That’s what Prim’s algorithm is doing in step 2a. It looks for the non-connected node. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 118 Context: ```markdown # Chapter 5: Structures ## Figure 5.14 The stages of Prim's minimal connecting edge set algorithm. Heavy lines indicate edges that have been (irrevocably) added to the set. | Step | Details | Value | Connections | |------|---------------|--------|--------------| | 1 | Node A | 98 | Node B, Node C | | 2 | Node A | 100 | Node D | | 3 | Node C | 150 | Node E, Node F | | 4 | Node D | 200 | Node G | | 5 | Node B | 300 | Node H | | 6 | Node E | 400 | Node I | | 7 | Node F | 500 | Node J | ... ``` #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 119 Context: 5.2 Trees ========= you think that the shortest path between any two nodes would land right on this Prim network? Yet if you compare Figure 5.14 with Figure 5.13 you'll see that the quickest way from Bordeaux to Strasbourg is through Marseille, not Vichy. So we end up with the remarkable fact that the shortest route between two points has nothing whatsoever to do with the shortest total distance between all points. Who knew? 5.2 Trees --------- A tree is really nothing but a simplification of a graph. There are two kinds of trees in the world: free trees, and rooted trees. ### Free trees A free tree is just a connected graph with no cycles. Every node is reachable from the others, and there’s only one way to get anywhere. Take a look at Figure 5.15. It looks just like a graph (and it is) but unlike the WWII France graph, it’s more skeletal. This is because in some sense, a free tree doesn’t contain anything “extra.” If you have a free tree, the following interesting facts are true: 1. There’s exactly one path between any two nodes. (Check it!) 2. If you remove any edge, the graph becomes disconnected. (Try it!) 3. If you add any new edge, you end up adding a cycle. (Try it!) 4. If there are n nodes, there are n - 1 edges. (Think about it!) So basically, if your goal is connecting all the nodes, and you have a free tree, you’re all set. Adding anything is redundant, and taking away anything breaks it. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 123 Context: # 5.2 TREES parent is A, and A has no parent. ## child Some nodes have children, which are nodes connected directly below it. A’s children are F and B, C’s are D and E, B’s only child is C, and E has no children. ## sibling A node with the same parent. E’s sibling is D, B’s is F, and none of the other nodes have siblings. ## ancestor Your parent, grandparent, great-grandparent, etc., all the way back to the root. B’s only ancestor is A, while E’s ancestors are C, B, and A. Note that F is not C’s ancestor, even though it’s above it on the diagram; there’s no connection from C to F, except back through the root (which doesn’t count). ## descendant Your children, grandchildren, great-grandchildren, etc., all the way to the leaves. B’s descendants are C, D, and E, while A’s are F, B, C, D, and E. ## leaf A node with no children. F, D, and E are leaves. Note that in a (very) small tree, the root could itself be a leaf. ## internal node Any node that’s not a leaf. A, B, and C are the internal nodes in our example. ## depth (of a node) A node’s depth is the distance (in number of nodes) from it to the root. The root itself has depth zero. In our example, B is of depth 1, E is of depth 3, and A is of depth 0. ## height (of a tree) A rooted tree’s height is the maximum depth of any of its nodes; i.e., the maximum distance from the root to any node. Our example has a height of 3, since the “deepest” nodes are D and E, each with a depth of 3. A tree with just one node is considered to have a height of 0. Bizarrely, but to be consistent, we’ll say that the empty tree has height -1! Strange, but what else could it be? To say it has height 0 seems inconsistent with a one-node tree also having height 0. At any rate, this won’t come up much. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 139 Context: # 5.2. TREES subtree – namely, Jim – and promote him in place of her. Figure 5.28 shows the result. Note that we replaced her with Jim not because it’s okay to blindly promote her right child, but because Jim had no left descendants, and hence he was the left-most node in her right subtree. (If he had left descendants, promoting him would have been just as wrong as promoting Ben. Instead, we would have gone left from Jim until we couldn’t go left anymore, and promoted that node.) ![Figure 5.28: The BST after removing Jessica correctly.](path_to_image) As another example, let’s go whole-hog and remove the root node, Mitch. The result is as shown in Figure 5.29. It’s rags-to-riches for Molly: she got promoted from a leaf all the way to the top. Why Molly? Because she was the left-most descendant of Mitch’s right subtree. To see why this works, just consider that Molly was immediately after Mitch in alphabetical order. The fact that he was a king and she a peasant was misleading. The two of them were actually very close: consecutive, in fact, with in-order traversal. So replacing Mitch with Molly avoids shuffling anybody out of alphabetical order and preserves the all-important BST property. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 142 Context: # CHAPTER 5. STRUCTURES Thirdly, there are specialized data structures you may learn about in future courses, such as AVL trees and red-black trees, which are binary search trees that add extra rules to prevent imbalancing. Basically, the idea is that when a node is inserted (or removed), certain metrics are checked to make sure that the change didn’t cause too great an imbalance. If it did, the tree is adjusted to minimize the imbalance. This comes at a slight cost every time the tree is changed, but prevents any possibility of a lopsided tree that would cause slow lookups in the long run. ## 5.3 Final Word Whew, that was a lot of information about structures. Before we continue our walk in the next chapter with a completely different topic, I’ll leave you with this summary thought. Let \( BST \) be the set of Binary Search Trees, and \( BT \) be the set of Binary Trees. Let \( RT \) be the set of rooted trees, and \( T \) be the set of trees (free or rooted). Finally, let \( CG \) be the set of connected graphs, and \( G \) the set of all graphs. Then we have: \[ BST \subseteq BT \subseteq RT \subseteq CG \subseteq G \] It’s a beautiful thing. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 144 Context: # CHAPTER 5. STRUCTURES 10. How many vertices and edges are there in the graph below? ![Graph](image_reference_placeholder) **7 and 10, respectively.** 11. What's the degree of vertex L? **It has an in-degree of 2, and an out-degree of 1.** 12. Is this graph directed? **Yes.** 13. Is this graph connected? **Depends on what we mean. There are two different notions of “connectedness” for directed graphs. One is strongly connected, which means every vertex is reachable from any other by following the arrows in their specified directions. By that definition, this graph is not connected: there’s no way to get to H from L, for example. It is weakly connected, however, which means that if you ignore the arrowheads and consider it like an undirected graph, it would be connected.** 14. Is it a tree? **No. For one thing, it can’t have any “extra” edges beyond what’s necessary to make it connected, and there’s redundancy here.** 15. Is it a DAG? **All in all, if you look very carefully, you’ll see that there is indeed a cycle: I→J→K→L→I. So if this graph were to represent a typical project workflow, it would be impossible to complete.** 16. If we reversed the direction of the I→J edge, would it be a DAG? **Yes. The edges could now be completed in this order: H, G, L, I, M, K, and finally J.** #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 145 Context: # 5.4. EXERCISES 17. If this graph represented an endorelation, how many ordered pairs would it have? **Answer:** 10. 18. Suppose we traversed the graph below in depth-first fashion, starting with node P. In what order would we visit the nodes? ![Graph](attachment_link_here) **Answer:** There are two possible answers: P, Q, R, S, T, N, O, or else P, O, N, T, S, R, Q. (The choice just depends on whether we go "left" or "right" initially.) Note in particular that either O or Q is at the very end of the list. 19. Now we traverse the same graph breadth-first fashion, starting with node P. Now in what order would we visit the nodes? **Answer:** Again, two possible answers: P, O, Q, N, R, T, S, or else P, Q, R, N, S, T. Note in particular that both O and Q are visited very early. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 148 Context: # CHAPTER 5. STRUCTURES 29. If we wanted to add a new node called “Shepherd” to this tree, where would he go? To Simon’s left. We can put the left-most node of Mal’s right subtree (that would be River) in Mal’s place, and then make Simon (and everything under him) become Wash’s left child. The result would look like this: ``` River / \ Jayne Zoe / \ \ Inara Kaylee Simon / Wash ``` 30. If we wanted to remove the “Mal” node from this tree, how would we do that? Take a moment and convince yourself that this Mal-less tree does in fact satisfy the BST property. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 158 Context: # CHAPTER 6. COUNTING and so on. Once we have the **RISK** permutations, we can generate the **BRISK** permutations in the same way: | B | R | I | S | K | |---|---|---|---|---| | R | B | I | S | K | | R | I | B | S | K | | R | I | S | B | K | | B | I | R | S | K | | T | B | R | I | S | K | | I | R | B | S | K | | I | R | S | K | | I | S | R | K | | B | S | K | | . . . | Another algorithm to achieve the same goal (though in a different order) is as follows: ## Algorithm #2 for enumerating permutations 1. Begin with a set of **n** objects. a. If **n = 1**, there is only one permutation; namely, the object itself. b. Otherwise, remove each of the objects in turn, and prepend that object to the permutations of all the others, creating another permutation each time. I find this one a little easier to get my head around, but in the end it’s personal preference. The permutations of **BRISK** are: “B followed by all the permutations of **RISK**, plus **I** followed by all the permutations of **BISK**, plus **R** followed by all the permutations of **BRSK**, etc.” So the first few permutations of a 4-letter word are: - R I S K - R I K S - R S I K - R S K I - R K I S - R K S I - I R S K - I R K S - I S R K - I S K R - I K R S - I K S R - S R I K - S R K I - S I R K - S I K R - S K R I - S K I R - K R I S - K R S I - K I R S - K I S R - K S R I - K S I R #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 167 Context: # 6.4 Summary Most of the time, counting problems all boil down to a variation of one of the following three basic situations: - \( n^k \) — this is when we have \( k \) different things, each of which is free to take on one of \( n \) completely independent choices. - \( \binom{n}{k} \) — this is when we’re taking a sequence of \( k \) different things from a set of \( n \), but no repeats are allowed. (A special case of this is \( n! \), when \( k = n \).) - \( \binom{n}{k} \) — this is when we’re taking \( k \) different things from a set of \( n \), but the order doesn’t matter. Sometimes it’s tricky to deduce exactly which of these three situations apply. You have to think carefully about the problem and ask yourself whether repeated values would be allowed and whether it matters what order the values appear in. This is often subtle. As an example, suppose my friend and I work out at the same gym. This gym has 18 different weight machines to choose from, each of which exercises a different muscle group. Each morning, #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 206 Context: # CHAPTER 8. LOGIC A statement that has a "truth value," which means that it is either true or false. The statement "all plants are living beings" could be a proposition, as could "Barack Obama was the first African-American President" and "Kim Kardashian will play the title role in Thor: Love and Thunder." By contrast, questions like "are you okay?" cannot be propositions, nor can commands like "hurry up and answer already!" or phrases like "Lynn's newborn schnauzer," because they are not statements that can be true or false. (Linguistically speaking, propositions have to be in the indicative mood.) We normally use capital letters (what else?) to denote propositions, like: - Let **A** be the proposition that UMW is in Virginia. - Let **B** be the proposition that the King of England is female. - Let **C** be the proposition that dogs are carnivores. Don't forget that a proposition doesn't have to be true in order to be a valid proposition (B is still a proposition, for example). It just matters that it is labeled and that it has the potential to be true or false. Propositions are considered atomic. This means that they are indivisible: to the logic system itself, or to a computer program, they are simply an opaque chunk of truth (or falsity) called "A" or whatever. When we humans read the description of A, we realize that it has to do with the location of a particular institution of higher education, and with the state of the union that it might reside (or not reside) in. All this is invisible to an artificially intelligent agent, however, which treats "A" as nothing more than a stand-in label for a statement that has no further discernible structure. So things are pretty boring so far. We can define and label propositions, but none of them have any connections to the others. We change that by introducing logical operators (also called logical connectives) with which we can build up compound constructions out of multiple propositions. The six connectives we’ll learn are: #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 220 Context: # CHAPTER 8. LOGIC - **TraveledToByModeInYear**(Rachel, Germany, plane, 2014) - **¬TraveledToByModeInYear**(Johnny, Mars, spaceship, 1776) Defining multiple inputs gives us more precision in defining relationships. Imagine creating the predicate **AteWithAttitude** and then asserting: - **AteWithAttitude**(lonely boy, spaghetti, gusto) - **¬AteWithAttitude**(Johnny, broccoli, gusto) - **AteWithAttitude**(Johnny, broccoli, trepidation) ## Predicates and Relations The astute reader may have noticed that the **IsFanOf** predicate, above, seems awfully similar to an **isFanOf** relation defined between sets \( P \) (the set of people) and \( R \) (the set of rock bands), where **isFanOf** \( \subseteq P \times R \). In both cases, we have pairs of people/bands for which it’s true, and pairs for which it’s false. Indeed these concepts are identical. In fact, a relation can be defined as the set of ordered pairs (or tuples) for which a predicate is true. Saying **IsFanOf(Rachel, The Beatles)** and **¬IsFanOf(Stephen, The Rolling Stones)** is really just another way of saying “Rachel isFanOf The Beatles” and “Stephen isFanOf The Rolling Stones.” ## Quantifiers One powerful feature of predicate logic is the ability to make grandiose statements about many things at once. Suppose we did want to claim that every state had a governor. How can we do it? We’ll add to our repertoire the notion of quantifiers. There are two kinds of quantifiers in predicate logic, the first of which is called the **universal quantifier**. It’s written **∀** and pronounced “for all.” Here’s an example: - **∀ HasGovernor(z)**. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 225 Context: # 8.2. Predicate Logic Deduce further statements, and you’ll learn about them when you study artificial intelligence later on. Most of them are formalized versions of common sense. “If you know A is true, and you know A=B is true, then you can conclude B is true.” Or “if you know X∧Y is false, and then you discover that Y is true, you can then conclude that X is false.” Etc. The power to produce new truth from existing truth is the hallmark of AI systems, and why this stuff really matters. If you can imagine a program doing this sort of automated reasoning, it will become clear why the precision of something like predicate logic—instead of the sloppiness of English—becomes important. English is a beautiful and poetic language, but its ambiguity is notorious. For example, back in Chapter 3 we used the phrase “some employee belongs to every department” when describing relations. Now consider that English sentence. What does “some employee belongs to every department” actually mean? Does it mean that there is some special employee who happens to hold membership in every department in the company? Or does it mean that no department is empty: all departments have at least one person in them, for crying out loud? The English could mean either. In predicate logic, we’re either asserting: - \(\exists x \, \text{Employee}(x) \land \forall y \, \text{BelongsTo}(x, y)\) or - \(\forall y \, \text{Employee}(x) \land \text{BelongsTo}(x, y)\) These are two very different things. A human being would realize that it’s the third one on the speaker’s mind, drawing from a whole range of experience and common sense and context clues. But a bot has available none of these, and so it demands that the language clearly and unambiguously state exactly what’s meant. English is rife with these ambiguities, especially involving pronouns. > “After John hit George he ran away.” What happened? Did John run away after striking George, fearing that George would retaliate? Or did George run away after getting hit, fearing additional abuse? It’s unclear what “he” refers to, so we can’t say from the sentence alone. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 226 Context: # Chapter 8: Logic Here's a funny one I’ll end with. Consider the sentence **“He made her duck.”** What is intended here? Did some guy reach out with his hand and forcefully push a woman’s head down out of the way of a screaming projectile? Or did he prepare a succulent dish of roasted fowl to celebrate her birthday? Oh, if the computer could only know. If we’d used predicate logic instead of English, it could! #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 233 Context: # 9.1. PROOF CONCEPTS The phone is in my pocket, and has not rung, and I conclude that the plan has not changed. I look at my watch, and it reads 5:17 pm, which is earlier than the time they normally leave, so I know I'm not going to catch them walking out the door. This is, roughly speaking, my thought process that justifies the conclusion that the house will be empty when I pull into the garage. Notice, however, that this prediction depends precariously on several facts. What if I spaced out the day of the week, and this is actually Thursday? All bets are off. What if my cell phone battery has run out of charge? Then perhaps they did try to call me but couldn’t reach me. What if I set my watch wrong and it’s actually 4:17 pm? Etc. Just like a chain is only as strong as its weakest link, a whole proof falls apart if even one step isn’t reliable. Knowledge bases in artificial intelligence systems are designed to support these chains of reasoning. They contain statements expressed in formal logic that can be examined to deduce only the new facts that logically follow from the old. Suppose, for instance, that we had a knowledge base that currently contained the following facts: 1. \( A = C \) 2. \( \neg (C \land D) \) 3. \( (F \lor E) \to D \) 4. \( A \lor B \) These facts are stated in propositional logic, and we have no idea what any of the propositions really mean, but then neither does the computer, so hey. Fact #1 tells us that if proposition A (whatever that may be) is true, then we know C is true as well. Fact #2 tells us that we know CAD is false, which means at least one of the two must be false. And so on. Large knowledge bases can contain thousands or even millions of such expressions. It’s a complete record of everything the system “knows.” Now suppose we learn an additional fact: \( \neg B \). In other words, the system interacts with its environment and comes to the conclusion. #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 236 Context: # CHAPTER 9. PROOF ## COLD You start with one word (like **WARM**) and you have to come up with a sequence of words, each of which differs from the previous by only one letter, such that you eventually reach the ending word (like **COLD**). It's sort of like feeling around in the dark: - **WARM** - **WART** - **WALT** - **WILT** - **WILD** - ... This attempt seemed promising at first, but now it looks like it's going nowhere. ("**WOLD?**" "**CILD?**" Hmm...) After starting over and playing around with it for a while, you might stumble upon: - **WARM** - **WORM** - **WORD** - **CORD** - **COLD** This turned out to be a pretty direct path: for each step, the letter we changed was exactly what we needed it to be for the target word **COLD**. Sometimes, though, you have to meander away from the target a little bit to find a solution, like going from **BLACK** to **WHITE**: - **BLACK** - **CLACK** - **CRACK** - **TRACK** - **TRICK** - **TRICE** #################### File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf Page: 255 Context: # 9.4. FINAL WORD enqueuing, 95 enumerating, 141, 149 equality (for sets), 11 equity (logical operator), 202, 203, 293 Euclid, 229 events, 60, 61 exclusive or, 63, 199 existential quantifier (∃), 214 exponential growth, 123, 145 exponential notation, 169 extensional, 10, 11, 37, 47, 92, 203 Facebook, 87, 97 factorial, 147, 152 Federalist Papers, 77 FIFO, 95 filesystems, 113 Fisher, Ronald, 67 floor operator (⌊⌋), 174 Foreman, George, 92 France, 103 Frazier, Joe, 92 free trees, 11, 242 frequentist, 66 full binary trees, 46 function calls, 46 functions, 65, 210 Fundamental Theorem of Computing, 142, 147 Gauss, Carl Friedrich, 236 Goldbach, Christian, 241 golf, 151, 154 graphs, 55, 86 greedy algorithm, 107 Haken, Wolfgang, 244 Hamilton, Alexander, 77 handedness, 79 Harry Potter, 35, 93 heap, 123 height (of a tree), 115 Here Before, 73 heterogeneous, 13, 15 hexadecimal numbers, 171, 173, 176, 180 homogeneous, 14 HTML, 114 identity laws (of sets), 21 image (of a function), 48 imaginary numbers, 17 implies (logical operator), 200, 203, 206 in-order traversal, 119, 131 inclusive or, 63, 199 independence (of events), 78 indirect proof, 229 inductive hypothesis, 234, 240 inductive step, 233, 234, 240 infinite, countably, 17 infinite, uncountably, 17 infinity, 12, 16, 17 injective (function), 48 integers (ℤ), 17, 24 intensional, 10, 11, 37, 47, 93 internal nodes, 115, 239 Internet, 87 intersection (of sets), 19, 199, 208, 227 interval, 61 #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 1 Context: # A Brief Introduction to Machine Learning for Engineers (2018), “A Brief Introduction to Machine Learning for Engineers”, Vol. XX, No. XX, pp 1–231. DOI: XXX. Osvaldo Simeone Department of Informatics King’s College London osvaldo.simeone@kcl.ac.uk #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 2 Context: # Contents I Basics 5 1 Introduction 6 1.1 What is Machine Learning? 6 1.2 When to Use Machine Learning? 8 1.3 Goals and Outline 11 2 A Gentle Introduction through Linear Regression 15 2.1 Supervised Learning 15 2.2 Inference 17 2.3 Frequentist Approach 19 2.4 Bayesian Approach 36 2.5 Minimum Description Length (MDL) 42 2.6 Information-Theoretic Metrics 44 2.7 Interpretation and Causality 47 2.8 Summary 49 3 Probabilistic Models for Learning 51 3.1 Preliminaries 52 3.2 The Exponential Family 53 3.3 Frequentist Learning 59 #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 4 Context: ``` 6.6 Discriminative Models........................................159 6.7 Autoencoders...............................................163 6.8 Ranking*...................................................164 6.9 Summary....................................................164 IV Advanced Modelling and Inference..............................165 7 Probabilistic Graphical Models................................166 7.1 Introduction...............................................167 7.2 Bayesian Networks..........................................170 7.3 Markov Random Fields.......................................178 7.4 Bayesian Inference in Probabilistic Graphical Models......182 7.5 Summary....................................................185 8 Approximate Inference and Learning............................186 8.1 Monte Carlo Methods........................................187 8.2 Variational Inference.......................................189 8.3 Monte Carlo-Based Variational Learning*....................197 8.4 Approximate Learning*......................................199 8.5 Summary....................................................201 V Conclusions...................................................202 9 Concluding Remarks............................................203 Appendices.......................................................206 A Appendix A: Information Measures..............................207 A.1 Entropy....................................................207 A.2 Conditional Entropy and Mutual Information................210 A.3 Divergence Measures.......................................212 B Appendix B: KL Divergence and Exponential Family..........215 Acknowledgements...............................................217 ``` #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 9 Context: # Acronyms - **AI:** Artificial Intelligence - **AMP:** Approximate Message Passing - **BN:** Bayesian Network - **DAG:** Directed Acyclic Graph - **ELBO:** Evidence Lower Bound - **EM:** Expectation Maximization - **ERM:** Empirical Risk Minimization - **GAN:** Generative Adversarial Network - **GLM:** Generalized Linear Model - **HMM:** Hidden Markov Model - **i.i.d.:** independent identically distributed - **KL:** Kullback-Leibler - **LASSO:** Least Absolute Shrinkage and Selection Operator - **LBP:** Loopy Belief Propagation - **LL:** Log-Likelihood - **LLR:** Log-Likelihood Ratio - **LS:** Least Squares - **MC:** Monte Carlo - **MCMC:** Markov Chain Monte Carlo - **MDL:** Minimum Description Length - **MFVI:** Mean Field Variational Inference - **MI:** Maximum Likelihood - **NRF:** Markov Random Field - **NLL:** Negative Log-Likelihood - **PAC:** Probably Approximately Correct - **pdf:** probability density function #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 11 Context: # Part I ## Basics #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 13 Context: # 1.1. What is Machine Learning? This starts with an in-depth analysis of the problem domain, which culminates with the definition of a mathematical model. The mathematical model is meant to capture the key features of the problem under study and is typically the result of the work of a number of experts. The mathematical model is finally leveraged to derive hand-crafted solutions to the problem. For instance, consider the problem of defining a chemical process to produce a given molecule. The conventional flow requires chemists to leverage their knowledge of models that predict the outcome of individual chemical reactions, in order to craft a sequence of suitable steps that synthesize the desired molecule. Another example is the design of speech translation or image/video compression algorithms. Both of these tasks involve the definition of models and algorithms by teams of experts, such as linguists, psychologists, and signal processing practitioners, not infrequently during the course of long standardization meetings. The engineering design flow outlined above may be too costly and inefficient for problems in which faster or less expensive solutions are desirable. The machine learning alternative is to collect large data sets, e.g., of labeled speech, images, or videos, and to use this information to train general-purpose learning machines to carry out the desired task. While the standard engineering flow relies on domain knowledge and on design optimized for the problem at hand, machine learning lets large amounts of data dictate algorithms and solutions. To this end, rather than requiring a precise model of the set-up under study, machine learning requires the specification of an objective, of a model to be trained, and of an optimization technique. Returning to the first example above, a machine learning approach would proceed by training a general-purpose machine to predict the outcome of known chemical reactions based on a large data set, and then by using the trained algorithm to explore ways to produce more complex molecules. In a similar manner, large data sets of images or videos would be used to train a general-purpose algorithm with the aim of obtaining compressed representations from which the original input can be recovered with some distortion. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 16 Context: # Introduction representing the data in a smaller or more convenient space; and generative modelling, which is the problem of learning a generating mechanism to produce artificial examples that are similar to available data in the data set \( D \). As a generalization of both supervised and unsupervised learning, **semi-supervised learning** refers to scenarios in which not all examples are labelled, with the unlabelled examples providing information about the distribution of the covariates \( x \). ## 3. Reinforcement learning Reinforcement learning refers to the problem of inferring optimal sequential decisions based on rewards or punishments received as a result of previous actions. Under supervised learning, the “label” refers to an action to be taken when the learner is in an informational state about the environment given by a variable \( x \). Upon taking an action \( a \) in a state \( x \), the learner is provided with feedback on the immediate reward accrued via this decision, and the environment moves on to a different state. As an example, an agent can be trained to navigate a given environment in the presence of obstacles by penalizing decisions that result in collisions. Reinforcement learning is hence neither supervised, since the learner is not provided with the optimal actions \( t \) to select in a given state \( x \); nor is it fully unsupervised, given the availability of feedback on the quality of the chosen action. Reinforcement learning is also distinguished from supervised and unsupervised learning due to the influence of previous actions on future states and rewards. This monograph focuses on supervised and unsupervised learning. These general tasks can be further classified along the following dimensions: - **Passive vs. active learning**: A passive learner is given the training examples, while an active learner can affect the choice of training examples on the basis of prior observations. - **Offline vs. online learning**: Offline learning operates over a batch of training samples, while online learning processes samples in a streaming fashion. Note that reinforcement learning operates inherently in an online manner, while supervised and unsupervised learning can be carried out by following either offline or online formulations. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 17 Context: # 1.3 Goals and Outline This monograph considers only passive and offline learning. ## 1.3 Goals and Outline This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces fundamental concepts and algorithms by building on first principles, while also exposing the reader to more advanced topics with extensive pointers to the literature, within a unified notation and mathematical framework. Unlike other texts that are focused on one particular aspect of the field, an effort has been made here to provide a broad but concise overview in which the main ideas and techniques are systematically presented. Specifically, the material is organized according to clearly defined categories, such as discriminative and generative models, frequentist and Bayesian approaches, exact and approximate inference, as well as directed and undirected models. This monograph is meant as an entry point for researchers with a background in probability and linear algebra. A prior exposure to information theory is useful but not required. Detailed discussions are provided on basic concepts and ideas, including overfitting and generalization, Maximum Likelihood and regularization, and Bayesian inference. The text also endeavors to provide intuitive explanations and pointers to advanced topics and research directions. Sections and subsections containing more advanced material that may be regarded at a first reading are marked with a star (+). The reader will find here further discussions on computing platform or programming frameworks, such as map-reduce, nor details on specific applications involving large data sets. These can be easily found in a vast and growing body of work. Furthermore, rather than providing exhaustive details on the existing myriad solutions in each specific category, techniques have been selected that are useful to illustrate the most salient aspects. Historical notes have also been provided only for a few selected milestone events. Finally, the monograph attempts to strike a balance between the algorithmic and theoretical viewpoints. In particular, all learning algorithms are presented in a manner that emphasizes their theoretical foundations while also providing practical insights. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 19 Context: # 1.3 Goals and Outline Generative Adversarial Networks (GANs) are also introduced. The fourth part covers more advanced modelling and inference approaches. Chapter 7 provides an introduction to probabilistic graphical models, namely Bayesian Networks (BNs) and Markov Random Fields (MRFs), as means to encode more complex probabilistic dependencies than the models studied in previous chapters. Approximate inference and learning methods are introduced in Chapter 8 by focusing on Monte Carlo (MC) and Variational Inference (VI) techniques. The chapter briefly introduces a unified way techniques such as variational EM, Variational AutoEncoders (VAE), and black-box inference. Some concluding remarks are provided in the last part, consisting of Chapter 9. We conclude this chapter by emphasizing the importance of probability as a common language for the definition of learning algorithms [34]. The centrality of the probabilistic viewpoint was not always recognized, but has deep historical roots. This is demonstrated by the following two quotes, the first from the first AI textbook published by P. H. Winston in 1977, and the second from an unfinished manuscript by J. von Neumann (see [126, 64] for more information): “Many ancient Greeks supported Socrates upon that deep, inexplicable thoughts came from the gods. Today’s equivalent to those gods is the erratic, even probabilistic neuron. It is more likely that increased randomness of neural behavior is the problem of the epileptic and the drunk, not the advantage of the brilliant.” from *Artificial Intelligence*, 1977. “Of all this will lead to theories of computation which are much less rigidly of an all-or-none nature than past and present formal logic. There are numerous indications to make us believe that this new system of formal logic will move closer to another discipline which has been little linked in the past with logic. This is thermodynamics primarily in the form it was received from Boltzmann.” #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 20 Context: # Introduction from *The Computer and the Brain*, 1958. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 22 Context: ```markdown variables \( t_n \) are assumed to be dependent on \( x_n \), and are referred to as dependent variables, labels, or responses. An example is illustrated in Fig. 2.1. We use the notation \( \mathbf{x}_D = (x_1, \ldots, x_N)^T \) for the covariates and \( \mathbf{t}_D = (t_1, \ldots, t_N)^T \) for the labels in the training set \( D \). Based on this data, the goal of supervised learning is to identify an algorithm to predict the label \( t \) for a new, that is, as of yet unobserved, domain point \( x \). ![Figure 2.1: Example of a training set \( D \) with \( N = 10 \) points \( (x_n, t_n) \), \( n = 1, \ldots, N \).](path_to_image) The outlined learning task is clearly impossible in the absence of additional information on the mechanism relating variables \( z \) and \( t \). With reference to Fig. 2.1, unless we assume, say, that \( z \) and \( t \) are related by a function \( t = f(z) \) with some properties, such as smoothness, we have no way of predicting the label \( t \) for an unobserved domain point \( z \). This observation is formalized by the no free lunch theorem to be reviewed in Chapter 5: one cannot learn rules that generalize to unseen examples without making assumptions about the mechanism generating the data. The set of all assumptions made by the learning algorithm is known as the inductive bias. This discussion points to a key difference between memorizing and learning. While the former amounts to mere retrieval of a value \( t_n \), ``` #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 50 Context: ``` 44 # A Gentle Introduction through Linear Regression Produces a description of approximately $$ - \sum_{n=1}^{N} \log{p(y_n | x_n, w_{ML}, \beta_{ML})} $$ bits. This description is, however, not sufficient, since the decoder of the description should also be informed about the parameters $(w_{ML}, \beta_{ML})$. Using quantization, the parameters can be described by means of a number $C(M)$ of bits that is proportional to the number of parameters, here $M + 2$. Concatenating these bits with the description produced by the ML model yields the overall description length $$ - \sum_{n=1}^{N} \log{p[y_n | x_n, w_{ML}, \beta_{ML}]} + C(M). \quad (2.39) $$ MDL – in the simplified form discussed here – selects the model order $M$ that minimizes the description length (2.39). Accordingly, the term $C(M)$ acts as a regularizer. The optimal value of $M$ for the MDL criterion is hence the result of the trade-off between the minimization of the overhead $C(M)$, which calls for a small value of $M$, and the minimization of the NLL, which decreases with $M$. Under some technical assumptions, the overhead term can be often evaluated in the form $$ \left( \frac{K}{2} \right) N \log N + c, $$ where $K$ is the number of parameters in the model and $c$ is a constant. This expression is not quite useful in practice, but it provides intuition about the mechanism used by MDL to combat overfitting. ## 2.6 Information-Theoretic Metrics We now provide a brief introduction to information-theoretic metrics by leveraging the example studied in this chapter. As we will see in the following chapters, information-theoretic metrics are used extensively in the definition of learning algorithms. Appendix A provides a detailed introduction to information-theoretic measures in terms of inferential tasks. Here we introduce the key metrics of Kullback-Leibler (KL) divergence and entropy by examining the asymptotic behavior of ML in the regime of large $N$. The case with finite $N$ is covered in Chapter 6 (see Sec. 6.4.3). ``` #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 54 Context: # A Gentle Introduction through Linear Regression ![Figure 2.10: Illustration of Simpson's paradox [113]](path_to_image) The way out of this conundrum is to leverage prior information we have about the problem domain. In fact, we can explain away this spurious correlation by including another measurable variable in the model, namely age. To see this, consider the same data, now redrawn by highlighting the age of the individual corresponding to each data point. The resulting plot, seen in Fig. 2.10 (bottom), reveals that older people — within the observed bracket — tend to have a higher cholesterol as well as to exercise more. Therefore, age is a common cause of both exercise and cholesterol levels. In order to capture the causal relationship between the latter variables, we hence need to adjust for age. Doing this requires to consider the trend within each age separately, recovering the expected conclusion that exercising is useful to lower one’s cholesterol. We conclude that in this example the correlation between \( z \) and \( t \), while useful for prediction, should not be acted upon for decision making. When assessing the causal relationship between \( z \) and \( t \), we should first understand which other variables may explain the observations and then discount any spurious correlations. This discussion reveals an important limitation of most existing machine learning algorithms when it comes to identifying causality relationships, or, more generally, to answering counterfactual queries [112]. The study of causality can be carried out within the elegant framework of footnotes. 7 This example is an instance of the so-called Simpson's paradox: patterns visible in the data disappear, or are even reversed, on the segregated data. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 55 Context: # 2.8 Summary In this chapter, we have reviewed three key learning frameworks, namely frequentist, Bayesian and MDL, within a parametric probabilistic setup. The frequentist viewpoint postulates the presence of a true unknown distribution for the data and aims at learning a predictor that generalizes well on unseen data drawn from this distribution. This can be done either by learning a probabilistic model to be plugged into the expression of the optimal predictor or by directly solving the ERM problem over the predictor. The Bayesian approach outputs a predictive distribution that combines prior information with the data by solving the inference problem of computing the posterior distribution over the unseen label. Finally, the MDL method aims at selecting a model that allows the data to be described with the smallest number of bits, hence doing away with the need to define the task of generalizing over unobserved examples. The chapter has also focused extensively on the key problem of overfitting, demonstrating how the performance of a learning algorithm can be understood in terms of bias and estimation error. In particular, while choosing a hypothesis class is essential in order to enable learning, choosing the “wrong” class constitutes an irrecoverable bias that can make learning impossible. As a real-world example, as reported in [109], including as independent variables in the ZIP code of an individual seeking credit at a bank may disadvantage certain applicants or minorities. Another example of this phenomenon is the famous experiment by B. F. Skinner on pigeons [133]. We conclude this chapter by emphasizing an important fact about the probabilistic models that are used in modern machine learning applications. In frequentist methods, typically at least two (possibly conditional) distributions are involved: the empirical data distribution and the model distribution. The former amounts to the histogram of the data which, by the law of large numbers, tends to the real distribution. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 58 Context: # 3.1 Preliminaries We start with a brief review of some definitions that will be used throughout the chapter and elsewhere in the monograph (see [28] for more details). Readers with a background in convex analysis and calculus may just review the concept of sufficient statistic in the last paragraph. First, we define a **convex set** as a subset of \(\mathbb{R}^D\), for some \(D\), that contains all segments between any two points in the set. Geometrically, convex sets hence cannot have “indentations.” Function \(f(x)\) is convex if its domain is a convex set and if it satisfies the inequality \(f(\lambda x_1 + (1 - \lambda)x_2) \leq \lambda f(x_1) + (1 - \lambda)f(x_2)\) for all \(x_1\) and \(x_2\) in its domain and for all \(0 \leq \lambda \leq 1\). Geometrically, this condition says that the function is “U”-shaped: the curve defining the function cannot be above the segment obtained by connecting any two points on the curve. A function is strictly convex if the inequality above is strict except for \(\lambda = 0\) or \(\lambda = 1\); a concave, or strictly concave, function is defined by reversing the inequality above – it is hence “n-shaped.” The minimization of a convex (“U”) function over a convex constraint set or the maximization of a concave (“n”) function over a convex constraint set are known as convex optimization problems. For these problems, there exist powerful analytical and algorithmic tools to obtain globally optimal solutions [28]. We also introduce two useful concepts from calculus. The **gradient** of a differentiable function \(f(x)\) with \(x = [x_1, \ldots, x_D]^T \in \mathbb{R}^D\) is defined as the \(D \times 1\) vector \(\nabla f(x) = [\frac{\partial f(x)}{\partial x_1}, \ldots, \frac{\partial f(x)}{\partial x_D}]^T\) containing all partial derivatives. At any point \(x\) in the domain of the function, the gradient is a vector that points to the direction of locally maximal increase of the function. The Hessian \(\nabla^2 f(x)\) is the \(D \times D\) matrix with \((i,j)\) element given by the second-order derivative \(\frac{\partial^2 f(x)}{\partial x_i \partial x_j}\). It captures the local curvature of the function. 1. A statistic is a function of the data. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 76 Context: To elaborate, let us denote as `exponential(η)` a probabilistic model in the exponential family, that is, a model of the form (3.1) with natural parameters η. We also write `exponential(μ|θ)` for a probabilistic model in the exponential family with mean parameters μ. Using the notation adopted in the previous chapter, in its most common form, a GLM defines the probability of a target variable as: \[ p(y|x, W) = \text{exponential}(η|Wz) \tag{3.53} \] where we recall that \( z \) is the vector of explanatory variables, and \( W \) here denotes a matrix of learnable weights of suitable dimensions. According to (3.53), GLMs posit that the response variable \( y \) has a conditional distribution from the exponential family, with natural parameter vector \( η = Wz \) given by a linear function of the given explanatory variables \( z \) with weights \( W \). More generally, we may have the parametrization: \[ p(y|x, W) = \text{exponential}(η|Wϕ(x)) \tag{3.54} \] for some feature vector \( ϕ(x) \) obtained as a function of the input variables \( x \) (see next chapter). While being the most common, the definition (3.54) is still not the most general for GLMs. More broadly, GLMs can be interpreted as a generalization of the linear model considered in the previous chapter, whereby the mean parameters are defined as a linear function of a feature vector. This viewpoint, described next, may also provide a more intuitive understanding of the modeling assumptions made by GLMs. Recall that, in the recurring example of Chapter 2, the target variable was modeled as Gaussian distributed with mean given by a linear function of the covariates \( z \). Extending the example, GLMs posit the model: \[ p(y|x, W) = \text{exponential}(η|g(Wϕ(x))) \tag{3.55} \] where the mean parameter vector is parametrized as a function of the feature vector \( ϕ(x) \) through a generally non-linear vector function \( g(·) \) of suitable dimensions. In words, GLMs assume that the target variable is a “noisy” measure of the mean \( μ = g(Wϕ(x)) \). When the function \( g(·) \) is selected as the gradient of the partition function of the selected model, e.g., \( g(·) = \nabla A(·) \), then, by (3.10), we... #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 77 Context: # 3.6 Maximum Entropy Property* In this most technical section, we review the maximum entropy property of the exponential family. Besides providing a compelling motivation for adopting models in this class, this property also illuminates the relationship between natural and mean parameters. The key result is the following: The distribution \(p(x|\theta)\) in (3.1) obtains the maximum entropy over all distributions \(p(x)\) that satisfy the constraints \(\mathbb{E}_{x \sim p(x)} [u_k(x)] = \mu_k\) for all \(k = 1, \ldots, K\). Recall that, as mentioned in Chapter 2 and discussed in more detail in Appendix A, the entropy is a measure of randomness of a random variable. Mathematically, the distribution \(p(x | \eta)\) solves the optimization problem \[ \max_{p} H(p) \quad \text{s.t.} \quad \mathbb{E}_{x \sim p(x)} [u_k(x)] = \mu_k \quad \text{for } k = 1, \ldots, K. \] Each natural parameter \(\eta_k\) turns out to be the optimal Lagrange multiplier associated with the \(k\)th constraint (see [45, Ch. 6-7]). To see the practical relevance of this result, suppose that the only information available about some data \(x\) is given by the means of given functions \(u(x)\), \(k = 1, \ldots, K\). The probabilistic model (3.1) can then be interpreted as encoding the least additional information about the data, in the sense that it is the "most random" distribution under the given constraints. This observation justifies the adoption of this model by the maximum entropy principle. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 82 Context: # 4 ## Classification The previous chapters have covered important background material on learning and probabilistic models. In this chapter, we use the principles and ideas covered so far to study the supervised learning problem of classification. Classification is arguably the quintessential machine learning problem, with the most advanced state of the art and the most extensive application to problems as varied as email spam detection and medical diagnosis. Due to space limitations, this chapter cannot provide an exhaustive review of all existing techniques and latest developments, particularly in the active field of neural network research. For instance, we do not cover decision trees here (see, e.g., [155]). Rather, we will provide a principled taxonomy of approaches, and offer a few representative techniques for each category within a unified framework. We will specifically proceed by first introducing as preliminary material the Stochastic Gradient Descent optimization method. Then, we will discuss deterministic and probabilistic discriminative models, and finally we will cover probabilistic generative models. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 87 Context: # 4.3 Discriminative Deterministic Models where the activation, or decision variable, is given as $$ a(x, \tilde{w}) = \sum_{d=1}^{D} w_d x + w_0 = \tilde{w}^T x $$ and we have defined the weight vectors \( w = [w_1, \ldots, w_D]^T \) and \( \tilde{w} = [w_0, w_1, \ldots, w_D]^T \), as well as the extended domain point \( \tilde{x} = [1, x^T]^T \), with \( x = [x_1, \ldots, x_D]^T \). The sigmoid function in decision rule (4.5) outputs 1 if its argument is positive, and 0 if the argument is negative depending on the assumed association rule in (4.4). ## Geometric Interpretation: Classification, Geometric and Functional Margins The decision rule (4.5) defines a hyperplane that separates the domain points classified as belonging to either of the two classes. A hyperplane is a line when \( D = 2 \); a plane when \( D = 3 \); and, more generally, a \( D - 1 \)-dimensional affine subspace in the domain space. The hyperplane is defined by the equation \( a(x, \tilde{w}) = 0 \), with points on either side characterized by either positive or negative values of the activation \( a(x, \tilde{w}) \). The decision hyperplane can be identified as described in Fig. 4.2: the vector \( w \) defines the direction perpendicular to the hyperplane and \( -w_0 / \|w\| \) is the bias of the decision surface in the decision space. --- ![Figure 4.2: Key definitions for a binary linear classifier.](#) $$ a(x, \tilde{w}) > 0 $$ $$ |a(x, \tilde{w})| $$ $$ a(x, \tilde{w}) = 0 $$ $$ a(x, \tilde{w}) < 0 $$ #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 107 Context: # 4.5. Discriminative Probabilistic Models: Beyond GLM of deep neural networks appears to defy one of the principles laid out in Chapter 2, which will be formalized in the next chapter: Highly over-parameterized models trained via ML suffer from overfitting and hence do not generalize well. Recent results suggest that the use of SGD, based on the gradient (4.37), may act as a regularizer following an MDL perspective. In fact, SGD favors the attainment of flat local maxima of the likelihood function. Flat maxima require a smaller number of bits to be specified with respect to sharp maxima; since, in flat maxima, the parameter vector can be described with limited precision as long as it remains within the flat region of the likelihood function [66, 79, 71] (see also [78] for a different perspective). Another important aspect concerns the hardware implementation of backprop. This is becoming extremely relevant given the practical applications of deep neural networks for consumers' devices. In fact, a key aspect of backprop is the need to propagate the error, which is measured at the last layer, to each synapse via the backward pass. This is needed in order to evaluate the gradient (4.37). While a software implementation of this rule does not present any conceptual difficulty, realizing the computations in (4.36) in hardware, or even on a biological neural system, is faced with a number of issues. 1. A first problem is the non-locality of the update (4.37). An update rule is local if it only uses information available at each neuron. In contrast, as seen, rule (4.37) requires back-propagation through the neural network. 2. Another issue is the need for the backward pass to use a different neural path with respect to the forward pass, given that, unlike the backward pass, the forward pass includes also non-linearities. A useful discussion of hardware implementation aspects can be found in [12] (see also references therein). To obviate at least some of these practical issues, a number of variations of the rule (4.37) have been proposed [12]. For instance, the feedback alignment rule modifies (4.36) by using fixed random matrices in lieu of the current weight matrices \( W' \); while the broadcast alignment rule rewrites the vectors \( \mathbf{y} - \mathbf{t} \) as a linear function with fixed random coefficients of the error ( \( \mathbf{y} - \mathbf{t} \) ), hence removing the need for back-propagation [129]. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 111 Context: # 4.6 Generative Probabilistic Models ![Figure 4.9](path/to/image.png) **Figure 4.9:** Probability that the class label is the same as for the examples marked with circles according to the output of the generative model QDA. The probability is represented by the color map illustrated by the bar on the right of the figure. For this example, it can be seen that LDA fails to separate the two classes (not shown). ### Example 4.4 We continue the example in Sec. 4.5 by showing in Fig. 4.9 the probability (4.43) that the class label is the same as for the examples marked with circles according to the output of QDA. Given that the covariates have a structure that is well modeled by a mixture of Gaussians with different covariance matrices, QDA is seen to perform well, arguably better than the discriminative models studied in Sec. 4.5. It is important to note, however, that LDA would fail in this example. This is because a model with equal class-dependent covariance matrices, as assumed by LDA, would entail a significant bias for this example. #### 4.6.3 Multi-Class Classification* As an example of a generative probabilistic model with multiple classes, we briefly consider the generalization of QDA to \( K \geq 2 \) classes. Extending (4.41) to multiple classes, the model is described as: \[ x \sim \text{Cat}(\pi) \tag{4.44a} \] \[ x | t \sim \mathcal{N}(\mu_k, \Sigma_k) \tag{4.44b} \] Image Analysis: ### Analysis of the Visual Content #### 1. Localization and Attribution - **Image Number**: Image 1 - This is a single image located near the top of the page, integrated within the textual content. #### 2. Object Detection and Classification - **Objects Identified**: - **Graph**: The main object is a scatter plot with mathematical notations and a probability color map. - **Text**: The text below the graph and the paragraphs containing examples and explanations. #### 3. Scene and Activity Analysis - **Scene Description**: Image 1 depicts a scatter plot graph illustrating the results of a statistical model used in generative probabilistic models. The graph is surrounded by explanatory text describing the significance of the plotted data. - **Activities**: The scene represents an educational or informational setting, focusing on explaining a statistical concept. #### 4. Text Analysis - **Text Extracted**: - **Figure Caption**: "Figure 4.9: Probability that the class label is the same as for the examples marked with circles according to the output of the generative model QDA. The probability is represented by the color map illustrated by the bar on the right of the figure. For this example, it can be seen that LDA fails to separate the two classes (not shown)." - **Example Text**: "Example 4.4. We continue the example in Sec. 4.5 by showing in Fig. 4.9 the probability (4.43) that the class label is the same as for the examples marked with circles according to the output of QDA. Given that the covariates have a structure that is well modelled by a mixture of Gaussians with different covariance matrices, QDA is seen to perform well, arguably better than the discriminative models studied in Sec. 4.5. It is important to note, however, that LDA would fail in this example. This is because a model with equal class-dependent covariance matrices, as assumed by LDA, would entail a significant bias for this example." - **Section Text**: "4.6.3 Multi-Class Classification* As an example of a generative probabilistic model with multiple classes, we briefly consider the generalization of QDA to \(K \geq 2\) classes. Extending (4.41) to multiple classes, the model is described as \(t \sim Cat(\pi)\) (4.44a) \(x|t = k \sim \mathcal{N}(\mu_k, \Sigma_k)\) (4.44b)" - **Significance**: The text explains concepts related to generative probabilistic models, highlighting the differences between Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). It offers an example and discusses performance with an emphasis on covariance structures. #### 5. Diagram and Chart Analysis - **Graph Analysis**: - **Axes and Scales**: - **X-axis**: Z1 ranging from -4 to 4 - **Y-axis**: Z2 ranging from -3 to 3 - **Color Bar**: Probability scale ranging from 0.01 to 1 - **Data Presented**: - Red crosses and blue circles represent data points, with the color map indicating the probability that a data point's class label matches the model's output. - **Key Insights**: - The scatter plot illustrates that QDA accurately assigns class labels to the data points marked with circles. - The color map indicates higher probabilities in regions with concentrated data points, showing QDA's effectiveness. - The caption notes LDA's failure to separate the two classes, which suggests the necessity of QDA for this dataset. #### 9. Perspective and Composition - **Perspective Description**: The image is viewed directly from the front, capturing a clear view of the scatter plot and accompanying text. - **Composition**: The scatter plot is centered in the image with explanatory text surrounding it, ensuring clarity and focus on the visual data representation. ### Contextual Significance - **Image Contribution**: This image and its associated text play a crucial role in explaining a key concept within the broader context of generative probabilistic models. It provides a visual and theoretical comparison between LDA and QDA, supporting the reader's understanding of the material. ### Diagram and Trend Analysis - **Trend Observation**: The scatter plot indicates clusters where QDA performs well in classifying data points. The color gradient reveals areas with varying probabilities, underscoring where QDA's classification aligns with actual data labels. ### Tables and Graph Numbers - This image does not include tables or specific graph data points for listing. ### Process Flows and Descriptions - No process flows or descriptions are presented in this image. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 113 Context: # 4.8 Summary When training the \(k\)th model, the outputs \(a_1(x, \mathbf{w}_1), \ldots, a_k(x, \mathbf{w}_{k-1})\) of the previously trained models, as well as their weights \(\pi_1, \ldots, \pi_{k-1}\), are kept fixed. Excluding the models \(k + 1, \ldots, K\), the training loss can be written as \[ L_k = \sum_{n=1}^{N} \alpha_{n}^{(k)} \exp(-\pi_k t_n - \sum_{j=1}^{k-1} \pi_j a_j(x_n, \mathbf{w}_j)). \tag{4.47} \] with the weights \[ \alpha_{n}^{(k)} = \exp\left(-t_n - \sum_{j=1}^{k-1} \pi_j a_j(x_n, \mathbf{w}_j)\right). \tag{4.48} \] An important point is that the weights (4.48) are larger for training samples with smaller functional margin under the mixture model \(\sum_{j=1}^{k} \pi_j a_j(x_n, \mathbf{w}_j)\). Therefore, when training the \(k\)th model, we give more importance to examples that fare worse in terms of classification margins under the current mixture model. Note that, at each training step \(k\), one trains a simple model, which has the added advantage of reducing the computational complexity as compared to the direct learning of the full training model. We refer to [23, Ch. 14][33, Ch. 10] for further details. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 115 Context: 4.8. Summary ----------- Chapter 8 for details). Learning rate schedules that satisfy (4.49) include \(\gamma(t) = 1/t\). The intuitive reason for the use of diminishing learning rates is the need to limit the impact of the “noise” associated with the finite-sample estimate of the gradient [22]. The proof of convergence leverages the unbiasedness of the estimate of the gradient obtained by SGD. In practice, a larger mini-batch size \(S\) decreases the variance of the estimate of the gradient, hence improving the accuracy when close to a stationary point. However, choosing a smaller \(S\) can improve the speed of convergence when the current solution is far from the optimum [152, Chapter 8][22]. A smaller mini-batch size \(S\) is also known to improve the generalization performance of learning algorithms by avoiding sharp external points of the training loss function [66, 79] (see also Sec. 4.5). Furthermore, as an alternative to decreasing the step size, one can also increase the size of the mini-batch along the iterations of the SGD algorithm [136]. Variations and Generalizations ------------------------------- Many variations of the discussed basic SGD algorithm have been proposed and routinely used. General principles motivating these schedule variants include [56, Chapter 8]: 1. **Momentum**, or heavy-ball, memory: correct the direction suggested by the stochastic gradient by considering the “momentum” acquired during the last update; 2. **Adapitivity**: use a different learning rate for different parameters depending on an estimate of the curvature of the loss function with respect to each parameter; 3. **Control variates**: in order to reduce the variance of the SGD updates, and control variates that do not affect the unbiasedness of the stochastic gradient; and 4. **Second-order updates**: include information about the curvature of the cost or objective function in the parameter update. As detailed in [56, Chapter 8][76, 43], to which we refer for further discussions, methods in the first category include Nesterov momentum; in the second category, we find AdaGrad, RMSProp, and Adam; and the third encompasses SVRG and SAGA. Finally, the fourth features Newton's method. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 119 Context: # 5 ## Statistical Learning Theory* Statistical learning theory provides a well-established theoretical framework in which to study the trade-off between the number \( N \) of available data points and the generalization performance of a trained machine. The approach formalizes the notions of model capacity, estimation error (or generalization gap), and bias that underlie many of the design choices required by supervised learning, as we have seen in the previous chapters. This chapter is of mathematical nature, and it departs from the algorithmic focus of the text so far. While it may be skipped at a first reading, the chapter sheds light on the key empirical observations made in the previous chapters relative to learning in a frequentist setup. It does so by covering the theoretical underpinnings of supervised learning within the classical framework of statistical learning theory. To this end, the chapter contains a number of formal statements with proofs. The proofs have been carefully selected in order to highlight and clarify the key theoretical ideas. This chapter follows mostly the treatment in [133]. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 123 Context: # 5.1. A Formal Framework for Supervised Learning illustrated in the next example, this expected behavior is a consequence of the law of large numbers, since a larger \( N \) allows an increasingly accurate estimation of the true general loss. ![Figure 5.2: High-probability interval (dashed arrow) for the generalization error \( L_y(f_N) \) versus the number \( N \) of data points for a model \( \mathcal{H} \).](path/to/image) ## Example 5.2 Consider the problem of binary classification using the model of threshold functions, namely \[ \mathcal{H} = \left\{ h(x) = \begin{cases} 0, & \text{if } x < \theta \\ 1, & \text{if } x \geq \theta \end{cases} \right\}, \] where \( \theta \) is a real number (D = 1). Note that the model is parametrized by the threshold \( \theta \). Make the realizability assumption that the true distribution is within the hypothesis allowed by the model, i.e., \( p(x, y) = p(y|x) \cdot p(x) \) and hence the optimal hypothesis is \( h^* = h_{\theta^*} \). Assuming a uniform distribution \( p(x) = \frac{1}{|a - b|} \) for the domain points, Fig. 5.3 shows the generalization error \( \Pr(h(x) \neq y | x) = |h(x) - y| \), as well as the training loss \( L_p(f_N) \) for the training set shown on the horizontal axis, for various values of \( N \). Note that the training loss \( L_p(f_N) \) is simply #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 129 Context: ### 5.3. PAC Learnability for Finite Hypothesis Classes We can now write the following sequence of equalities and inequalities, which prove the lemma and hence conclude the proof: \[ \Pr_{D \sim P_{data}} \left[ \exists h \in \mathcal{H} : |L_D(h) - L_D(\phi)| > \frac{\epsilon}{2} \right] \] \[ = \Pr_{D \sim P_{data}} \left[ \bigcup_{h \in \mathcal{H}} \left\{ |L_D(h) - L_D(\phi)| > \frac{\epsilon}{2} \right\} \right] \] \[ \leq \sum_{h \in \mathcal{H}} \Pr_{D \sim P_{data}} \left[ |L_D(h) - L_D(\phi)| > \frac{\epsilon}{2} \right] \] \[ \leq 2 \sum_{h \in \mathcal{H}} \exp \left( -\frac{N \epsilon^2}{2} \right) = 2|\mathcal{H}| \exp \left( -\frac{N \epsilon^2}{2} \right) \leq \delta, \] where the first inequality follows by the union bound; the second by Hoeffding's inequality; and the third can be verified to be true as long as the inequality \( N \geq \frac{2}{\epsilon^2} \log \frac{2|\mathcal{H}|}{\delta} \) is satisfied. ### 5.3.2 Structural Risk Minimization The result proved above is useful also to introduce the Structural Risk Minimization (SRM) learning approach. SRM is a method for joint model selection and hypothesis learning that is based on the minimization of an upper bound on the generalization loss. In principle, the approach avoids the use of validation and has deep theoretical properties in terms of generalization. In practical applications, the approach is rarely used, and validation is often preferable. It is nevertheless conceptually and theoretically a cornerstone of statistical learning theory. To elaborate, assume that we have a nested set of hypothesis classes \(\mathcal{H}_1 \subset \mathcal{H}_2 \subset \ldots \subset \mathcal{H}_M\). From Lemma 5.2, we can obtain the following bound: \[ L_D(h) \leq L_D(\phi) + \sqrt{ \frac{2\mathcal{H} |\mathcal{W}|}{N}} \quad (5.20) \] #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 133 Context: # 5.5. Summary To elaborate, consider a probabilistic model \( \mathcal{H} \) defined as the set of all pmfs \( p(x|\theta) \) parameterized by \( \theta \) in a given set. With some abuse of notation, we take \( \mathcal{H} \) to be also the domain of parameter \( \theta \). To fix the ideas, assume that \( \theta \) takes values over a finite alphabet. We know from Section 2.5 that a distribution \( q(x) \) is associated with a lossless compression scheme that requires around \( -\log q(x) \) bits to describe a value. Furthermore, if we were informed about the true parameter \( \theta \), the minimum average coding length would be the entropy \( H(p(\cdot|\theta)) \), which requires setting \( q(x) = p(x|\theta) \) (see Appendix A). Assume now that we only know that the parameter \( \theta \) lies in set \( \mathcal{H} \), and hence the true distribution \( p(x|\theta) \) is not known. In this case, we cannot select the true parameter distribution, and we need instead to choose a generally different distribution \( q(x) \) to define a compression scheme. With a given distribution \( q(x) \), the average coding length is given by \[ \bar{L}(q(x), \theta) = -\sum_{x} p(x|\theta) \log q(x). \] Therefore, the choice of a generally different distribution \( q(x) \) entails a redundancy of \[ \Delta R(q(x), \theta) = -\sum_{x} p(x|\theta) \log \frac{q(x)}{p(x|\theta)} \geq 0 \tag{5.22} \] bits. The redundancy \( \Delta R(q(x), \theta) \) in (5.22) depends on the true value of \( \theta \). Since the latter is not known, this quantity cannot be computed. We can instead obtain a computable metric by maximizing over all values of \( \theta \in \mathcal{H} \), which yields the worst-case redundancy \[ \Delta R(q(x), \mathcal{H}) = \max_{\theta \in \mathcal{H}} \Delta R(q(x), \theta). \tag{5.23} \] This quantity can be minimized over \( q(x) \) yielding the so-called min-max redundancy: \[ \Delta R(q(x)) = \min_{q} \Delta R(q(x), \mathcal{H}) \tag{5.24} \] \[ = \min_{q} \max_{\theta} \sum_{x} p(x|\theta) \log \frac{p(x|\theta)}{q(x)} \tag{5.25} \] \[ = \min_{q} \max_{\theta} \sum_{x} p(x|\theta) \log \frac{p(x|\theta)}{q(x)}. \tag{5.26} \] The minimum redundancy can be taken as a measure of the capacity of model \( \mathcal{H} \), since a richer model tends to yield a larger \( \Delta R(\mathcal{H}) \). In fact, #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 138 Context: # Unsupervised Learning may wish to cluster a set \( D \) of text documents according to their topics, by modeling the latter as an unobserved label \( z_n \). Broadly speaking, this requires to group together documents that are similar according to some metric. It is important at this point to emphasize the distinction between classification and clustering: While the former assumes the availability of a labelled set of training examples and evaluates its (generalization) performance on a separate set of unlabelled examples, the latter works with a single, unlabelled, set of examples. The different notation used for the labels \( z_n \) in lieu of \( t_n \) is meant to provide a reminder of this key difference. ## Dimensions reduction and representation: Given the set \( D \), we would like to represent the data points \( x_n \in D \) in a space of lower dimensionality. This makes it possible to highlight independent explanatory factors, and/or to ease visualization and interpretation [93], e.g., for text analysis via vector embedding (see, e.g., [124]). ## Feature extraction: Feature extraction is the task of deriving functions of the data points \( x_n \) that provide useful lower-dimensional inputs for tasks such as supervised learning. The extracted features are unobserved, and hence latent, variables. As an example, the hidden layer of a deep neural network extracts features from the data for use by the output layer (see Sec. 4.5). ## Generation of new samples: The goal here is to train a machine that is able to produce samples that are approximately distributed according to the true distribution \( p(x) \). For example, in computer graphics for filmmaking or gaming, one may want to train a software that is able to produce artistic scenes based on a given description. The variety of tasks and the difficulty in providing formal definitions, e.g., on the realism of an artificially generated image, may hinder unsupervised learning, at least in its current state, a less formal field than supervised learning. Often, loss criteria in unsupervised learning measure the divergence between the learned model and the empirical data distribution, but there are important exceptions, as we will see. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 169 Context: # 6.8 Ranking ## Denoising autoencoders An alternative approach to facilitate the learning of useful features is that taken by denoising autoencoders [150]. Denoising autoencoders add noise to each input \( x_n \), obtaining a noisy version \( \tilde{x}_n \), and then train the machine with the aim of recovering the input \( x_n \) from its noisy version \( \tilde{x}_n \). Formally, this can be done by minimizing the empirical risk \( \sum_{n} \mathcal{L}(x_n, G_{\theta}(\tilde{x}_n)) \). ## Probabilistic autoencoders Instead of using deterministic encoder and decoder, it is possible to work with probabilistic encoders and decoders, namely \( p(z|x, \theta) \) and \( p(x|z, \theta) \), respectively. Treating the decoder as a variational distribution, learning can then be done by a variant of the EM algorithm. The resulting algorithm, known as Variational AutoEncoder (VAE), will be mentioned in Chapter 8. ## 6.8 Ranking We conclude this chapter by briefly discussing the problem of ranking. When one has available ranked examples, ranking can be formulated as a supervised learning problem [133]. Here, we focus instead on the problem of ranking a set of webpages based only on the knowledge of their underlying graph of mutual hyperlinks. This set-up may be considered as a special instance of unsupervised learning. We specifically describe a representative, popular, scheme known as PageRank [110], which uses solely the web of hyperlinks as a form of supervision signal to guide the ranking. To elaborate, we define the connectivity graph by including a vertex for each webpage and writing the adjacency matrix as: \[ L_{ij} = \begin{cases} 1 & \text{if page } j \text{ links to page } i \\ 0 & \text{otherwise} \end{cases} \] The outgoing degree of a webpage is given as: \[ C_j = \sum_{i} L_{ij}. \] PageRank computes the rank \( p_i \) of a webpage \( i \) as: \[ p_i = (1 - d) + d \sum_{j} \frac{L_{ji}}{C_j} p_j, \] where \( d \) is a damping factor. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 191 Context: When the undirected graph is not a tree, one can use the **junction tree algorithm** for exact Bayesian inference. The idea is to group subsets of variables together in cliques, in such a way that the resulting graph is a tree. The complexity depends on the treewidth of the resulting graph. When this complexity is too high for the given application, approximate inference methods are necessary. This is the subject of the next chapter. ## 7.5 Summary Probabilistic graphical models encode a priori information about the structure of the data in the form of causality relationships — via directed graphs and Bayesian networks (BNs) — or mutual affinities — via undirected graphs and Markov Random Fields (MRFs). This structure translates into conditional independence conditions. The structural properties encoded by probabilistic graphical models have the potential advantage of controlling the capacity of a model, hence contributing to the reduction of overfitting at the expense of possible bias effects (see Chapter 5). They also facilitate Bayesian inference (Chapters 2-4), at least in graphs with tree-like structures. Probabilistic graphical models can be used as the underlying probabilistic framework for supervised, unsupervised, and semi-supervised learning problems, depending on which subsets of random variables (rvs) are observed or latent. While graphical models can reduce the complexity of Bayesian inference, this generally remains computationally infeasible for most models of interest. To address this problem, the next chapter discusses approximate Bayesian inference, as well as associated learning problems (Chapter 6). #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 207 Context: # 8.5 Summary Having observed in previous chapters that learning in probabilistic models is often held back by the complexity of exact Bayesian inference for hidden variables, this chapter has provided an overview of approximate, lower-complexity inference techniques. We have focused on MC and VI methods, which are most commonly used. The treatment stressed the impact of design choices in the selection of different types of approximation criteria, such as M- and I-Projection. It also covered the use of approximate inference in learning problems. Topics that improve over the state of the art discussed in this chapter are being actively investigated. Some additional topics for future research are covered in the next chapter. ## Appendix: M-Projection with the Exponential Family In this appendix, we consider the problem of obtaining the M-projection of a distribution \( p(z \mid \varphi) = Z(\varphi)^{-1} \exp\{u^T(\varphi)\} \) from the exponential family with sufficient statistics \( u(X) \). We will prove that, if there exists a value \( \varphi^* \) of the natural parameter vector that satisfies the moment matching condition (8.14), then \( q(z \mid \varphi) \) is the M-projection. We first write the KL divergence as: \[ KL(q(z \| \varphi)) = \ln Z(\varphi) - \varphi^T E_{p(z \mid u)}[u(z)] - H(p). \tag{8.31} \] The difference between the KL divergence for a generic parameter vector \( \varphi \) and the vector \( \varphi^* \) satisfying (8.14) can be written as: \[ KL(q(z \mid \varphi) \| q(z \mid \varphi^*)) = E_{q(z \mid \varphi)}\left[ \ln\left(\frac{q(z \mid \varphi)}{q(z \mid \varphi^*)}\right) \right] = -KL(q(z \mid \varphi) \| q(z \mid \varphi^*)). \tag{8.32} \] Since the latter inequality is non-negative and equal to zero when \( \varphi = \varphi^* \), this choice of the natural parameters minimizes the KL divergence. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 208 Context: # Part V ## Conclusions - This section summarizes the key findings of the study. - Conclusions drawn from the data. - Implications of the findings for future research. 1. **First conclusion point.** 2. **Second conclusion point.** 3. **Third conclusion point.** ### Recommendations - **Recommendation 1:** Description of recommendation. - **Recommendation 2:** Description of recommendation. - **Recommendation 3:** Description of recommendation. ### References 1. Author, A. (Year). *Title of the work*. Publisher. 2. Author, B. (Year). *Title of the work*. Publisher. ### Acknowledgements - Acknowledgement of contributions from individuals or organizations. #################### File: A%20Brief%20Introduction%20to%20Machine%20Learning%20for%20Engineers%20-%20Osvaldo%20Simeone%20%28PDF%29.pdf Page: 210 Context: # Concluding Remarks The data, producing the wrong response upon minor, properly chosen, changes in the explanatory variables. Note that such adversarially chosen examples, which cause a specific machine to fail, are conceptually different from the randomly selected examples that are assumed when defining the generalization properties of a network. There is evidence that finding such examples is possible even without knowing the internal structure of a machine, but solely based on black-box observations [11]. Modifying the training procedure in order to ensure robustness to adversarial examples is an active area of research with important practical implications [55]. ## Computing Platforms and Programming Frameworks In order to scale up machine learning applications, it is necessary to leverage distributed computing architectures and related standard programming frameworks [17, 7]. As a complementary and more futuristic approach, recent work has been proposed to leverage the capabilities of an annealing-based quantum computers as samplers [82] or discrete optimizers [103]. ## Transfer Learning Machines trained for a certain task currently need to be re-trained in order to be re-purposed for a different task. For instance, a machine that learned how to drive a car would need to be retrained in order to learn how to drive a truck. The field of transfer learning covers scenarios in which one wishes to transfer the expertise acquired from some tasks to others. Transfer learning includes different related paradigms, such as multitask learning, lifelong learning, zero-shot learning, and domain adaptation [149]. In multitask learning, several tasks are learned simultaneously. Typical solutions for multitask learning based on neural networks preserve the presence of common representations among neural networks trained for different tasks [19]. Lifelong learning utilizes a machine trained on a number of tasks to carry out a new task by leveraging the knowledge accumulated during the previous training phases [143]. Zero-shot learning refers to models capable of recognizing unseen classes with training examples available only for related, but different, classes. This often entails the task of learning representation of classes, such as prototype vectors, that generate data in the class through a fixed probabilistic mechanism [52]. Domain adaptation will be discussed separately in the next point. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 1 Context: # A First Encounter with Machine Learning **Max Welling** *Donald Bren School of Information and Computer Science* *University of California Irvine* November 4, 2011 #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 6 Context: # PREFACE about 60% correct on 100 categories, the fact that we pull it off seemingly effortlessly serves as a “proof of concept” that it can be done. But there is no doubt in my mind that building truly intelligent machines will involve learning from data. The first reason for the recent successes of machine learning and the growth of the field as a whole is rooted in its multidisciplinary character. Machine learning emerged from AI but quickly incorporated ideas from fields as diverse as statistics, probability, computer science, information theory, convex optimization, control theory, cognitive science, theoretical neuroscience, physics, and more. To give an example, the main conference in this field is called: **advances in neural information processing systems**, referring to information theory and theoretical neuroscience and cognitive science. The second, perhaps more important reason for the growth of machine learning is the exponential growth of both available data and computer power. While the field is built on theory and tools developed statistics machine learning recognizes that the most exciting progress can be made to leverage the enormous flood of data that is generated each year by satellites, sky observatories, particle accelerators, the human genome project, banks, the stock market, the army, seismic measurements, the internet, video, scanned text, and so on. It is difficult to appreciate the exponential growth of data that our society is generating. To give an example, a modern satellite generates roughly the same amount of data all previous satellites produced together. This insight has shifted the attention from highly sophisticated modeling techniques on small datasets to more basic analysis on such larger data-sets (the latter sometimes called **data-mining**). Hence the emphasis shifted to algorithmic efficiency and as a result many machine learning faculty (like myself) can typically be found in computer science departments. To give some examples of recent successes of this approach one would only have to turn on one computer and perform an internet search. Modern search engines do not run terribly sophisticated algorithms, but they manage to store and sift through almost the entire content of the internet to return sensible search results. There has also been much success in the field of machine translation, not because a new model was invented but because many more translated documents became available. The field of machine learning is multifaceted and expanding fast. To sample a few sub-disciplines: statistical learning, kernel methods, graphical models, artificial neural networks, fuzzy logic, Bayesian methods, and so on. The field also covers many types of learning problems, such as supervised learning, supervised learning, semi-supervised learning, active learning, reinforcement learning, etc. I will only cover the most basic approaches in this book from a highly per #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 7 Context: # MEANT FOR INDUSTRY AS WELL AS BACKGROUND READING This book was written during my sabbatical at the Radboud University in Nijmegen (Netherlands). I would like to thank Hans for discussion on intuition. I also thank Prof. Bert Kappen, who leads an excellent group of postdocs and students for his hospitality. Marga, kids, UCI,... --- There are a few main aspects I want to cover from a personal perspective. Instead of trying to address every aspect of the entire field, I have chosen to present a few popular and perhaps useful tools and approaches. What will (hopefully) be significantly different than most other scientific books is the manner in which I will present these methods. I have always been frustrated by the lack of proper explanation of equations. Many times, I have been staring at a formula without the slightest clue where it came from or how it was derived. Many books also excel in stating facts in an almost encyclopedic style, without providing the proper intuition of the method. This is my primary mission: to write a book that conveys intuition. The first chapter will be devoted to why I think this is important. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 9 Context: # Learning and Intuition We have all experienced the situation that the solution to a problem presents itself while riding your bike, walking home, “relaxing” in the washroom, waking up in the morning, taking your shower, etc. Importantly, it did not appear while banging your head against the problem in a conscious effort to solve it, staring at the equations on a piece of paper. In fact, I would claim that all my bits and pieces of progress have occurred while taking a break and “relaxing out of the problem.” Greek philosophers walked in circles when thinking about a problem; most of us stare at a computer screen all day. The purpose of this chapter is to make you more aware of where your creative mind is located and to interact with it in a fruitful manner. My general thesis is that, contrary to popular belief, creative thinking is not performed by conscious thinking. It is rather an interplay between your conscious mind, which prepares the seeds to be planted into the unconscious part of your mind. The unconscious mind will munch on the problem “out of sight” and return promising roads to solutions to the consciousness. This process iterates until the conscious mind decides the problem is sufficiently solved, intractable, or plain dull and moves on to the next. It may be a little unsettling to learn that at least part of your thinking goes on in a part of your mind that seems inaccessible and has a very limited interface with what you think of as yourself. But it is undeniable that it is there, and it is also undeniable that it plays a role in the creative thought process. To become a creative thinker, one should learn to play this game more effectively. To do so, we should think about the language in which to represent knowledge that is most effective in terms of communication with the unconscious. In other words, what type of “interface” between conscious and unconscious mind should we use? It is probably not a good idea to memorize all the details of a complicated equation or problem. Instead, we should extract the abstract idea and capture the essence of it in a picture. This could be a movie with colors and other elements. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 11 Context: Many people may find this somewhat experimental way to introduce students to new topics counter-productive. Undoubtedly for many it will be. If you feel under-challenged and become bored, I recommend moving on to the more advanced textbooks, of which there are many excellent samples on the market (for a list see [books](#)). But I hope that for most beginning students, this intuitive style of writing may help to gain a deeper understanding of the ideas that I will present in the following. Above all, have fun! #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 13 Context: # Chapter 1 ## Data and Information Data is everywhere in abundant amounts. Surveillance cameras continuously capture video; every time you make a phone call your name and location get recorded. Often, your clicking pattern is recorded when surfing the web. Most financial transactions are recorded; satellites and observatories generate terabytes of data every year; the FBI maintains a DNA database of most convicted criminals. Soon, all written text from our libraries is digitized. Need I go on? But data in itself is useless. Hidden inside the data is valuable information. The objective of machine learning is to pull the relevant information from the data and make it available to the user. What do we mean by "relevant information"? When analyzing data, we typically have a specific question in mind, such as: - “How many types of car can be discerned in this video?” - “What will be the weather next week?” The answer can take the form of a single number (there are 5 cars), or a sequence of numbers (or the temperature next week) or a complicated pattern (the cloud configuration next week). If the answer to our query is complex, we like to visualize it using graphs, bar plots, or even little movies. One should keep in mind that the particular analysis depends on the task one has in mind. Let me spell out a few tasks that are typically considered in machine learning: ### Prediction Here we ask ourselves whether we can extrapolate the information in the data to new use cases. For instance, if I have a database of attributes of Hummers such as weight, color, number of people it can hold, etc., and another database of attributes for Ferraris, then we can try to predict the type of car (Hummer or Ferrari) from a new set of attributes. Another example is predicting the weather given all the recorded weather patterns in the past: can we predict the weather next week, or the stock prices? #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 15 Context: # 1.1. DATA REPRESENTATION Most datasets can be represented as a matrix, \( X = [X_{n,k}] \), with rows indexed by "attribute-index" \( i \) and columns indexed by "data-index" \( n \). The value \( X_{n,k} \) for attribute \( i \) and data-case \( n \) can be binary, real, discrete, etc., depending on what we measure. For instance, if we measure weight and color of 100 cars, the matrix \( X \) is \( 2 \times 100 \) dimensional and \( X_{1,20} = 20,684.57 \) is the weight of car nr. 20 in some units (a real value) while \( X_{2,20} = 2 \) is the color of car nr. 20 (say one of 6 predefined colors). Most datasets can be cast in this form (but not all). For documents, we can give each distinct word of a prespecified vocabulary a number and simply count how often a word was present. Say the word "book" is defined to have nr. 10, 568 in the vocabulary then \( X_{1,1005} = 4 \) would mean: the word book appeared 4 times in document 5076. Sometimes the different data-cases do not have the same number of attributes. Consider searching the internet for images about cats. You’ll retrieve a large variety of images most with a different number of pixels. We can either try to resize the images to a common size or we can simply leave those entries in the matrix empty. It may also occur that a certain entry is supposed to be there but it couldn’t be measured. For instance, if we run an optical character recognition system on a scanned document some letters will not be recognized. We’ll use a question mark “?” to indicate that that entry wasn’t observed. It is very important to realize that there are many ways to represent data and not all are equally suitable for analysis. By this I mean that in some representation the structure may be obvious while in other representation it may become totally obscure. It is still there, but just harder to find. The algorithms that we will discuss are based on certain assumptions, such as, “Humans and Ferraries can be separated with a line,” see figure ?. While this may be true if we measure weight in kilograms and height in meters, it is no longer true if we decide to re-code these numbers into bit-strings. The structure is still in the data, but we would need a much more complex assumption to discover it. A lesson to be learned is thus to spend some time thinking about in which representation the structure is as obvious as possible and transform the data if necessary before applying standard algorithms. In the next section we’ll discuss some standard preprocessing operations. It is often advisable to visualize the data before preprocessing and analyzing it. This will often tell you if the structure is a good match for the algorithm you had in mind for further analysis. Chapter ? will discuss some elementary visualization techniques. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 17 Context: # 1.2. PREPROCESSING THE DATA Attribute separately) and then added and divided by \( N \). You have perhaps noticed that variance does not have the same units as \( X \) itself. If \( X \) is measured in grams, then variance is measured in grams squared. To scale the data to have the same scale in every dimension, we divide by the square-root of the variance, which is usually called the sample standard deviation: \[ X_{n}^{m} = \frac{X_{n}}{\sqrt{V[|X|_{i}]}} \quad \forall n \] Note again that sphering requires centering, implying that we always have to perform these operations in this order, first center, then sphere. Figure ??a,b,c illustrates this process. You may now be asking, “well what if the data where elongated in a diagonal direction?”. Indeed, we can also deal with such a case by first centering, then rotating such that the elongated direction points in the direction of one of the axes, and then scaling. This requires quite a bit more math, and will postpone this issue until chapter ?? on “principal components analysis”. However, the question is in fact a very deep one, because one could argue that one could keep changing the data using more and more sophisticated transformations until all the structure was removed from the data and there would be nothing left to analyze! It is indeed true that the pre-processing steps can be viewed as part of the modeling process in that it identifies structure (and then removes it). By remembering the sequence of transformations you performed, you have implicitly built a model. Reversely, many algorithms can be easily adapted to model the mean and scale of the data. Now, the preprocessing is no longer necessary and becomes integrated into the model. Just as preprocessing can be viewed as building a model, we can use a model to transform structured data into (more) unstructured data. The details of this process will be left for later chapters but a good example is provided by compression algorithms. Compression algorithms are based on models for the redundancy in data (e.g. text, images). The compression consists in removing this redundancy and transforming the original data into a less redundant (and hence more succinct) code. Models and structure reducing data transformations are in essence each others reverse: we often associate with a model an understanding of how the data was generated, starting from random noise. Reversely, pre-processing starts with the data and understands how we can get back to the unstructured random state of the data. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 18 Context: # CHAPTER 1. DATA AND INFORMATION The origin. If data happens to be just positive, it doesn’t fit this assumption very well. Taking the following logarithm can help in that case: $$ X'_{nm} = \log(\alpha + X_{nm}) \tag{1.5} $$ #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 20 Context: # CHAPTER 2. DATA VISUALIZATION An example of such a scatter plot is given in Figure ??. Note that we have a total of \( d(d - 1) / 2 \) possible two-dimensional projections which amounts to 4950 projections for 100-dimensional data. This is usually too many to manually inspect. How do we cut down on the number of dimensions? Perhaps random projections may work? Unfortunately, that turns out to be not a great idea in many cases. The reason is that data projected on a random subspace often looks distributed according to what is known as a Gaussian distribution (see Figure ??). The deeper reason behind this phenomenon is the **central limit theorem** which states that the sum of a large number of independent random variables is (under certain conditions) distributed as a Gaussian distribution. Hence, if we denote with \( \mathbf{w} \) a vector in \( \mathbb{R}^d \) and by \( x \) the d-dimensional random variable, then \( y = \mathbf{w}^T \mathbf{x} \) is the value of the projection. This is clearly a weighted sum of the random variables \( x_i, \; i = 1, \ldots, d \). If we assume that \( x_i \) are approximately independent, then we can see that their sum will be governed by its central limit theorem. Analogously, a dataset \( \{X_n\} \) can thus be visualized in one dimension by "histogramming" the values of \( y = \mathbf{w}^T \mathbf{x} \), see Figure ??. In this figure we clearly recognize the characteristic "Bell-shape" of the Gaussian distribution of projected and histogrammed data. In one sense the central limit theorem is a rather helpful quirk of nature. Many variables follow Gaussian distributions and the Gaussian distribution is one of the few distributions which have very nice analytical properties. Unfortunately, the Gaussian distribution is also the most uninformative distribution. This notion of "uninformative" can actually be made very precise using information theory and states: > Given a fixed mean and variance, the Gaussian density represents the least amount of information among all densities with the same mean and variance. This is rather unfortunate for our purposes because Gaussian projections are the least revealing dimensions to look at. So in general we have to work a bit harder to see interesting structure. A large number of algorithms has been devised to search for informative projections. The simplest being "principal component analysis" or PCA for short ??. Here, interesting means dimensions of high variance. However, the fact that high variance is not always a good measure of interestingness and one should rather search for dimensions that are non-Gaussian. For instance, "independent components analysis" (ICA) ?? and "projection pursuit" ?? search for dimen- #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 23 Context: # Chapter 3 ## Learning This chapter is without question the most important one of the book. It concerns the core, almost philosophical question of what learning really is (and what it is not). If you want to remember one thing from this book, you will find it here in this chapter. Ok, let’s start with an example. Alice has a rather strange ailment. She is not able to recognize objects by their visual appearance. At home, she is doing just fine: her mother explained to Alice for every object in her house what it is and how you use it. When she is home, she recognizes these objects (if they have not been moved too much), but when she enters a new environment she is lost. For example, if she enters a new meeting room, she needs a long time to infer what the chairs and the table are in the room. She has been diagnosed with a severe case of "overfitting." What is the matter with Alice? Nothing is wrong with her memory because she remembers the objects once she has seen them. In fact, she has a fantastic memory. She remembers every detail of the objects she has seen. And every time she sees a new object, she reasons that the object in front of her is surely not a chair because it doesn’t have all the features she has seen in earlier chairs. The problem is that Alice cannot generalize the information she has observed from one instance of a visual object category to other, yet unobserved members of the same category. The fact that Alice’s disease is so rare is understandable; there must have been a strong selection pressure against this disease. Imagine our ancestors walking through the savanna one million years ago. A lion appears on the scene. Ancestral Alice has seen lions before, but not this particular one, and it does not induce a fear response. Of course, she has no time to infer the possibility that this animal may be dangerous logically. Alice’s contemporaries noticed that the animal was yellow-brown, had manes, etc., and immediately un- #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 25 Context: # Information Theory and Image Compression The part of the information that does not carry over to the future, the unpredictable information, is called "noise." There is also the information that is predictable, the learnable part of the information stream. The task of any learning algorithm is to separate the predictable part from the unpredictable part. Now imagine Bob wants to send an image to Alice. He has to pay 1 dollar cent for every bit that he sends. If the image were completely white, it would be really stupid of Bob to send the message: ``` pixel 1: white, pixel 2: white, pixel 3: white, .... ``` He could just have sent the message: ``` all pixels are white! ``` The blank image is completely predictable but carries very little information. Now imagine a image that consists of white noise (your television screen if the cable is not connected). To send the exact image Bob will have to send: ``` pixel 1: white, pixel 2: black, pixel 3: black, .... ``` Bob cannot do better because there is no predictable information in that image, i.e., there is no structure to be modeled. You can imagine playing a game and revealing one pixel at a time to someone and paying him 15 for every next pixel he predicts correctly. For the white image, you can do perfect for the noisy picture you would be random guessing. Real pictures are in between: some pixels are very hard to predict, while others are easier. To compress the image, Bob can extract rules such as: always predict the same color as the majority of the pixels next to you, except when there is an edge. These rules constitute the model for the regularities of the image. Instead of sending the entire image pixel by pixel, Bob will now first send his rules and ask Alice to apply the rules. Every time the rule fails, Bob also sends a correction: ``` pixel 103: white, pixel 245: black. ``` A few rules and two corrections are obviously cheaper than 256 pixel values and no rules. There is one fundamental tradeoff hidden in this game. Since Bob is sending only a single image, it does not pay to send an incredibly complicated model that would require more bits to explain than simply sending all pixel values. If he were sending 1 billion images it would pay off to first send the complicated model because he would be saving a fraction of all bits for every image. On the other hand, if Bob wants to send 2 pixels, there really is no need in sending a model whatsoever. Therefore, the size of Bob's model depends on the amount of data he wants to transmit. In this context, the boundary between what is model and what is noise depends on how much data we are dealing with! If we use a model that is too complex, we overfit to the data at hand, i.e., part of the model represents noise. On the other hand, if we use a too simple model we "underfit" (over-generalize) and valuable structure remains unmodeled. Both lead to sub-optimal compression of the image. The compression game can therefore be used to find the right size of model complexity for a given dataset. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 26 Context: # CHAPTER 3. LEARNING Now let’s think for a moment what we really mean with “a model”. A model represents our prior knowledge of the world. It imposes structure that is not necessarily present in the data. We call this the **inductive bias**. Our inductive bias often comes in the form of a parameterized model. That is to say, we define a family of models but let the data determine which of these models is most appropriate. A strong inductive bias means that we don’t leave flexibility in the model for the data to work on. We are so convinced of ourselves that we basically ignore the data. The downside is that if we are creating a “bad bias” towards to wrong model. On the other hand, if we are correct, we can learn the remaining degrees of freedom in our model from very few data-cases. Conversely, we may leave the door open for a huge family of possible models. If we now let the data zoom in on the model that best explains the training data it will overfit to the peculiarities of that data. Now imagine you sampled 10 datasets of the same size \(N\) and train these very flexible models separately on each of these datasets (note that in reality you only have access to one such dataset but please play along in this thought experiment). Let’s say we want to determine the value of some parameter \(\theta\). Because the models are so flexible, we can actually model the idiosyncrasies of each dataset. The result is that the value for \(\theta\) is likely to be very different for each dataset. But because we didn’t impose much inductive bias the average of many of such estimates will be about right. We say that the bias is small, but the variance is high. In the case of very restrictive models the opposite happens: the bias is potentially large but the variance small. Note that not only is a large bias bad (for obvious reasons), a large variance is bad as well: because we only have one dataset of size \(N\), our estimate could be very far off simply we were unlucky with the dataset we were given. What we should therefore strive for is to inject all our prior knowledge into the learning problem (this makes learning easier) but avoid injecting the wrong prior knowledge. If we don’t trust our prior knowledge we should let the data speak. However, letting the data speak too much might lead to overfitting, so we need to find the boundary between too complex and too simple a model and get its complexity just right. Access to more data means that the data can speak more relative to prior knowledge. That, in a nutshell, is what machine learning is all about. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 27 Context: # 3.1 In a Nutshell Learning is all about generalizing regularities in the training data to new, yet unobserved data. It is not about remembering the training data. Good generalization means that you need to balance prior knowledge with information from data. Depending on the dataset size, you can entertain more or less complex models. The correct size of the model can be determined by playing a compression game. Learning = generalization = abstraction = compression. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 28 Context: # CHAPTER 3. LEARNING ## Introduction Learning involves acquiring knowledge, skills, attitudes, or competencies. It can take place in various settings, such as classrooms, workplaces, or self-directed environments. This chapter discusses the types and processes of learning. ## Types of Learning 1. **Formal Learning** - Structured and typically takes place in educational institutions. - Includes degrees, diplomas, and certifications. 2. **Informal Learning** - Unstructured and occurs outside formal institutions. - Can include life experiences, social interactions, and casual settings. 3. **Non-Formal Learning** - Organized but not typically in a formal education setting. - Often community-based, such as workshops and training sessions. ## Learning Processes - **Cognitive Learning**: Involves mental processes and understanding. - **Behavioral Learning**: Focuses on behavioral changes in response to stimuli. - **Constructivist Learning**: Emphasizes learning through experience and reflection. ## Table of Learning Theories | Theory | Key Contributor | Description | |--------------------------|----------------------|--------------------------------------------------| | Behaviorism | B.F. Skinner | Learning as a change in behavior due to reinforcement. | | Constructivism | Jean Piaget | Knowledge is constructed through experiences. | | Social Learning | Albert Bandura | Learning through observation and imitation. | ## Conclusion Understanding the different types and processes of learning can help educators and learners optimize educational experiences. Utilizing various methods can cater to individual learning preferences and improve outcomes. ## References - Smith, J. (2020). *Learning Theories: A Comprehensive Approach*. Educational Press. - Doe, A. (2019). *Techniques for Effective Learning*. Learning Publishers. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 31 Context: fall under the name "reinforcement learning". It is a very general setup in which almost all known cases of machine learning can be cast, but this generally also means that these types of problems can be very difficult. The most general RL problems do not even assume that you know what the world looks like (i.e. the maze for the mouse), so you have to simultaneously learn a model of the world and solve your task in it. This dual task induces interesting trade-offs: should you invest time now to learn machine learning and reap the benefit later in terms of a high salary working for Yahoo!, or should you stop investing now and start exploiting what you have learned so far? This is clearly a function of age, or the time horizon that you still have to take advantage of these investments. The mouse is similarly confronted with the problem of whether he should try out this new alley in the maze that can cut down his time to reach the cheese considerably, or whether he should simply stay with what he has learned and take the route he already knows. This clearly depends on how often he thinks he will have to run through the same maze in the future. We call this the exploration versus exploitation trade-off. The reason that RL is a very exciting field of research is because of its biological relevance. Do we not also have to figure out how the world works and survive in it? Let's go back to the news-articles. Assume we have control over what article we will label next. Which one would be pick? Surely the one that would be most informative in some suitably defined sense. Or the mouse in the maze. Given that it needs to explore, where does he explore? Surely he will try to seek out alleys that look promising, i.e. alleys that he expects to maximize his reward. We call the problem of finding the next best data-case to investigate "active learning". One may also be faced with learning multiple tasks at the same time. These tasks are related but not identical. For instance, consider the problem if recommending movies to customers of Netflix. Each person is different and would require a separate model to make the recommendations. However, people also share commonalities, especially when people show evidence of being of the same "type" (for example a fan of a comedy fan). We can learn personalized models but share features between them. Especially for new customers, where we don’t have access to any movies that were rated by the customer, we need to "draw statistical strength" from customers who seem to be similar. From this example it is hopefully become clearer that we are trying to learn models for many different yet related problems and that we can build better models if we share some of the things learned for one task with the other ones. The trick is not to share too much nor too little and how much we should share depends on how much data and prior knowledge we have access to for each task. We call this subfield of machine learning: "multi-task learning". #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 32 Context: # Chapter 4: Types of Machine Learning ## 4.1 In a Nutshell There are many types of learning problems within machine learning. Supervised learning deals with predicting class labels from attributes, unsupervised learning tries to discover interesting structure in data, semi-supervised learning uses both labeled and unlabeled data to improve predictive performance, reinforcement learning can handle simple feedback in the form of delayed reward, active learning optimizes the next sample to include in the learning algorithm, and multi-task learning deals with sharing common model components between related learning tasks. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 35 Context: 5.1 The Idea In a Nutshell ========================== To classify a new data item, you first look for the \( k \) nearest neighbors in feature space and assign it the same label as the majority of these neighbors. Because 98 noisy dimensions have been added, this effect is detrimental to the kNN algorithm. Once again, it is very important to choose your initial representation with much care and preprocess the data before you apply the algorithm. In this case, preprocessing takes the form of "feature selection" on which a whole book in itself could be written. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 40 Context: 28 # CHAPTER 6. THE NATIVE BAYESIAN CLASSIFIER For ham emails, we compute exactly the same quantity, $$ P_{ham}(X_i = j) = \frac{ \# \text{ ham emails for which the word } j \text{ was found} }{ \text{total } \# \text{ of ham emails} } $$ (6.5) $$ = \frac{\sum_{n} \mathbb{I}(X_{n} = j \land Y_{n} = 0)}{ \sum_{n} \mathbb{I}(Y_{n} = 0)} $$ (6.6) Both these quantities should be computed for all words or phrases (or more generally attributes). We have now finished the phase where we estimate the model from the data. We will often refer to this phase as "learning" or training a model. The model helps us understand how data was generated in some approximate setting. The next phase is that of prediction or classification of new email. ## 6.3 Class-Prediction for New Instances New email does not come with a label ham or spam (if it would we could throw spam in the spam-box right away). What we do see are the attributes $\{X_i\}$. Our task is to guess the label based on the model and the measured attributes. The approach we take is simple: calculate whether the email has a higher probability of being generated from the spam or the ham model. For example, because the word "viagra" has a tiny probability of being generated under the ham model it will end up with a higher probability under the spam model. But clearly, all words have a say in this process. It’s like a large committee of experts, one for each word; each member casts a vote and can say things like: "I am 99% certain its spam", or "It’s almost definitely not spam (0.1% spam)". Each of these opinions will be multiplied together to generate a final score. We then figure out whether ham or spam has the highest score. There is one little practical caveat with this approach, namely that the product of a large number of probabilities, each of which is necessarily smaller than one, very quickly gets so small that your computer can’t handle it. There is an easy fix though. Instead of multiplying probabilities as scores, we use the logarithms of those probabilities and add the logarithms. This is numerically stable and leads to the same conclusion because if \(a > b\) then we also have \(\log(a) > \log(b)\) and vice versa. In equations we compute the score as follows: $$ S_{spam} = \sum_{i} \log P_{spam}(X_i = e_i) + \log P(spam) $$ (6.7) #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 43 Context: # 6.5 Remarks One of the main limitations of the NB classifier is that it assumes independence between attributes (This is presumably the reason why we call it the naive Bayesian classifier). This is reflected in the fact that each classifier has an independent vote in the final score. However, imagine that I measure the words, "home" and "mortgage". Observing "mortgage" certainly raises the probability of observing "home". We say that they are positively correlated. It would therefore be more fair if we attributed a smaller weight to "home" if we already observed mortgage because they convey the same thing: this email is about mortgages for your home. One way to obtain a more fair voting scheme is to model these dependencies explicitly. However, this comes at a computational cost (a longer time before you receive your email in your inbox) which may not always be worth the additional accuracy. One should also note that more parameters do not necessarily improve accuracy because too many parameters may lead to overfitting. # 6.6 The Idea In a Nutshell Consider Figure ??. We can classify data by building a model of how the data was generated. For NB we first decide whether we will generate a data-item from class \( Y = 0 \) or class \( Y = 1 \). Given that decision we generate the values for \( D \) attributes independently. Each class has a different model for generating attributes. Classification is achieved by computing which model was more likely to generate the new data-point, biasing the outcome towards the class that is expected to generate more data. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 50 Context: # CHAPTER 7. THE PERCEPTRON The function `tanh(·)` is plotted in Figure ?? It shows that the cost can never be larger than 2, which ensures robustness against outliers. We leave it to the reader to derive the gradients and formulate the gradient descent algorithm. ## 7.3 The Idea In a Nutshell Figure ?? tells the story. One assumes that your data can be separated by a line. Any line can be represented by `w^T x = α`. Data cases from one class satisfy `w^T x_n ≤ α` while data cases from the other class satisfy `w^T x_n ≥ α`. To achieve that, you write down a cost function that penalizes data cases falling on the wrong side of the line and minimize it over `(w, α)`. For a test case, you simply compute the sign of `w^T x_{test} - α` to make a prediction as to which class it belongs to. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 54 Context: 42 # CHAPTER 8. SUPPORT VECTOR MACHINES The theory of duality guarantees that for convex problems, the dual problem will be concave, and moreover, that the unique solution of the primal problem corresponds to the unique solution of the dual problem. In fact, we have: \( L_P(w^*) = L_D(\alpha^*) \), i.e. the “duality-gap” is zero. Next we turn to the conditions that must necessarily hold at the saddle point and thus the solution of the problem. These are called the KKT conditions (which stands for Karush-Kuhn-Tucker). These conditions are necessary in general, and sufficient for convex optimization problems. They can be derived from the primal problem by setting the derivatives w.r.t to zero. Also, the constraints themselves are part of these conditions and we need that for inequality constraints the Lagrange multipliers are non-negative. Finally, an important constraint called “complementary slackness” needs to be satisfied. \[ \begin{align*} \partial_w L_P = 0 & \rightarrow \quad w - \sum_i \alpha_i y_i x_i = 0 \quad (8.12) \\ \partial_{\alpha} L_P = 0 & \rightarrow \sum_i \alpha_i y_i = 0 \quad (8.13) \\ \text{constraint - 1} & \quad y_i (w^T x_i - b) - 1 \geq 0 \quad (8.14) \\ \text{multiplier condition} & \quad \alpha_i \geq 0 \quad (8.15) \\ \text{complementary slackness} & \quad \alpha_i [y_i (w^T x_i - b) - 1] = 0 \quad (8.16) \end{align*} \] It is the last equation which may be somewhat surprising. It states that either the inequality constraint is satisfied, but not saturated: \( y_i (w^T x_i - b) - 1 > 0 \) in which case \( \alpha_i \) for that data-case must be zero, or the inequality constraint is saturated \( y_i (w^T x_i - b) - 1 = 0 \), in which case \( \alpha_i \) can be any value \( \geq 0 \). Inequality constraints which are saturated are said to be “active”, while unsaturated constraints are inactive. One could imagine the process of searching for a solution as a ball which runs down the primary objective function using gradient descent. At some point, it will hit a wall which is the constraint and although the derivative is still pointing partially towards the wall, the constraints prohibits the ball to go on. This is an active constraint because the ball is glued to that wall. When a final solution is reached, we could remove some constraints, without changing the solution, these are inactive constraints. One could think of the term \( \partial_{\alpha} L_P \) as the force acting on the ball. We see from the first equation above that only the forces with \( \alpha_i \neq 0 \) exert a force on the ball that balances with the force from the curved quadratic surface \( w \). The training cases with \( \alpha_i > 0 \), representing active constraints on the position of the support hyperplane are called support vectors. These are the vectors. #################### File: A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf Page: 81 Context: # Chapter 14 ## Kernel Canonical Correlation Analysis Imagine you are given 2 copies of a corpus of documents, one written in English, the other written in German. You may consider an arbitrary representation of the documents, but for definiteness we will use the “vector space” representation where there is an entry for every possible word in the vocabulary and a document is represented by count values for every word, i.e., if the word “the” appeared 12 times and the first word in the vocabulary we have \( X_1(doc) = 12 \), etc. Let’s say we are interested in extracting low dimensional representations for each document. If we had only one language, we could consider running PCA to extract directions in word space that carry most of the variance. This has the ability to infer semantic relations between the words such as synonymy, because if words tend to co-occur often in documents, i.e., they are highly correlated, they tend to be combined into a single dimension in the new space. These spaces can often be interpreted as topic spaces. If we have two translations, we can try to find projections of each representation separately such that the projections are maximally correlated. Hopefully, this implies that they represent the same topic in two different languages. In this way we can extract language independent topics. Let \( x \) be a document in English and \( y \) a document in German. Consider the projections: \( u = a^Tx \) and \( v = b^Ty \). Also assume that the data have zero mean. We now consider the following objective: \[ \rho = \frac{E[uv]}{\sqrt{E[u^2]E[v^2]}} \tag{14.1} \] ########## """QUERY: in 5 sentences summarize the context""" Consider the chat history for relevant information. Use all information included. Use as much tokens as needed. Important: If you find information separated by a | in the context, it is a table formatted in Markdown. 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Page 50, A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf - Page 54, A%20First%20Encounter%20with%20Machine%20Learning%20-%20Max%20Welling%20%28PDF%29.pdf - Page 81 ================================================== **Elapsed Time: 3.98 seconds** ================================================== FINAL ANSWER Answer: In the tome titled "A Cool Brisk Walk Through Discrete Mathematics," authored by Stephen Davies, Ph.D., the reader embarks upon a journey through the foundational concepts of discrete mathematics, encompassing sets, functions, and relations, which serve as the bedrock for various applications in fields such as computer science and combinatorial design. The text is structured methodically, beginning with an introduction that lays the groundwork, followed by detailed explorations of each concept, including definitions, properties, and operations associated with sets, functions, and relations, ultimately culminating in practical applications and a reflective conclusion. The author emphasizes the importance of these mathematical principles in understanding and solving real-world problems, particularly in the realm of technology and computation. Additionally, the document includes visual analyses and exercises that enhance comprehension and engagement with the material, making it accessible to learners at various levels. Through this scholarly work, Davies aims to demystify discrete mathematics, presenting it as an inviting and essential discipline for aspiring mathematicians and computer scientists alike. ================================================== **Elapsed Time: 4.08 seconds** ==================================================