Learning from Data

layout: true
<div class="my-header">
  <p class="align_left"><img src="images/yonsei_logo.png" style="height: 30px;"/></p>
  <p class="align_right"><b>Chapter 1 - The Learning Problem</b></p>
</div>
<div class="my-footer">
  <p class="align_right"><b>2022.01.09 @ SYTEARK</b></p>
  <p class="align_left"><b>Tae Geun Kim</b></p>
</div>

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# The Learning Problem

<h4 style="color:brown">Part 1 of Chapter 1</h4>

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## Table of Contents

* Learning from Data

* A Simple Learning Model - Perceptron

* Learning versus Design

* Types of Learning

* Is Learning Feasible?

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# Learning from Data

---

### A tree

.left-column[
<div id="boxshadow" style="width:90%">
  <img src="images/tree.jpg" alt="simple" style="width:100%"/>
  <figcaption style="text-align: center;"><b>Fig.1</b> A tree</figcaption>
</div>
]

.right-column[
.center[
  It is because we learned it by looking at trees.
]
]
--
.right-column[
.center[
In other words, we learned from **data**.
]
]

---

### Netflix @ 2006

<div id="boxshadow" style="width:100%">
  <img src="images/netflix.webp" alt="netflix" style="width:100%"/>
  <figcaption style="text-align: center;"><b>Fig.2</b> Netflix Prize</figcaption>
</div>

---

### Netflix @ 2006

* Winner takes all : $1,000,000

* It can only be received when there is 10% improvement.

.center[
<div id="boxshadow" style="width:100%">
  <img src="images/movie.png" alt="netflix" style="width:60%;"/>
  <figcaption style="text-align: center;"><b>Fig.3</b> A model for how a viewer rates a movie</figcaption>
</div>
]

---
class: split-50

### Components of Learning

**Metaphor**: Credit approval

<div id="boxshadow" style="width:95%">
  <img src="images/credit.png" alt="netflix" style="width:100%;"/>
  <figcaption style="text-align: center;"><b>Fig.4</b> Applicant information</figcaption>
</div>

.right-column[
* Input: $\mathbf{x}$ (Customer information)
]
--
.right-column[
* Output: $y$ (good/bad customer?)
]
--
.right-column[
* Target function: $f: \mathcal{X} \rightarrow \mathcal{Y}$  
(ideal credit approval formula)
]
--
.right-column[
* Data: $(\mathbf{x}_1, y_1),\,\cdots\,(\mathbf{x}_N, y_N)$
]
--
.right-column[
* Hypothesis set: $\mathcal{H}$ s.t. $h \in \mathcal{H}$  
where $h: \mathcal{X} \rightarrow \mathcal{Y}$
]
--
.right-column[
* Final hypothesis: $g: \mathcal{X} \rightarrow \mathcal{Y}$
]

---

### Components of Learning

.center[
<div id="boxshadow" style="width:100%">
  <img src="images/basic.png" alt="basic" style="width:70%;"/>
  <figcaption style="text-align: center;"><b>Fig.5</b> Basic setup of the learning problem</figcaption>
</div>
]

---

# A Simple Learning Model - Perceptron

---

### A simple hypothesis set - the perceptron

* Input space : $~\mathcal{X} = \mathbb{R}^d$

* Output space: $~\mathcal{Y} = \left\\{ -1,\,1 \right\\}$

* Hypothesis:
$$
h(\mathbf{x}) =
\begin{cases}
+1&\text{ if } \displaystyle \sum\_{i=1}^d w\_i x\_i > \text{threshold} \\\\
-1&\text{ if } \displaystyle \sum\_{i=1}^d w\_i x\_i < \text{threshold}
\end{cases}
$$

$$
{\color{red}h}(\mathbf{x}) = \text{sign}\left\\{ \left(\sum\_{i=1}^d \color{red}{w\_i} x\_i \right) - \color{red}{\text{thresold}}\right \\}
$$

---

### A simple hypothesis set - the perceptron

If we denote $\text{thresold}$ as $w_0$ & $x_0 = 1$, then

$$
{\color{red}h}(\mathbf{x}) = \text{sign}\left(\sum\_{i=0}^d \color{red}{w\_i} x\_i \right)
$$

In a vector form,

$$
\color{red}{h}(\mathbf{x}) = \text{sign}(\color{red}{\mathbf{w}}^T \mathbf{x}) \\\\
\text{where }~ \mathbf{w} = [w_0,\cdots,w_d]^T,~~\mathbf{x} = [1,x_1,\cdots,x_d]^T
$$

The corresponding hypothesis set is

$$
\mathcal{H} = \left\\{ h\_\mathbf{w} \,|\, \mathbf{w} \in \mathbb{R}^{d+1} \right\\}
$$

This model of $\mathcal{H}$ is called the **perceptron**.

---

### A simple learning algorithm - PLA

**Q. How to determine which hypothesis is the best?**

A. For perceptron, there is the algorithm - *Perceptron Learning Algorithm (PLA)*.

.center[
<div class="animated-border-quote" style="width:100%">
  <blockquote style="width:80%">
    <p>At iteration $t$, where $t = 0,1,2,\cdots$</p>
    <ul>
      <li>$\mathbf{w}(t)$: a current value of the weight vector</li>
      <li>$(\mathbf{x}(t),y(t))$: a random picked misclassified data
        $$
        y(t) \neq \text{sign}(\mathbf{w}^T \mathbf{x}(t))
        $$
      </li>
      <li>Update $\mathbf{w}(t)$ with following rule:
        $$
        \mathbf{w}(t+1) = \mathbf{w}(t) + y(t) \mathbf{x}(t)
        $$
      </li>
    </ul>
  </blockquote>
</div>
]

---

### A simple learning algorithm - PLA

**Q. Why does PLA work?**

.center[
<div class="animated-border-quote" style="width:100%">
  <blockquote style="width:80%">
    <p><b>Exercise 1.3</b></p>
    <p style="text-align:left">
      The weight update rule in previous slide has the nice interpretation that it moves in the direction of classifying
      $\mathbf{x}(t)$ correctly.
      <ol>
        <li>Show that $~y(t) \mathbf{w}^T(t)\mathbf{x}(t) < 0$.</li>
        <li>Show that $~y(t) \mathbf{w}^T(t+1)\mathbf{x}(t) > y(t) \mathbf{w}^T(t)\mathbf{x}(t)$.</li>
        <li>As far as classifying $\mathbf{x}(t)$ is concerned, argue that the move from $\mathbf{w}(t)$ to $\mathbf{w}(t+1)$
          is a move <i>in the right direction</i>.
      </ol>
    </p>
  </blockquote>
</div>
]

**Q. Can PLA converge?**
--
$~~\Rightarrow~$ Problem 1.3 (Homework)

---

### Significance of Perceptron

.left-column[
.center[
<div id="boxshadow" style="width:90%">
  <img src="images/pla.png" alt="PLA" style="width:100%">
  <figcaption style="text-align:center";><b>Fig.6</b> PLA</figcaption>
</div>
]
]

.right-column[
  * Within the infinite space of all weight vectors, the perceptron algorithm manages
  to find a weight vector that works, using a simple iterative process.
]

.right-column[
  * This illustrates how a learning algorithm can effectively search an infinite hypothesis set
  using a finite number of simple steps.
]

.right-column[
  * This feature is characteristics of many techniques that are used in learning,
  some of which are far more sophisticated than the perceptron.
]

---

# Learning versus Design

---

### Learning

.center[
<div id="boxshadow" style="width: 100%;">
  <img src="images/learn.png" alt="learning" style="width:100%;">
  <figcaption style="text-align:center";><b>Fig.7</b> The learning approach to coin classification</figcaption>
</div>
]

---

### Design

.center[
<div id="boxshadow" style="width: 100%;">
  <img src="images/design.png" alt="design" style="width:100%;">
  <figcaption style="text-align:center";><b>Fig.8</b> The design approach to coin classification</figcaption>
</div>
]

---

### Learning vs Design

.center[
<div class="animated-border-quote" style="width:100%">
  <blockquote style="width:90%">
    <p><b>Exercise 1.5</b></p>
    <p style="text-align:left">
      Which of the following problems are more suited for the learning approach and which are more suited for the design approach?
    </p>
    <ol>
      <li>Determining the age at which a particular medical test should be performed.</li>
      <li>Classifying numbers into primes and non-primes.</li>
      <li>Detecting potential fraud in credit card charges.</li>
      <li>Determining the time it would take a falling object to hit the ground.</li>
      <li>Determining the optimal cycle for traffic lights in a busy intersection.</li>
    </ol>
  </blockquote>
</div>
]

---

# Types of Learning

---

### Types of Learning

There are three types of learning:

* **Supervised learning**

* **Reinforcement learning**

* **Unsupervised learning**

If you want to detail, then read the 1.2 section of LFD.

---

### Types of Learning

.center[
<div class="animated-border-quote" style="width:100%">
  <blockquote style="width:90%">
    <p><b>Exercise 1.6</b></p>
    <p style="text-align: left;">
      For each of the following tasks, identify which type fo learning is involved
      (supervised, reinforcement, or unsupervised) and the training data to be used.
      If a task can fit more than one type, explain how and describe the training data for each type.
    </p>
    <ol>
      <li>Recommending a book to a user in an online bookstore.</li>
      <li>Playing tic-tac-toe.</li>
      <li>Categorizing movies into different types.</li>
      <li>Learning to play music.</li>
      <li>Credit limit: Deciding the maximum allowed debt for each bank customer.</li>
    </ol>
  </blockquote>
</div>
]

---

# Is Learning Feasible?

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### Is Learning Feasible?

* Our goal is to approximate the target function $f(x)$.

* But the target function is *unknown*.

**Q. How could a limited data set reveal enough information to pin down the entire target function?**

---

### Is Learning Feasible?

**Q. How could a limited data set reveal enough information to pin down the entire target function?**

.center[
<div id="boxshadow" style="width:100%">
  <img src="images/feasible.png" alt="feasible" style="width:70%">
  <figcaption style="text-align:center";><b>Fig.9</b> A visual learning problem.</figcaption>
</div>
]

---

### Outside the Data Set

* When we get the training data $\mathcal{D}$, we know the value of $f$ on all the points in $\mathcal{D}$.

* But it doesn't mean that we have learned $f$, since it doesn't guarantee that we know anything about $f$ outside of $\mathcal{D}$.

**Q. Does the data set $\mathcal{D}$ tell us anything outside of $\mathcal{D}$ that we didn't know before?**

* If answer is yes, then we have learned *something*.

* If answer is no, we can conclude that learning is not feasible.

---

### Outside the Data Set

.center[
<div id="boxshadow" style="width:100%">
  <img src="images/feasible2.png" alt="feasible2" style="width:100%">
  <figcaption style="text-align:center";><b>Fig.10</b> Which should be the final hypothesis?</figcaption>
</div>
]

As long as $f$ is an unknown function, knowing $\mathcal{D}$ cannot exclude any pattern of values for $f$ outside of $\mathcal{D}$.  
--
Thus, the predictions of $g$ outside of $\mathcal{D}$ are meaningless.

But there is the **probability**.

---

### Probability to the Rescue

.center[
<div id="boxshadow" style="width:100%">
  <img src="images/bin.png" alt="bin" style="width:100%">
  <figcaption style="text-align:center";><b>Fig.11</b> A random sample is picked from a bin of red and green marbles.</figcaption>
</div>
]

---

### Probability to the Rescue

.center[
<div class="animated-border-quote" style="width:100%">
  <blockquote style="width:80%">
    <p><b>Exercise 1.8</b></p>
    <p style="text-align:left">
      If $\mu=0.9$, what is the probability that a sample of 10 marbles will have $\nu \leq 0.1$?
      [Hints: 1. Use binomial distribution. 2. The answer is a very small number.]
    </p>
  </blockquote>
</div>
]

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### References

* Y. S. Abu-Mostafa, M. Magdon-Ismail, and H.-T. Lin, *Learning from data: A short course*. Seattle: AMLbook.com, 2012.

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# Thank you!