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CS229 Lecture Notes

Andrew Ng and Tengyu Ma

June 11, 2023

Contents

• I Supervised learning
• 1 Linear regression
  • 1.1 LMS algorithm
  • 1.2 The normal equations
    • 1.2.1 Matrix derivatives
    • 1.2.2 Least squares revisited
  • 1.3 Probabilistic interpretation
  • 1.4 Locally weighted linear regression (optional reading)
• 2 Classification and logistic regression
  • 2.1 Logistic regression
  • 2.2 Digression: the perceptron learning algorithm
  • 2.3 Multi-class classification
  • 2.4 Another algorithm for maximizing `(θ)
• 3 Generalized linear models
  • 3.1 The exponential family
  • 3.2 Constructing GLMs
    • 3.2.1 Ordinary least squares
    • 3.2.2 Logistic regression
• 4 Generative learning algorithms
  • 4.1 Gaussian discriminant analysis
    • 4.1.1 The multivariate normal distribution
    • 4.1.2 The Gaussian discriminant analysis model
    • 4.1.3 Discussion: GDA and logistic regression
  • 4.2 Naive bayes (Option Reading)
    • 4.2.1 Laplace smoothing
    • 4.2.2 Event models for text classification
• CS229 Spring
• 5 Kernel methods
  • 5.1 Feature maps
  • 5.2 LMS (least mean squares) with features
  • 5.3 LMS with the kernel trick
  • 5.4 Properties of kernels
• 6 Support vector machines
  • 6.1 Margins: intuition
  • 6.2 Notation (option reading)
  • 6.3 Functional and geometric margins (option reading)
  • 6.4 The optimal margin classifier (option reading)
  • 6.5 Lagrange duality (optional reading)
  • 6.6 Optimal margin classifiers: the dual form (option reading)
  • 6.7 Regularization and the non-separable case (optional reading)
  • 6.8 The SMO algorithm (optional reading)
    • 6.8.1 Coordinate ascent
    • 6.8.2 SMO
• II Deep learning
• 7 Deep learning
  • 7.1 Supervised learning with non-linear models
  • 7.2 Neural networks
  • 7.3 Modules in Modern Neural Networks
  • 7.4 Backpropagation
    • 7.4.1 Preliminaries on partial derivatives
    • 7.4.2 General strategy of backpropagation
    • 7.4.3 Backward functions for basic modules
    • 7.4.4 Back-propagation for MLPs
  • 7.5 Vectorization over training examples
• III Generalization and regularization
• 8 Generalization
  • 8.1 Bias-variance tradeoff
    • 8.1.1 A mathematical decomposition (for regression)
  • 8.2 The double descent phenomenon
  • 8.3 Sample complexity bounds (optional readings)
• CS229 Spring - 8.3.1 Preliminaries - 8.3.2 The case of finite H - 8.3.3 The case of infinite H
• 9 Regularization and model selection
  • 9.1 Regularization
  • 9.2 Implicit regularization effect (optional reading)
  • 9.3 Model selection via cross validation
  • 9.4 Bayesian statistics and regularization
• IV Unsupervised learning
• 10 Clustering and the k-means algorithm
• 11 EM algorithms
  • 11.1 EM for mixture of Gaussians
  • 11.2 Jensen’s inequality
  • 11.3 General EM algorithms
    • 11.3.1 Other interpretation of ELBO
  • 11.4 Mixture of Gaussians revisited
    • reading) 11.5 Variational inference and variational auto-encoder (optional
• 12 Principal components analysis
• 13 Independent components analysis
  • 13.1 ICA ambiguities
  • 13.2 Densities and linear transformations
  • 13.3 ICA algorithm
• 14 Self-supervised learning and foundation models
  • 14.1 Pretraining and adaptation
  • 14.2 Pretraining methods in computer vision
  • 14.3 Pretrained large language models
    • 14.3.1 Open up the blackbox of Transformers
    • 14.3.2 Zero-shot learning and in-context learning
• CS229 Spring
• V Reinforcement Learning and Control
• 15 Reinforcement learning
  • 15.1 Markov decision processes
  • 15.2 Value iteration and policy iteration
  • 15.3 Learning a model for an MDP
  • 15.4 Continuous state MDPs
    • 15.4.1 Discretization
    • 15.4.2 Value function approximation
  • 15.5 Connections between Policy and Value Iteration (Optional)
• 16 LQR, DDP and LQG
  • 16.1 Finite-horizon MDPs
  • 16.2 Linear Quadratic Regulation (LQR)
  • 16.3 From non-linear dynamics to LQR
    • 16.3.1 Linearization of dynamics
    • 16.3.2 Differential Dynamic Programming (DDP)
  • 16.4 Linear Quadratic Gaussian (LQG)
• 17 Policy Gradient (REINFORCE)

Part I

Supervised learning

Let’s start by talking about a few examples of supervised learning prob- lems. Suppose we have a dataset giving the living areas and prices of 47 houses from Portland, Oregon:

Living area (feet^2 ) Price (1000$s) 2104 400 1600 330 2400 369 1416 232 3000 540 .. .

..

.

We can plot this data:

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

100

200

300

400

500

600

700

800

900

1000

housing prices

square feet

price (in $1000)

Given data like this, how can we learn to predict the prices of other houses in Portland, as a function of the size of their living areas? To establish notation for future use, we’ll use x(i)^ to denote the “input” variables (living area in this example), also called input features, and y(i) to denote the “output” or target variable that we are trying to predict (price). A pair (x(i), y(i)) is called a training example, and the dataset that we’ll be using to learn—a list of n training examples {(x(i), y(i)); i = 1 ,... , n}—is called a training set. Note that the superscript “(i)” in the notation is simply an index into the training set, and has nothing to do with exponentiation. We will also use X denote the space of input values, and Y the space of output values. In this example, X = Y = R. To describe the supervised learning problem slightly more formally, our goal is, given a training set, to learn a function h : X 7 → Y so that h(x) is a “good” predictor for the corresponding value of y. For historical reasons, this

function h is called a hypothesis. Seen pictorially, the process is therefore like this:

Training set

house.)

(living area of

Learning algorithm

x (^) h predicted y(predicted price) of house)

When the target variable that we’re trying to predict is continuous, such as in our housing example, we call the learning problem a regression prob- lem. When y can take on only a small number of discrete values (such as if, given the living area, we wanted to predict if a dwelling is a house or an apartment, say), we call it a classification problem.

Chapter 1

Linear regression

To make our housing example more interesting, let’s consider a slightly richer dataset in which we also know the number of bedrooms in each house:

Living area (feet^2 ) #bedrooms Price (1000$s) 2104 3 400 1600 3 330 2400 3 369 1416 2 232 3000 4 540 .. .

..

.

..

.

Here, the x’s are two-dimensional vectors in R^2. For instance, x (i) 1 is the living area of the i-th house in the training set, and x (i) 2 is its number of bedrooms. (In general, when designing a learning problem, it will be up to you to decide what features to choose, so if you are out in Portland gathering housing data, you might also decide to include other features such as whether each house has a fireplace, the number of bathrooms, and so on. We’ll say more about feature selection later, but for now let’s take the features as given.) To perform supervised learning, we must decide how we’re going to rep- resent functions/hypotheses h in a computer. As an initial choice, let’s say we decide to approximate y as a linear function of x:

hθ(x) = θ 0 + θ 1 x 1 + θ 2 x 2

Here, the θi’s are the parameters (also called weights) parameterizing the space of linear functions mapping from X to Y. When there is no risk of

confusion, we will drop the θ subscript in hθ(x), and write it more simply as h(x). To simplify our notation, we also introduce the convention of letting x 0 = 1 (this is the intercept term), so that

h(x) =

∑^ d

i=

θixi = θT^ x,

where on the right-hand side above we are viewing θ and x both as vectors, and here d is the number of input variables (not counting x 0 ). Now, given a training set, how do we pick, or learn, the parameters θ? One reasonable method seems to be to make h(x) close to y, at least for the training examples we have. To formalize this, we will define a function that measures, for each value of the θ’s, how close the h(x(i))’s are to the corresponding y(i)’s. We define the cost function:

J(θ) =

∑^ n

i=

(hθ(x(i)) − y(i))^2.

If you’ve seen linear regression before, you may recognize this as the familiar least-squares cost function that gives rise to the ordinary least squares regression model. Whether or not you have seen it previously, let’s keep going, and we’ll eventually show this to be a special case of a much broader family of algorithms.

1.1 LMS algorithm

We want to choose θ so as to minimize J(θ). To do so, let’s use a search algorithm that starts with some “initial guess” for θ, and that repeatedly changes θ to make J(θ) smaller, until hopefully we converge to a value of θ that minimizes J(θ). Specifically, let’s consider the gradient descent algorithm, which starts with some initial θ, and repeatedly performs the update:

θj := θj − α

∂

∂θj

J(θ).

(This update is simultaneously performed for all values of j = 0,... , d.) Here, α is called the learning rate. This is a very natural algorithm that repeatedly takes a step in the direction of steepest decrease of J. In order to implement this algorithm, we have to work out what is the partial derivative term on the right hand side. Let’s first work it out for the

case of if we have only one training example (x, y), so that we can neglect the sum in the definition of J. We have:

∂ ∂θj

J(θ) =

∂

∂θj

(hθ(x) − y)^2

= 2 ·

(hθ(x) − y) ·

∂

∂θj

(hθ(x) − y)

= (hθ(x) − y) ·

∂

∂θj

( (^) d ∑

i=

θixi − y

)

= (hθ(x) − y) xj

For a single training example, this gives the update rule:^1

θj := θj + α

(

y(i)^ − hθ(x(i))

)

x( ji ).

The rule is called the LMS update rule (LMS stands for “least mean squares”), and is also known as the Widrow-Hoff learning rule. This rule has several properties that seem natural and intuitive. For instance, the magnitude of the update is proportional to the error term (y(i)^ − hθ(x(i))); thus, for in- stance, if we are encountering a training example on which our prediction nearly matches the actual value of y(i), then we find that there is little need to change the parameters; in contrast, a larger change to the parameters will be made if our prediction hθ(x(i)) has a large error (i.e., if it is very far from y(i)). We’d derived the LMS rule for when there was only a single training example. There are two ways to modify this method for a training set of more than one example. The first is replace it with the following algorithm:

Repeat until convergence {

θj := θj + α

∑^ n

i=

(

y(i)^ − hθ(x(i))

)

x( ji ), (for every j) (1.1)

}

(^1) We use the notation “a := b” to denote an operation (in a computer program) in

which we set the value of a variable a to be equal to the value of b. In other words, this operation overwrites a with the value of b. In contrast, we will write “a = b” when we are asserting a statement of fact, that the value of a is equal to the value of b.

By grouping the updates of the coordinates into an update of the vector θ, we can rewrite update (1.1) in a slightly more succinct way:

θ := θ + α

∑^ n

i=

(

y(i)^ − hθ(x(i))

)

x(i)

The reader can easily verify that the quantity in the summation in the update rule above is just ∂J(θ)/∂θj (for the original definition of J). So, this is simply gradient descent on the original cost function J. This method looks at every example in the entire training set on every step, and is called batch gradient descent. Note that, while gradient descent can be susceptible to local minima in general, the optimization problem we have posed here for linear regression has only one global, and no other local, optima; thus gradient descent always converges (assuming the learning rate α is not too large) to the global minimum. Indeed, J is a convex quadratic function. Here is an example of gradient descent as it is run to minimize a quadratic function.

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

The ellipses shown above are the contours of a quadratic function. Also shown is the trajectory taken by gradient descent, which was initialized at (48,30). The x’s in the figure (joined by straight lines) mark the successive values of θ that gradient descent went through. When we run batch gradient descent to fit θ on our previous dataset, to learn to predict housing price as a function of living area, we obtain θ 0 = 71.27, θ 1 = 0.1345. If we plot hθ(x) as a function of x (area), along with the training data, we obtain the following figure:

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

100

200

300

400

500

600

700

800

900

1000

housing prices

square feet

price (in $1000)

If the number of bedrooms were included as one of the input features as well, we get θ 0 = 89. 60 , θ 1 = 0.1392, θ 2 = − 8 .738. The above results were obtained with batch gradient descent. There is an alternative to batch gradient descent that also works very well. Consider the following algorithm:

Loop {

for i = 1 to n, {

θj := θj + α

(

y(i)^ − hθ(x(i))

)

x( ji ), (for every j) (1.2)

}

}

By grouping the updates of the coordinates into an update of the vector θ, we can rewrite update (1.2) in a slightly more succinct way:

θ := θ + α

(

y(i)^ − hθ(x(i))

)

x(i)

In this algorithm, we repeatedly run through the training set, and each time we encounter a training example, we update the parameters according to the gradient of the error with respect to that single training example only. This algorithm is called stochastic gradient descent (also incremental gradient descent). Whereas batch gradient descent has to scan through the entire training set before taking a single step—a costly operation if n is large—stochastic gradient descent can start making progress right away, and

continues to make progress with each example it looks at. Often, stochastic gradient descent gets θ “close” to the minimum much faster than batch gra- dient descent. (Note however that it may never “converge” to the minimum, and the parameters θ will keep oscillating around the minimum of J(θ); but in practice most of the values near the minimum will be reasonably good approximations to the true minimum.^2 ) For these reasons, particularly when the training set is large, stochastic gradient descent is often preferred over batch gradient descent.

1.2 The normal equations

Gradient descent gives one way of minimizing J. Let’s discuss a second way of doing so, this time performing the minimization explicitly and without resorting to an iterative algorithm. In this method, we will minimize J by explicitly taking its derivatives with respect to the θj ’s, and setting them to zero. To enable us to do this without having to write reams of algebra and pages full of matrices of derivatives, let’s introduce some notation for doing calculus with matrices.

1.2.1 Matrix derivatives

For a function f : Rn×d^7 → R mapping from n-by-d matrices to the real numbers, we define the derivative of f with respect to A to be:

∇Af (A) =







∂f ∂A 11 · · ·^

∂f ∂A 1 d .. .

... ..

.

∂f ∂An 1 · · ·^

∂f ∂And







Thus, the gradient ∇Af (A) is itself an n-by-d matrix, whose (i, j)-element is

∂f /∂Aij. For example, suppose A =

[

A 11 A 12 A 21 A 22

]

is a 2-by-2 matrix, and

the function f : R^2 ×^2 7 → R is given by

f (A) =

A 11 + 5A^212 + A 21 A 22.

(^2) By slowly letting the learning rate α decrease to zero as the algorithm runs, it is also

possible to ensure that the parameters will converge to the global minimum rather than merely oscillate around the minimum.

Here, Aij denotes the (i, j) entry of the matrix A. We then have

∇Af (A) =

[ 3

2 10 A^12

A 22 A 21

]

.

1.2.2 Least squares revisited

Armed with the tools of matrix derivatives, let us now proceed to find in closed-form the value of θ that minimizes J(θ). We begin by re-writing J in matrix-vectorial notation. Given a training set, define the design matrix X to be the n-by-d matrix (actually n-by-d + 1, if we include the intercept term) that contains the training examples’ input values in its rows:

X =











— (x(1))T^ — — (x(2))T^ — .. . — (x(n))T^ —











.

Also, let ~y be the n-dimensional vector containing all the target values from the training set:

~y =











y(1) y(2) .. . y(n)











.

Now, since hθ(x(i)) = (x(i))T^ θ, we can easily verify that

Xθ − ~y =







(x(1))T^ θ .. . (x(n))T^ θ





 −







y(1) .. . y(n)







=







hθ(x(1)) − y(1) .. . hθ(x(n)) − y(n)





.

Thus, using the fact that for a vector z, we have that zT^ z =

∑

i z

2 i :

1 2

(Xθ − ~y)T^ (Xθ − ~y) =

∑^ n

i=

(hθ(x(i)) − y(i))^2

= J(θ)

Finally, to minimize J, let’s find its derivatives with respect to θ. Hence,

∇θJ(θ) = ∇θ

(Xθ − ~y)T^ (Xθ − ~y)

=

∇θ

(

(Xθ)T^ Xθ − (Xθ)T^ ~y − ~yT^ (Xθ) + ~yT^ ~y

)

=

∇θ

(

θT^ (XT^ X)θ − ~yT^ (Xθ) − ~yT^ (Xθ)

)

=

∇θ

(

θT^ (XT^ X)θ − 2(XT^ ~y)T^ θ

)

=

(

2 XT^ Xθ − 2 XT^ ~y

)

= XT^ Xθ − XT^ ~y

In the third step, we used the fact that aT^ b = bT^ a, and in the fifth step used the facts ∇xbT^ x = b and ∇xxT^ Ax = 2Ax for symmetric matrix A (for more details, see Section 4.3 of “Linear Algebra Review and Reference”). To minimize J, we set its derivatives to zero, and obtain the normal equations:

XT^ Xθ = XT^ ~y

Thus, the value of θ that minimizes J(θ) is given in closed form by the equation θ = (XT^ X)−^1 XT^ ~y.^3

1.3 Probabilistic interpretation

When faced with a regression problem, why might linear regression, and specifically why might the least-squares cost function J, be a reasonable choice? In this section, we will give a set of probabilistic assumptions, under which least-squares regression is derived as a very natural algorithm. Let us assume that the target variables and the inputs are related via the equation y(i)^ = θT^ x(i)^ + (i), (^3) Note that in the above step, we are implicitly assuming that XT (^) X is an invertible

matrix. This can be checked before calculating the inverse. If either the number of linearly independent examples is fewer than the number of features, or if the features are not linearly independent, then XT^ X will not be invertible. Even in such cases, it is possible to “fix” the situation with additional techniques, which we skip here for the sake of simplicty.

where (i)^ is an error term that captures either unmodeled effects (such as if there are some features very pertinent to predicting housing price, but that we’d left out of the regression), or random noise. Let us further assume that the (i)^ are distributed IID (independently and identically distributed) according to a Gaussian distribution (also called a Normal distribution) with mean zero and some variance σ^2. We can write this assumption as “(i)^ ∼ N (0, σ^2 ).” I.e., the density of (i)^ is given by

p((i)) =

√

2 πσ

exp

(

−

((i))^2 2 σ^2

)

.

This implies that

p(y(i)|x(i); θ) =

√

2 πσ

exp

(

−

(y(i)^ − θT^ x(i))^2 2 σ^2

)

.

The notation “p(y(i)|x(i); θ)” indicates that this is the distribution of y(i) given x(i)^ and parameterized by θ. Note that we should not condition on θ (“p(y(i)|x(i), θ)”), since θ is not a random variable. We can also write the distribution of y(i)^ as y(i)^ | x(i); θ ∼ N (θT^ x(i), σ^2 ). Given X (the design matrix, which contains all the x(i)’s) and θ, what is the distribution of the y(i)’s? The probability of the data is given by p(~y|X; θ). This quantity is typically viewed a function of ~y (and perhaps X), for a fixed value of θ. When we wish to explicitly view this as a function of θ, we will instead call it the likelihood function:

L(θ) = L(θ; X, ~y) = p(~y|X; θ).

Note that by the independence assumption on the (i)’s (and hence also the y(i)’s given the x(i)’s), this can also be written

L(θ) =

∏^ n

i=

p(y(i)^ | x(i); θ)

=

∏^ n

i=

√

2 πσ

exp

(

−

(y(i)^ − θT^ x(i))^2 2 σ^2

)

.

Now, given this probabilistic model relating the y(i)’s and the x(i)’s, what is a reasonable way of choosing our best guess of the parameters θ? The principal of maximum likelihood says that we should choose θ so as to make the data as high probability as possible. I.e., we should choose θ to maximize L(θ).

Instead of maximizing L(θ), we can also maximize any strictly increasing function of L(θ). In particular, the derivations will be a bit simpler if we instead maximize the log likelihood `(θ):

`(θ) = log L(θ)

= log

∏^ n

i=

√

2 πσ

exp

(

−

(y(i)^ − θT^ x(i))^2 2 σ^2

)

=

∑^ n

i=

log

√

2 πσ

exp

(

−

(y(i)^ − θT^ x(i))^2 2 σ^2

)

= n log

√

2 πσ

−

σ^2

·

∑^ n

i=

(y(i)^ − θT^ x(i))^2.

Hence, maximizing `(θ) gives the same answer as minimizing

1 2

∑^ n

i=

(y(i)^ − θT^ x(i))^2 ,

which we recognize to be J(θ), our original least-squares cost function. To summarize: Under the previous probabilistic assumptions on the data, least-squares regression corresponds to finding the maximum likelihood esti- mate of θ. This is thus one set of assumptions under which least-squares re- gression can be justified as a very natural method that’s just doing maximum likelihood estimation. (Note however that the probabilistic assumptions are by no means necessary for least-squares to be a perfectly good and rational procedure, and there may—and indeed there are—other natural assumptions that can also be used to justify it.) Note also that, in our previous discussion, our final choice of θ did not depend on what was σ^2 , and indeed we’d have arrived at the same result even if σ^2 were unknown. We will use this fact again later, when we talk about the exponential family and generalized linear models.

1.4 Locally weighted linear regression (optional

reading)

Consider the problem of predicting y from x ∈ R. The leftmost figure below shows the result of fitting a y = θ 0 + θ 1 x to a dataset. We see that the data doesn’t really lie on straight line, and so the fit is not very good.

(^00 1 2 3 4 5 6 )

1

2

3

4

x

y

(^00 1 2 3 4 5 6 )

1

2

3

4

x

y

(^00 1 2 3 4 5 6 )

1

2

3

4

x

y

Instead, if we had added an extra feature x^2 , and fit y = θ 0 + θ 1 x + θ 2 x^2 , then we obtain a slightly better fit to the data. (See middle figure) Naively, it might seem that the more features we add, the better. However, there is also a danger in adding too many features: The rightmost figure is the result of fitting a 5-th order polynomial y =

∑ 5

j=0 θj^ x

j (^). We see that even though the

fitted curve passes through the data perfectly, we would not expect this to be a very good predictor of, say, housing prices (y) for different living areas (x). Without formally defining what these terms mean, we’ll say the figure on the left shows an instance of underfitting—in which the data clearly shows structure not captured by the model—and the figure on the right is an example of overfitting. (Later in this class, when we talk about learning theory we’ll formalize some of these notions, and also define more carefully just what it means for a hypothesis to be good or bad.) As discussed previously, and as shown in the example above, the choice of features is important to ensuring good performance of a learning algorithm. (When we talk about model selection, we’ll also see algorithms for automat- ically choosing a good set of features.) In this section, let us briefly talk about the locally weighted linear regression (LWR) algorithm which, assum- ing there is sufficient training data, makes the choice of features less critical. This treatment will be brief, since you’ll get a chance to explore some of the properties of the LWR algorithm yourself in the homework. In the original linear regression algorithm, to make a prediction at a query point x (i.e., to evaluate h(x)), we would:

1. Fit θ to minimize

∑

i(y (i) (^) − θT (^) x(i)) (^2).

2. Output θT^ x.

In contrast, the locally weighted linear regression algorithm does the fol- lowing:

1. Fit θ to minimize

∑

i w (i)(y(i) (^) − θT (^) x(i)) (^2).

2. Output θT^ x.

Here, the w(i)’s are non-negative valued weights. Intuitively, if w(i)^ is large for a particular value of i, then in picking θ, we’ll try hard to make (y(i)^ − θT^ x(i))^2 small. If w(i)^ is small, then the (y(i)^ − θT^ x(i))^2 error term will be pretty much ignored in the fit. A fairly standard choice for the weights is^4

w(i)^ = exp

(

−

(x(i)^ − x)^2 2 τ 2

)

Note that the weights depend on the particular point x at which we’re trying to evaluate x. Moreover, if |x(i)^ − x| is small, then w(i)^ is close to 1; and if |x(i)^ − x| is large, then w(i)^ is small. Hence, θ is chosen giving a much higher “weight” to the (errors on) training examples close to the query point x. (Note also that while the formula for the weights takes a form that is cosmetically similar to the density of a Gaussian distribution, the w(i)’s do not directly have anything to do with Gaussians, and in particular the w(i) are not random variables, normally distributed or otherwise.) The parameter τ controls how quickly the weight of a training example falls off with distance of its x(i)^ from the query point x; τ is called the bandwidth parameter, and is also something that you’ll get to experiment with in your homework. Locally weighted linear regression is the first example we’re seeing of a non-parametric algorithm. The (unweighted) linear regression algorithm that we saw earlier is known as a parametric learning algorithm, because it has a fixed, finite number of parameters (the θi’s), which are fit to the data. Once we’ve fit the θi’s and stored them away, we no longer need to keep the training data around to make future predictions. In contrast, to make predictions using locally weighted linear regression, we need to keep the entire training set around. The term “non-parametric” (roughly) refers to the fact that the amount of stuff we need to keep in order to represent the hypothesis h grows linearly with the size of the training set.

(^4) If x is vector-valued, this is generalized to be w(i) (^) = exp(−(x(i) (^) − x)T (^) (x(i) (^) − x)/(2τ 2 )),

or w(i)^ = exp(−(x(i)^ − x)T^ Σ−^1 (x(i)^ − x)/(2τ 2 )), for an appropriate choice of τ or Σ.

Chapter 2

Classification and logistic

regression

Let’s now talk about the classification problem. This is just like the regression problem, except that the values y we now want to predict take on only a small number of discrete values. For now, we will focus on the binary classification problem in which y can take on only two values, 0 and 1. (Most of what we say here will also generalize to the multiple-class case.) For instance, if we are trying to build a spam classifier for email, then x(i) may be some features of a piece of email, and y may be 1 if it is a piece of spam mail, and 0 otherwise. 0 is also called the negative class, and 1 the positive class, and they are sometimes also denoted by the symbols “-” and “+.” Given x(i), the corresponding y(i)^ is also called the label for the training example.

2.1 Logistic regression

We could approach the classification problem ignoring the fact that y is discrete-valued, and use our old linear regression algorithm to try to predict y given x. However, it is easy to construct examples where this method performs very poorly. Intuitively, it also doesn’t make sense for hθ(x) to take values larger than 1 or smaller than 0 when we know that y ∈ { 0 , 1 }. To fix this, let’s change the form for our hypotheses hθ(x). We will choose

hθ(x) = g(θT^ x) =

1 + e−θT^ x^

,

where

g(z) =

1 + e−z

20

is called the logistic function or the sigmoid function. Here is a plot showing g(z):

−5^0 −4 −3 −2 −1 0 1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z

g(z)

Notice that g(z) tends towards 1 as z → ∞, and g(z) tends towards 0 as z → −∞. Moreover, g(z), and hence also h(x), is always bounded between 0 and 1. As before, we are keeping the convention of letting x 0 = 1, so that θT^ x = θ 0 +

∑d j=1 θj^ xj^. For now, let’s take the choice of g as given. Other functions that smoothly increase from 0 to 1 can also be used, but for a couple of reasons that we’ll see later (when we talk about GLMs, and when we talk about generative learning algorithms), the choice of the logistic function is a fairly natural one. Before moving on, here’s a useful property of the derivative of the sigmoid function, which we write as g′:

g′(z) =

d dz

1 + e−z

(1 + e−z^ )^2

(

e−z^

)

=

(1 + e−z^ )

·

(

1 −

(1 + e−z^ )

)

= g(z)(1 − g(z)).

So, given the logistic regression model, how do we fit θ for it? Following how we saw least squares regression could be derived as the maximum like- lihood estimator under a set of assumptions, let’s endow our classification model with a set of probabilistic assumptions, and then fit the parameters via maximum likelihood.

Let us assume that

P (y = 1 | x; θ) = hθ(x) P (y = 0 | x; θ) = 1 − hθ(x)

Note that this can be written more compactly as

p(y | x; θ) = (hθ(x))y^ (1 − hθ(x))^1 −y

Assuming that the n training examples were generated independently, we can then write down the likelihood of the parameters as

L(θ) = p(~y | X; θ)

=

∏^ n

i=1

p(y(i)^ | x(i); θ)

=

∏^ n

i=1

(

hθ(x(i))

)y(i)^ ( 1 − hθ(x(i))

) 1 −y(i)

As before, it will be easier to maximize the log likelihood:

`(θ) = log L(θ) =

∑^ n

i=1

y(i)^ log h(x(i)) + (1 − y(i)) log(1 − h(x(i))) (2.1)

How do we maximize the likelihood? Similar to our derivation in the case of linear regression, we can use gradient ascent. Written in vectorial notation, our updates will therefore be given by θ := θ + α∇θ`(θ). (Note the positive rather than negative sign in the update formula, since we’re maximizing, rather than minimizing, a function now.) Let’s start by working with just one training example (x, y), and take derivatives to derive the stochastic gradient ascent rule:

∂ ∂θj

`(θ) =

(

y

g(θT^ x)

− (1 − y)

1 − g(θT^ x)

)

∂

∂θj

g(θT^ x)

=

(

y

g(θT^ x)

− (1 − y)

1 − g(θT^ x)

)

g(θT^ x)(1 − g(θT^ x))

∂

∂θj

θT^ x

=

(

y(1 − g(θT^ x)) − (1 − y)g(θT^ x)

)

xj = (y − hθ(x)) xj (2.2)

Above, we used the fact that g′(z) = g(z)(1 − g(z)). This therefore gives us the stochastic gradient ascent rule

θj := θj + α

(

y(i)^ − hθ(x(i))

)

x( ji)

If we compare this to the LMS update rule, we see that it looks identical; but this is not the same algorithm, because hθ(x(i)) is now defined as a non-linear function of θT^ x(i). Nonetheless, it’s a little surprising that we end up with the same update rule for a rather different algorithm and learning problem. Is this coincidence, or is there a deeper reason behind this? We’ll answer this when we get to GLM models.

Remark 2.1.1: An alternative notational viewpoint of the same loss func- tion is also useful, especially for Section 7.1 where we study nonlinear models. Let `logistic : R × { 0 , 1 } → R≥ 0 be the logistic loss defined as

`logistic(t, y) , y log(1 + exp(−t)) + (1 − y) log(1 + exp(t)). (2.3)

One can verify by plugging in hθ(x) = 1/(1 + e−θ

x ) that the negative log- likelihood (the negation of `(θ) in equation (2.1)) can be re-written as

−(θ) =logistic(θ>x, y). (2.4)

Oftentimes θ>x or t is called the logit. Basic calculus gives us that

∂`logistic(t, y) ∂t

= y

− exp(−t) 1 + exp(−t)

• (1 − y)

1 + exp(−t)

(2.5)

= 1/(1 + exp(−t)) − y. (2.6)

Then, using the chain rule, we have that

∂ ∂θj

`(θ) = −

∂`logistic(t, y) ∂t

·

∂t ∂θj

(2.7)

= (y − 1 /(1 + exp(−t))) · xj = (y − hθ(x))xj , (2.8)

which is consistent with the derivation in equation (2.2). We will see this viewpoint can be extended nonlinear models in Section 7.1.

2.2 Digression: the perceptron learning algo-

rithm

We now digress to talk briefly about an algorithm that’s of some historical interest, and that we will also return to later when we talk about learning

theory. Consider modifying the logistic regression method to “force” it to output values that are either 0 or 1 or exactly. To do so, it seems natural to change the definition of g to be the threshold function:

g(z) =

{

1 if z ≥ 0 0 if z < 0

If we then let hθ(x) = g(θT^ x) as before but using this modified definition of g, and if we use the update rule

θj := θj + α

(

y(i)^ − hθ(x(i))

)

x( ji ).

then we have the perceptron learning algorithn. In the 1960s, this “perceptron” was argued to be a rough model for how individual neurons in the brain work. Given how simple the algorithm is, it will also provide a starting point for our analysis when we talk about learning theory later in this class. Note however that even though the perceptron may be cosmetically similar to the other algorithms we talked about, it is actually a very different type of algorithm than logistic regression and least squares linear regression; in particular, it is difficult to endow the perceptron’s predic- tions with meaningful probabilistic interpretations, or derive the perceptron as a maximum likelihood estimation algorithm.

2.3 Multi-class classification

Consider a classification problem in which the response variable y can take on any one of k values, so y ∈ { 1 , 2 ,... , k}. For example, rather than classifying emails into the two classes spam or not-spam—which would have been a binary classification problem—we might want to classify them into three classes, such as spam, personal mails, and work-related mails. The label / response variable is still discrete, but can now take on more than two values. We will thus model it as distributed according to a multinomial distribution. In this case, p(y | x; θ) is a distribution over k possible discrete outcomes and is thus a multinomial distribution. Recall that a multinomial distribu- tion involves k numbers φ 1 ,... , φk specifying the probability of each of the outcomes. Note that these numbers must satisfy

∑k i=1 φi^ = 1. We will de- sign a parameterized model that outputs φ 1 ,... , φk satisfying this constraint given the input x. We introduce k groups of parameters θ 1 ,... , θk, each of them being a vector in Rd. Intuitively, we would like to use θ> 1 x,... , θ k> x to represent