CS 229 - Machine Learning

Supervised Learning cheatsheet Star

Introduction to Supervised Learning

Given a set of data points $\{x^{(1)}, ..., x^{(m)}\}$ associated to a set of outcomes $\{y^{(1)}, ..., y^{(m)}\}$, we want to build a classifier that learns how to predict $y$ from $x$.

Type of prediction The different types of predictive models are summed up in the table below:

 Regression Classification Outcome Continuous Class Examples Linear regression Logistic regression, SVM, Naive Bayes

Type of model The different models are summed up in the table below:

 Discriminative model Generative model Goal Directly estimate $P(y|x)$ Estimate $P(x|y)$ to then deduce $P(y|x)$ What's learned Decision boundary Probability distributions of the data Illustration  Examples Regressions, SVMs GDA, Naive Bayes

Notations and general concepts

Hypothesis The hypothesis is noted $h_\theta$ and is the model that we choose. For a given input data $x^{(i)}$ the model prediction output is $h_\theta(x^{(i)})$.

Loss function A loss function is a function $L:(z,y)\in\mathbb{R}\times Y\longmapsto L(z,y)\in\mathbb{R}$ that takes as inputs the predicted value $z$ corresponding to the real data value $y$ and outputs how different they are. The common loss functions are summed up in the table below:

 Least squared error Logistic loss Hinge loss Cross-entropy $\displaystyle\frac{1}{2}(y-z)^2$ $\displaystyle\log(1+\exp(-yz))$ $\displaystyle\max(0,1-yz)$ $\displaystyle-\Big[y\log(z)+(1-y)\log(1-z)\Big]$    Linear regression Logistic regression SVM Neural Network

Cost function The cost function $J$ is commonly used to assess the performance of a model, and is defined with the loss function $L$ as follows:

$\boxed{J(\theta)=\sum_{i=1}^mL(h_\theta(x^{(i)}), y^{(i)})}$

Gradient descent By noting $\alpha\in\mathbb{R}$ the learning rate, the update rule for gradient descent is expressed with the learning rate and the cost function $J$ as follows:

$\boxed{\theta\longleftarrow\theta-\alpha\nabla J(\theta)}$ Remark: Stochastic gradient descent (SGD) is updating the parameter based on each training example, and batch gradient descent is on a batch of training examples.

Likelihood The likelihood of a model $L(\theta)$ given parameters $\theta$ is used to find the optimal parameters $\theta$ through likelihood maximization. We have:

$\boxed{\theta^{\textrm{opt}}=\underset{\theta}{\textrm{arg max }}L(\theta)}$

Remark: in practice, we use the log-likelihood $\ell(\theta)=\log(L(\theta))$ which is easier to optimize.

Newton's algorithm Newton's algorithm is a numerical method that finds $\theta$ such that $\ell'(\theta)=0$. Its update rule is as follows:

$\boxed{\theta\leftarrow\theta-\frac{\ell'(\theta)}{\ell''(\theta)}}$

Remark: the multidimensional generalization, also known as the Newton-Raphson method, has the following update rule:

$\theta\leftarrow\theta-\left(\nabla_\theta^2\ell(\theta)\right)^{-1}\nabla_\theta\ell(\theta)$

Linear models

Linear regression

We assume here that $y|x;\theta\sim\mathcal{N}(\mu,\sigma^2)$

Normal equations By noting $X$ the design matrix, the value of $\theta$ that minimizes the cost function is a closed-form solution such that:

$\boxed{\theta=(X^TX)^{-1}X^Ty}$

LMS algorithm By noting $\alpha$ the learning rate, the update rule of the Least Mean Squares (LMS) algorithm for a training set of $m$ data points, which is also known as the Widrow-Hoff learning rule, is as follows:

$\boxed{\forall j,\quad \theta_j \leftarrow \theta_j+\alpha\sum_{i=1}^m\left[y^{(i)}-h_\theta(x^{(i)})\right]x_j^{(i)}}$

Remark: the update rule is a particular case of the gradient ascent.

LWR Locally Weighted Regression, also known as LWR, is a variant of linear regression that weights each training example in its cost function by $w^{(i)}(x)$, which is defined with parameter $\tau\in\mathbb{R}$ as:

$\boxed{w^{(i)}(x)=\exp\left(-\frac{(x^{(i)}-x)^2}{2\tau^2}\right)}$

Classification and logistic regression

Sigmoid function The sigmoid function $g$, also known as the logistic function, is defined as follows:

$\forall z\in\mathbb{R},\quad\boxed{g(z)=\frac{1}{1+e^{-z}}\in]0,1[}$

Logistic regression We assume here that $y|x;\theta\sim\textrm{Bernoulli}(\phi)$. We have the following form:

$\boxed{\phi=p(y=1|x;\theta)=\frac{1}{1+\exp(-\theta^Tx)}=g(\theta^Tx)}$

Remark: logistic regressions do not have closed form solutions.

Softmax regression A softmax regression, also called a multiclass logistic regression, is used to generalize logistic regression when there are more than 2 outcome classes. By convention, we set $\theta_K=0$, which makes the Bernoulli parameter $\phi_i$ of each class $i$ be such that:

$\boxed{\displaystyle\phi_i=\frac{\exp(\theta_i^Tx)}{\displaystyle\sum_{j=1}^K\exp(\theta_j^Tx)}}$

Generalized Linear Models

Exponential family A class of distributions is said to be in the exponential family if it can be written in terms of a natural parameter, also called the canonical parameter or link function, $\eta$, a sufficient statistic $T(y)$ and a log-partition function $a(\eta)$ as follows:

$\boxed{p(y;\eta)=b(y)\exp(\eta T(y)-a(\eta))}$

Remark: we will often have $T(y)=y$. Also, $\exp(-a(\eta))$ can be seen as a normalization parameter that will make sure that the probabilities sum to one.

The most common exponential distributions are summed up in the following table:

 Distribution $\eta$ $T(y)$ $a(\eta)$ $b(y)$ Bernoulli $\log\left(\frac{\phi}{1-\phi}\right)$ $y$ $\log(1+\exp(\eta))$ $1$ Gaussian $\mu$ $y$ $\frac{\eta^2}{2}$ $\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{y^2}{2}\right)$ Poisson $\log(\lambda)$ $y$ $e^{\eta}$ $\displaystyle\frac{1}{y!}$ Geometric $\log(1-\phi)$ $y$ $\log\left(\frac{e^\eta}{1-e^\eta}\right)$ $1$

Assumptions of GLMs Generalized Linear Models (GLM) aim at predicting a random variable $y$ as a function of $x\in\mathbb{R}^{n+1}$ and rely on the following 3 assumptions:

$(1)\quad\boxed{y|x;\theta\sim\textrm{ExpFamily}(\eta)}$
$(2)\quad\boxed{h_\theta(x)=E[y|x;\theta]}$
$(3)\quad\boxed{\eta=\theta^Tx}$

Remark: ordinary least squares and logistic regression are special cases of generalized linear models.

Support Vector Machines

The goal of support vector machines is to find the line that maximizes the minimum distance to the line.

Optimal margin classifier The optimal margin classifier $h$ is such that:

$\boxed{h(x)=\textrm{sign}(w^Tx-b)}$

where $(w, b)\in\mathbb{R}^n\times\mathbb{R}$ is the solution of the following optimization problem:

$\boxed{\min\frac{1}{2}||w||^2}\quad\quad\textrm{such that }\quad \boxed{y^{(i)}(w^Tx^{(i)}-b)\geqslant1}$ Remark: the decision boundary is defined as $\boxed{w^Tx-b=0}$.

Hinge loss The hinge loss is used in the setting of SVMs and is defined as follows:

$\boxed{L(z,y)=[1-yz]_+=\max(0,1-yz)}$

Kernel Given a feature mapping $\phi$, we define the kernel $K$ as follows:

$\boxed{K(x,z)=\phi(x)^T\phi(z)}$

In practice, the kernel $K$ defined by $K(x,z)=\exp\left(-\frac{||x-z||^2}{2\sigma^2}\right)$ is called the Gaussian kernel and is commonly used. Remark: we say that we use the "kernel trick" to compute the cost function using the kernel because we actually don't need to know the explicit mapping $\phi$, which is often very complicated. Instead, only the values $K(x,z)$ are needed.

Lagrangian We define the Lagrangian $\mathcal{L}(w,b)$ as follows:

$\boxed{\mathcal{L}(w,b)=f(w)+\sum_{i=1}^l\beta_ih_i(w)}$

Remark: the coefficients $\beta_i$ are called the Lagrange multipliers.

Generative Learning

A generative model first tries to learn how the data is generated by estimating $P(x|y)$, which we can then use to estimate $P(y|x)$ by using Bayes' rule.

Gaussian Discriminant Analysis

Setting The Gaussian Discriminant Analysis assumes that $y$ and $x|y=0$ and $x|y=1$ are such that:

$(1)\quad\boxed{y\sim\textrm{Bernoulli}(\phi)}$
$(2)\quad\boxed{x|y=0\sim\mathcal{N}(\mu_0,\Sigma)}$
$(3)\quad\boxed{x|y=1\sim\mathcal{N}(\mu_1,\Sigma)}$

Estimation The following table sums up the estimates that we find when maximizing the likelihood:

 $\widehat{\phi}$ $\widehat{\mu_j}\quad{\small(j=0,1)}$ $\widehat{\Sigma}$ $\displaystyle\frac{1}{m}\sum_{i=1}^m1_{\{y^{(i)}=1\}}$ $\displaystyle\frac{\sum_{i=1}^m1_{\{y^{(i)}=j\}}x^{(i)}}{\sum_{i=1}^m1_{\{y^{(i)}=j\}}}$ $\displaystyle\frac{1}{m}\sum_{i=1}^m(x^{(i)}-\mu_{y^{(i)}})(x^{(i)}-\mu_{y^{(i)}})^T$

Naive Bayes

Assumption The Naive Bayes model supposes that the features of each data point are all independent:

$\boxed{P(x|y)=P(x_1,x_2,...|y)=P(x_1|y)P(x_2|y)...=\prod_{i=1}^nP(x_i|y)}$

Solutions Maximizing the log-likelihood gives the following solutions:

$\boxed{P(y=k)=\frac{1}{m}\times\#\{j|y^{(j)}=k\}}\quad\textrm{ and }\quad\boxed{P(x_i=l|y=k)=\frac{\#\{j|y^{(j)}=k\textrm{ and }x_i^{(j)}=l\}}{\#\{j|y^{(j)}=k\}}}$
with $k\in\{0,1\}$ and $l\in[\![1,L]\!]$

Remark: Naive Bayes is widely used for text classification and spam detection.

Tree-based and ensemble methods

These methods can be used for both regression and classification problems.

CART Classification and Regression Trees (CART), commonly known as decision trees, can be represented as binary trees. They have the advantage to be very interpretable.

Random forest It is a tree-based technique that uses a high number of decision trees built out of randomly selected sets of features. Contrary to the simple decision tree, it is highly uninterpretable but its generally good performance makes it a popular algorithm.

Remark: random forests are a type of ensemble methods.

Boosting The idea of boosting methods is to combine several weak learners to form a stronger one. The main ones are summed up in the table below:

 Adaptive boosting Gradient boosting • High weights are put on errors to improve at the next boosting step • Known as Adaboost • Weak learners are trained on residuals • Examples include XGBoost

Other non-parametric approaches

$k$-nearest neighbors The $k$-nearest neighbors algorithm, commonly known as $k$-NN, is a non-parametric approach where the response of a data point is determined by the nature of its $k$ neighbors from the training set. It can be used in both classification and regression settings.

Remark: the higher the parameter $k$, the higher the bias, and the lower the parameter $k$, the higher the variance. Learning Theory

Union bound Let $A_1, ..., A_k$ be $k$ events. We have:

$\boxed{P(A_1\cup ...\cup A_k)\leqslant P(A_1)+...+P(A_k)}$ Hoeffding inequality Let $Z_1, .., Z_m$ be $m$ iid variables drawn from a Bernoulli distribution of parameter $\phi$. Let $\widehat{\phi}$ be their sample mean and $\gamma>0$ fixed. We have:

$\boxed{P(|\phi-\widehat{\phi}|>\gamma)\leqslant2\exp(-2\gamma^2m)}$

Remark: this inequality is also known as the Chernoff bound.

Training error For a given classifier $h$, we define the training error $\widehat{\epsilon}(h)$, also known as the empirical risk or empirical error, to be as follows:

$\boxed{\widehat{\epsilon}(h)=\frac{1}{m}\sum_{i=1}^m1_{\{h(x^{(i)})\neq y^{(i)}\}}}$

Probably Approximately Correct (PAC) PAC is a framework under which numerous results on learning theory were proved, and has the following set of assumptions:

• the training and testing sets follow the same distribution
• the training examples are drawn independently

Shattering Given a set $S=\{x^{(1)},...,x^{(d)}\}$, and a set of classifiers $\mathcal{H}$, we say that $\mathcal{H}$ shatters $S$ if for any set of labels $\{y^{(1)}, ..., y^{(d)}\}$, we have:

$\boxed{\exists h\in\mathcal{H}, \quad \forall i\in[\![1,d]\!],\quad h(x^{(i)})=y^{(i)}}$

Upper bound theorem Let $\mathcal{H}$ be a finite hypothesis class such that $|\mathcal{H}|=k$ and let $\delta$ and the sample size $m$ be fixed. Then, with probability of at least $1-\delta$, we have:

$\boxed{\epsilon(\widehat{h})\leqslant\left(\min_{h\in\mathcal{H}}\epsilon(h)\right)+2\sqrt{\frac{1}{2m}\log\left(\frac{2k}{\delta}\right)}}$

VC dimension The Vapnik-Chervonenkis (VC) dimension of a given infinite hypothesis class $\mathcal{H}$, noted $\textrm{VC}(\mathcal{H})$ is the size of the largest set that is shattered by $\mathcal{H}$.

Remark: the VC dimension of ${\small\mathcal{H}=\{\textrm{set of linear classifiers in 2 dimensions}\}}$ is 3. Theorem (Vapnik) Let $\mathcal{H}$ be given, with $\textrm{VC}(\mathcal{H})=d$ and $m$ the number of training examples. With probability at least $1-\delta$, we have:

$\boxed{\epsilon(\widehat{h})\leqslant \left(\min_{h\in\mathcal{H}}\epsilon(h)\right) + O\left(\sqrt{\frac{d}{m}\log\left(\frac{m}{d}\right)+\frac{1}{m}\log\left(\frac{1}{\delta}\right)}\right)}$