Lecture 4: Logistic Regression Machine Learning CUNY Graduate Center

Today Linear Regression Logistic Regression Bayesians v. Frequentists Bayesian Linear Regression Logistic Regression Linear Model for Classification

Regularization: Penalize large weights Introduce a penalty term in the loss function. Regularized Regression (L2-Regularization or Ridge Regression)

More regularization The penalty term defines the styles of regularization L2-Regularization L1-Regularization L0-Regularization L0-norm is the optimal subset of features

Curse of dimensionality Increasing dimensionality of features increases the data requirements exponentially. For example, if a single feature can be accurately approximated with 100 data points, to optimize the joint over two features requires 100*100 data points. Models should be small relative to the amount of available data Dimensionality reduction techniques – feature selection – can help. L0-regularization is explicit feature selection L1- and L2-regularizations approximate feature selection.

Bayesians v. Frequentists What is a probability? Frequentists A probability is the likelihood that an event will happen It is approximated by the ratio of the number of observed events to the number of total events Assessment is vital to selecting a model Point estimates are absolutely fine Bayesians A probability is a degree of believability of a proposition. Bayesians require that probabilities be prior beliefs conditioned on data. The Bayesian approach “is optimal”, given a good model, a good prior and a good loss function. Don’t worry so much about assessment. If you are ever making a point estimate, you’ve made a mistake. The only valid probabilities are posteriors based on evidence given some prior

Bayesian Linear Regression The previous MLE derivation of linear regression uses point estimates for the weight vector, w. Bayesians say, “hold it right there”. Use a prior distribution over w to estimate parameters Alpha is a hyperparameter over w, where alpha is the precision or inverse variance of the distribution. Now optimize:

Optimize the Bayesian posterior As usual it’s easier to optimize after a log transform.

Optimize the Bayesian posterior As usual it’s easier to optimize after a log transform.

Optimize the Bayesian posterior Ignoring terms that do not depend on w IDENTICAL formulation as L2-regularization

Context Overfitting is bad. Bayesians vs. Frequentists Is one better? Machine Learning uses techniques from both camps.

Logistic Regression Linear model applied to classification Supervised: target information is available Each data point xi has a corresponding target ti. Goal: Identify a function

Target Variables In binary classification, it is convenient to represent ti as a scalar with a range of [0,1] Interpretation of ti as the likelihood that xi is the member of the positive class Used to represent the confidence of a prediction. For L > 2 classes, ti is often represented as a K element vector. tij represents the degree of membership in class j. |ti| = 1 E.g. 5-way classification vector:

Graphical Example of Classification

Decision Boundaries

Graphical Example of Classification

Classification approaches Generative Models the joint distribution between c and x Highest data requirements Discriminative Fewer parameters to approximate Discriminant Function May still be trained probabilistically, but not necessarily modeling a likelihood.

Treating Classification as a Linear model

Relationship between Regression and Classification Since we’re classifying two classes, why not set one class to ‘0’ and the other to ‘1’ then use linear regression. Regression: -infinity to infinity, while class labels are 0, 1 Can use a threshold, e.g. y >= 0.5 then class 1 y < 0.5 then class 2 f(x)>=0.5? Happy/Good/ClassA Sad/Not Good/ClassB 1

Odds-ratio Rather than thresholding, we’ll relate the regression to the class-conditional probability. Ratio of the odd of prediction y = 1 or y = 0 If p(y=1|x) = 0.8 and p(y=0|x) = 0.2 Odds ratio = 0.8/0.2 = 4 Use a linear model to predict odds rather than a class label.

Logit – Log odds ratio function LHS: 0 to infinity RHS: -infinity to infinity Use a log function. Has the added bonus of disolving the division leading to easy manipulation

Logistic Regression A linear model used to predict log-odds ratio of two classes Include image.

Logit to probability

Sigmoid function Squashing function to map the reals to a finite domain.

Gaussian Class-conditional Assume the data is generated from a gaussian distribution for each class. Leads to a bayesian formulation of logistic regression.

Bayesian Logistic Regression

Maximum Likelihood Extimation Logistic Regression Class-conditional Gaussian. Multinomial Class distribution. As ever, take the derivative of this likelihood function w.r.t.

Maximum Likelihood Estimation of the prior

Maximum Likelihood Estimation of the prior

Maximum Likelihood Estimation of the prior

Discriminative Training Take the derivatives w.r.t. Be prepared for this for homework. In the generative formulation, we need to estimate the joint of t and x. But we get an intuitive regularization technique. Discriminative Training Model p(t|x) directly.

What’s the problem with generative training Formulated this way, in D dimensions, this function has D parameters. In the generative case, 2D means, and D(D+1)/2 covariance values Quadratic growth in the number of parameters. We’d rather linear growth.

Discriminative Training

Optimization Take the gradient in terms of w

Optimization

Optimization

Optimization

Optimization: putting it together

Optimization We know the gradient of the error function, but how do we find the maximum value? Setting to zero is nontrivial Numerical approximation

Gradient Descent Take a guess. Move in the direction of the negative gradient Jump again. In a convex function this will converge Other methods include Newton-Raphson

Multi-class discriminant functions Can extend to multiple classes Other approaches include constructing K-1 binary classifiers. Each classifier compares cn to not cn Computationally simpler, but not without problems

Exponential Model Logistic Regression is a type of exponential model. Linear combination of weights and features to produce a probabilistic model.

Problems with Binary Discriminant functions

K-class discriminant

Entropy Measure of uncertainty, or Measure of “Information” High uncertainty equals high entropy. Rare events are more “informative” than common events.

Entropy How much information is received when observing ‘x’? If independent, p(x,y) = p(x)p(y). H(x,y) = H(x) + H(y) The information contained in two unrelated events is equal to their sum.

Entropy Binary coding of p(x): -log p(x) “How many bits does it take to represent a value p(x)?” How many “decimal” places? How many binary decimal places? Expected value of observed information

Examples of Entropy Uniform distributions have higher distributions.

Maximum Entropy Logistic Regression is also known as Maximum Entropy. Entropy is convex. Convergence Expectation. Constrain this optimization to enforce good classification. Increase maximum likelihood of the data while making the distribution of weights most even. Include as many useful features as possible.

Maximum Entropy with Constraints From Klein and Manning Tutorial

Optimization formulation If we let the weights represent likelihoods of value for each feature. For each feature i

Solving MaxEnt formulation Convex optimization with a concave objective function and linear constraints. Lagrange Multipliers Dual representation of the maximum likelihood estimation of Logistic Regression For each feature i

Summary Bayesian Regularization Logistic Regression Entropy Introduction of a prior over parameters serves to constrain weights Logistic Regression Log odds to construct a linear model Formulation with Gaussian Class Conditionals Discriminative Training Gradient Descent Entropy Logistic Regression as Maximum Entropy.

Next Time Graphical Models Read Chapter 8.1, 8.2