Lecture 4: Logistic Regression. Machine Learning CUNY Graduate Center. Today. Linear Regression Bayesians v. Frequentists Bayesian Linear Regression Logistic Regression Linear Model for Classification. Regularization: Penalize large weights. Regularized Regression
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation
Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement.Téléchargez la présentation en cliquant sur ce lien.Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.
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 likelihoodthat 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:
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 tias 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, tiis often represented as a K element vector. • tijrepresents the degree of membership in class j. • |ti| = 1 • E.g. 5-way classification vector:
Classification approaches • Generative • Models the joint distribution between c and x • Highest data requirements • Discriminative • Fewer parameters to approximate • DiscriminantFunction • May still be trained probabilistically, but not necessarily modeling a likelihood.
Relationship between Regression and Classification 1 Happy/Good/ClassA f(x)>=0.5? Sad/Not Good/ClassB • 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
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
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.
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.
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 notcn • 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.
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
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.
