Topics. Machine learning: lecture 5. Classification
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1 Machine learning: lecture Tommi S. Jaakkola MIT CSAIL Topics Classification and regression regression approach to classification Fisher discriminant elementary decision theory Logistic regression model, rationale estimation, stochastic gradient additive extension generalization Tommi Jaakkola, MIT CSAIL Classification Example: digit recognition (8x8 binary digits binary digit actual label target label in learning Classification via regression Suppose e ignore the fact that the target output y is binary (e.g., / rather than a continuous variable So e ill estimate a regression function f(x; = + x d x d = + x T, based on the available data as before. Assuming y = f(x; + ɛ, ɛ N(, σ, then the ML objective for the parameters reduces to least squares fitting: J n ( = n (y i f(x i ; n i= Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL Classification via regression cont d We can use the resulting regression function f(x; ŵ = + x T ŵ, to classify any ne (test example x according to label = if f(x; >., and label = otherise Classification via regression cont d Given the dissociation beteen the objective (classification and the estimation criterion (regression it is not clear that this approach leads to sensible results f(x; ŵ =. therefore defines a decision boundary that partitions the input space into to class specific regions (half spaces 6 sometimes good! sometimes bad Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL 6
2 Linear regression and projections A regression function (here in D f(x; = + x T projects each point x = [x x ] T to a line parallel to. point in R d projected point in R x z = x T x z = x T x n z n = x T n We can study ho ell the projected points {z,..., z n }, vieed as functions of, are separated across the classes. Linear regression and projections A regression function (here in D f(x; = + x T projects each point x = [x x ] T to a line parallel to.!! f(x; = f(x; =. f(x; =!! We can study ho ell the projected points {z,..., z n }, vieed as functions of, are separated across the classes. Tommi Jaakkola, MIT CSAIL 7 Tommi Jaakkola, MIT CSAIL 8 Projection and classification By varying e get different levels of separation beteen the projected points Optimizing the projection We ould like to find that someho maximizes the separation of the projected points across classes!!!!!!!!!!!!!!!! We can quantify the separation (overlap in terms of means and variances of the resulting -dimensional class distributions!!!! Tommi Jaakkola, MIT CSAIL 9 Tommi Jaakkola, MIT CSAIL Fisher discriminant: preliminaries!!!! Class descriptions in R d : class : n samples, mean µ, covariance Σ class : n samples, mean µ, covariance Σ Projected class descriptions in R: class : n samples, mean µ T, variance T Σ class : n samples, mean µ T, variance T Σ Fisher discriminant Estimation criterion: e find that maximizes J F isher ( = = The solution (class separation ŵ (n Σ + n Σ (µ µ is decision theoretically optimal for to normal populations ith equal covariances (Σ = Σ (Separation of projected means Sum of ithin class variances (µ T µ T n T Σ + n T Σ!!!! Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL
3 Background: simple decision theory Suppose e kno the class-conditional densities p(x y for y =, as ell as the overall class frequencies P (y. Ho do e decide hich class a ne example x belongs to so as to minimize the overall probability of error?.... class density P(x y= class density P(x y= Background: simple decision theory Suppose e kno the class-conditional densities p(x y for y =, as ell as the overall class frequencies P (y. Ho do e decide hich class a ne example x belongs to so as to minimize the overall probability of error? The minimum probability of. class density error decisions are given by... class density P(x y= P(x y= y = arg max y=, { p(x yp (y } = arg max y=, { P (y x }!!!! Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL Logistic regression The optimal decisions are based on the posterior class probabilities P (y x. For binary classification problems, e can rite these decisions as y = if and y = otherise. P (y = x P (y = x > Logistic regression The optimal decisions are based on the posterior class probabilities P (y x. For binary classification problems, e can rite these decisions as y = if and y = otherise. P (y = x P (y = x > We generally don t kno P (y x but e can parameterize the possible decisions according to P (y = x P (y = x = f(x; = + x T Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL 6 Logistic regression cont d Our -odds model P (y = x P (y = x = + x T gives rise to a specific form for the conditional probability over the labels (the istic model: P (y = x, = g ( + x T here g(z = ( + exp( z is a istic squashing.. function that turns. predictions into probabilities.!! g(z z Logistic regression: decisions Logistic regression models imply a decision boundary P (y = x P (y = x = + x T = + x T = Tommi Jaakkola, MIT CSAIL 7 Tommi Jaakkola, MIT CSAIL 8
4 Logistic regression: decisions Logistic regression models imply a decision boundary P (y = x P (y = x = + x T = + x T = class class Fitting istic regression models As ith the regression models e can fit the istic models using the maximum (conditional -likelihood criterion n l(d; = P (y i x i, here i= P (y = x, = g ( + x T The -likelihood function l(d; is a jointly concave function of the parameters ; a number of optimization techniques are available for finding the maximizing parameters Tommi Jaakkola, MIT CSAIL 9 Tommi Jaakkola, MIT CSAIL About the ML solution If e set the derivatives of the -likelihood ith respect to the parameters to zero l(d; = j l(d; = yi P (y i = x i, = i= yi P (y i = x i, x ij = i= the optimality conditions again require that the prediction errors ɛ i = ( y i P (y i = x i,, i =,..., n corresponding to the optimal setting of the parameters are uncorrelated ith any function of the inputs. Stochastic gradient ascent We can try to maximize the -likelihood in an on-line or incremental fashion. Given each training input x i and the binary (/ label y i, e can change the parameters slightly to increase the corresponding -probability + η P (y i x i, = + η ( y i P (y i = x i, }{{} prediction error here η is the learning rate. [ The resulting update is similar to the mistake driven algorithm discussed earlier; examples that are already confidently classified do not lead to any significant updates x i ] Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL Stochastic gradient ascent cont d To understand the procedure graphically e focus on a single example and omit the bias term + η ( y i P (y i = x i, x }{{} i prediction error η ( P (y = x, x ne Gradient ascent of the -likelihood We can also perform gradient ascent steps on the likelihood of all the training labels given examples at the same time. In other ords, + η l(d; = + η yi P (y i = x i, [ ] x i i= x, y = x, y = Still need to figure out a ay to set the learning rate to guarantee convergence. Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL
5 Setting the learning rate: Armijo rule The learning rate in + η l(d; should satisfy ( l D; ne {}}{ + η l(d; estimated actual η l(d; l(d; l(d; η l(d; The Armijo rule suggests finding the smallest integer m such that η = η q m, q < is a valid choice in this sense. Armijo rule is guaranteed to converge to a (local maximum under certain technical assumptions Additive models and classification Similarly to regression models, e can extend the istic regression models to additive (istic models P (y = x, = g ( + φ (x +... m φ m (x As before e are free to choose the basis functions φ i (x to capture relevant properties of any specific classification problem Since e also over-fit easily, e can use leave-one-out crossvalidation (in terms of -likelihood or classification error to estimate the generalization performance CV -likelihood = n P (y i x i, ŵ i n i= Tommi Jaakkola, MIT CSAIL Tommi Jaakkola, MIT CSAIL 6 Simple binary classification problem in R Simple binary classification problem in R Tommi Jaakkola, MIT CSAIL 7 Tommi Jaakkola, MIT CSAIL 8 Simple binary classification problem in R Simple binary classification problem in R quadratic quadratic CV =.6 CV =. Tommi Jaakkola, MIT CSAIL 9 Tommi Jaakkola, MIT CSAIL
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