Relevance Vector Machines


 Brenda Hampton
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1 LUT February 21, 2011
2 Support Vector Machines Model / Regression Marginal Likelihood Regression Relevance vector machines Exercise
3 Support Vector Machines The relevance vector machine (RVM) is a bayesian sparse kernel technique for regression and classification Solves some problems with the support vector machines (SVM) Used in detection and classification. Detecting cancer cells, classificating DNA sequences... etc.
4 Support Vector Machines Support Vector Machines (SVM) A nonprobabilistic decision machine. Returns point estimate for regression and binary decision for classification. Makes decisions based on the function: y(x; w) = w i K(x, x i ) + w 0 (1) where K is the kernel function and w 0 is the bias. Attempts to minimize the error while simultaneously maximize the margin between the two classes.
5 Support Vector Machines Support Vector Machines (SVM) y = 1 y = 0 y = 1 y = 1 y = 0 y = 1 margin
6 Support Vector Machines SVM Problems The number of required support vectors typically grows linearly with the size of the training set Nonprobabilistic predictions. Requires estimation of error/margin tradeoff parameters K(x, x i ) must satisfy mercel s condition.
7 Model / Regression Marginal Likelihood Apply bayesian treatment to SVM. Associates a prior over the model weights governed by a set of hyperparameters. Posterior distributions of the majority of weights are peaked around zero. Training vectors associated with the nonzero weights are the relevance vectors. Typically utilizes fewer kernel functions than SVM.
8 The model Outline Model / Regression Marginal Likelihood For given data set of inputtarget pairs {x n, t n } N n=1 t n = y(x n ; w) + ɛ n (2) where ɛ n are samples from some noise process which is assumed to be meanzero Gaussian with variance σ 2. Thus, p(t n x) = N (t n y(x n ), σ 2 ) (3)
9 The model (cont.) Outline Model / Regression Marginal Likelihood encode sparsity in the prior. p(w α) = N i=0 which is Gaussian, but conditioned on α. N (w i 0, α 1 i ) (4) we must define hyperpriors over all α m to complete the specification of hierarchical prior: p(w m ) = p(w m α m )p(α m )dα m (5)
10 Regression Outline Model / Regression Marginal Likelihood The model has independent Gaussian noise: t n N (y(x n ; w), σ 2 ) Corresponding likelihood: { p(t w, σ 2 ) = (2πσ 2 ) N/2 exp 1 } t Φw 2 2σ2 (6) where t = (t q,..., t N ), w = (w q,..., w M ) and Φ is the NxM design matrix with Φ n m = φ m (x n )
11 The model (cont.) Outline Model / Regression Marginal Likelihood The desired posterior over all unknowns: p(w, α, σ 2 t) = p(t w, α, σ2 )p(w, α, σ 2 ) p(t) (7) When given a new test point, x, predictions are made for the corresponding target t, in terms of predictive distribution: p(t t) = p(t w, α, σ 2 )p(w, α, σ 2 t)dwdαdσ 2 (8) But we have a problem here. We cannot perform these computations analytically. Approximations are needed.
12 The model (cont.) Outline Model / Regression Marginal Likelihood We need to decompose the posterior as: p(w, α, σ 2 t) = p(w t, α, σ 2 )p(α, σ 2 t) (9) And so, the posterior distribution over the weights is: p(w t, α, σ 2 ) = p(t w, α, σ2 )p(w α) p(t α, σ 2 ) N (w µ, Σ) (10) where Σ = (σ 2 Φ T Φ + A) 1 (11) µ = σ 2 ΣΦ T t (12)
13 Marginal Likelihood Outline Model / Regression Marginal Likelihood Marginal Likelihood can be written as p(t α, σ 2 ) = p(t w, σ 2 )p(w α)dw (13) Maximizing the marginal likelyhood function is known as the typeii maximum likelihood method. We must optimize p(t α, σ 2 ). There are a few ways to do this.
14 Marginal Likelihood optimization Model / Regression Marginal Likelihood Maximizes (13) with iterative reestimation. Differentiating logp(t α, σ 2 ) gives iterative reestimation approach: αi new = γ i µ 2 i (14) (σ 2 ) new t Φµ 2 = N Σ M i=1 γ i where we have defined quantities as γ i = 1 α i Σ ii. γ i is a measure of how welldetermined is the parameter w i (15)
15 Model / Regression Marginal Likelihood RVMs for classification The likelihood P(t w) is now Bernoulli: P(t w) = N g{y(x n ; w)} t n[1 g{y(x n ; w)}] 1 tn (16) n=1 with g(y) = 1/(1 + e y ) the sigmoid function. No noise variance, same sparse prior as regression. Unlike regression, The weight posteriors p(w t, α) cannot be obtained analytically. Approximations are once again needed.
16 Model / Regression Marginal Likelihood Gaussian posterior approximation Find posterior mode w M P for current values of α by using optimization Compute Hessian Negate and invert to give the covariance for a gaussian approximation p(w t, α) N (w M P, Σ) α are updated using µ and Σ.
17 Regression RVM Regression Example sinc function: sinc(x) = sin(x)/x Linear spline kernel: K(x m, x n ) = 1 + x m x n + x m x n min(x m, x n ) xm+xn 2 min(x m, x m ) 2 + min(xm,xn)3 3 with ɛ = 0.01, 100 uniform, noisefree samples.
18 RVM Regression Example Regression
19 RVM Regression Example Regression
20 Regression RVM Example Ripley s synthetic data Gaussian kernel: K(x m, x n ) = exp( r 2 ) x m x n 2 with r = 0.5
21 RVM Example Regression
22 Relevance vector machines Exercise Sparsity: the prediction of new inputs depend on the kernel function evaluated at a subset of the training data points. TODO More detailed explanation in the original publication: Tipping M., Sparse Bayesian Learning and the Relevance Vector Machine, Journal of Machine Learning Research 1, 2001, pp
23 Relevance vector machines Exercise Exercise Fetch Tipping s matlab toolbox for sparse bayes from http: // Try SparseBayesDemo.m with different likelihood models (Gaussian, Bernoulli...) and familiarize yourself with the toolbox Try to replicate results from the regression example.
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