Optimal PAC-Bayesian Posteriors for Stochastic Classifiers and their use for Choice of SVM Regularization Parameter
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PAC-Bayesian set up involves a stochastic classifier characterized by a posterior distribution on a classifier set, offers a high probability bound on its averaged true risk and is robust to the training sample used. For a given posterior, this bound captures the trade off between averaged empirical risk and KL-divergence based model complexity term. Our goal is to identify an optimal posterior with the least PAC-Bayesian bound. We consider a finite classifier set and 5 distance functions: KL-divergence, its Pinsker's and a sixth degree polynomial approximations; linear and squared distances. Linear distance based model results in a convex optimization problem. We obtain closed form expression for its optimal posterior. For uniform prior, this posterior has full support with weights negative-exponentially proportional to number of misclassifications. Squared distance and Pinsker's approximation bounds are possibly quasi-convex and are observed to have single local minimum. We derive fixed point equations (FPEs) using partial KKT system with strict positivity constraints. This obviates the combinatorial search for subset support of the optimal posterior. For uniform prior, exponential search on a full-dimensional simplex can be limited to an ordered subset of classifiers with increasing empirical risk values. These FPEs converge rapidly to a stationary point, even for a large classifier set when a solver fails. We apply these approaches to SVMs generated using a finite set of SVM regularization parameter values on 9 UCI datasets. These posteriors yield stochastic SVM classifiers with tight bounds. KL-divergence based bound is the tightest, but is computationally expensive due to non-convexity and multiple calls to a root finding algorithm. Optimal posteriors for all 5 distance functions have lowest 10 values on most datasets, with linear distance being the easiest to obtain.
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