  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% test error
values on most datasets, with linear distance being the easiest to obtain.

Machine Learning;Machine Learning