A Strongly Quasiconvex PAC-Bayesian Bound

TL;DR

This paper introduces a new convex PAC-Bayesian bound and an optimization method for hypothesis space construction, demonstrating competitive results with cross-validation and revealing quasiconvex trade-offs.

Contribution

It presents a novel convex PAC-Bayesian bound and an alternating minimization procedure with conditions ensuring global optimality.

Findings

Bound is convex in posterior and trade-off parameter

Alternating minimization converges to global minimum under certain conditions

Experimental results show competitive performance with cross-validation

Abstract

We propose a new PAC-Bayesian bound and a way of constructing a hypothesis space, so that the bound is convex in the posterior distribution and also convex in a trade-off parameter between empirical performance of the posterior distribution and its complexity. The complexity is measured by the Kullback-Leibler divergence to a prior. We derive an alternating procedure for minimizing the bound. We show that the bound can be rewritten as a one-dimensional function of the trade-off parameter and provide sufficient conditions under which the function has a single global minimum. When the conditions are satisfied the alternating minimization is guaranteed to converge to the global minimum of the bound. We provide experimental results demonstrating that rigorous minimization of the bound is competitive with cross-validation in tuning the trade-off between complexity and empirical performance. In all our experiments the trade-off turned to be quasiconvex even when the sufficient conditions were violated.