Optimization for Gaussian Processes via Chaining

TL;DR

This paper introduces a generalized Gaussian process optimization method using localized chaining and covering numbers, achieving comparable regret bounds to GP-UCB while improving empirical efficiency across diverse input spaces.

Contribution

It extends the GP-UCB algorithm to arbitrary kernels and spaces with a novel chaining approach and a new optimization scheme based on covering numbers.

Findings

Theoretical regret bounds match those of GP-UCB.

Algorithm demonstrates improved empirical efficiency.

Applicable to complex and simple input spaces.

Abstract

In this paper, we consider the problem of stochastic optimization under a bandit feedback model. We generalize the GP-UCB algorithm [Srinivas and al., 2012] to arbitrary kernels and search spaces. To do so, we use a notion of localized chaining to control the supremum of a Gaussian process, and provide a novel optimization scheme based on the computation of covering numbers. The theoretical bounds we obtain on the cumulative regret are more generic and present the same convergence rates as the GP-UCB algorithm. Finally, the algorithm is shown to be empirically more efficient than its natural competitors on simple and complex input spaces.