Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design

TL;DR

This paper provides theoretical regret bounds for Gaussian process optimization in bandit problems, connecting it with experimental design and demonstrating favorable empirical performance on sensor data.

Contribution

It derives the first regret bounds for GP optimization with low RKHS norm functions, linking it to experimental design and spectral analysis.

Findings

Bounded cumulative regret in terms of information gain

Established explicit regret bounds for common covariance functions

GP-UCB outperforms other heuristics in sensor data experiments

Abstract

Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches.