Regret Bounds for Deterministic Gaussian Process Bandits

TL;DR

This paper establishes that for deterministic Gaussian process bandits, the regret decreases exponentially fast, significantly improving upon the sublinear rates known for noisy cases, under certain regularity conditions.

Contribution

It provides the first analysis showing exponential regret decay for deterministic GP bandits, contrasting with previous sublinear bounds for noisy observations.

Findings

Exponential regret decay rate for deterministic GP bandits.

Regret bound depends on the dimension and local behavior of the objective.

Results hold with high probability under regularity assumptions.

Abstract

This paper analyses the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Srinivas et al., 2010) proved that the regret vanishes at the approximate rate of $O(\frac{1}{\sqrt{t}})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret decreases asymptotically according to $O(e^{-\frac{\tau t}{(\ln t)^{d/4}}})$ with high probability. Here, d is the dimension of the search space and $\tau$ is a constant that depends on the behaviour of the objective function near its global maximum.