Exponential Regret Bounds for Gaussian Process Bandits with Deterministic Observations

TL;DR

This paper establishes exponential regret bounds for Gaussian process bandits with deterministic observations, showing faster convergence rates than the stochastic case under certain regularity conditions.

Contribution

It extends the analysis of GP bandits to deterministic observations, proving exponential regret decay, unlike the sublinear rates in noisy settings.

Findings

Exponential decay of regret in deterministic GP bandits

Regret bounds depend on the dimension and local behavior of the objective

High probability guarantees for the convergence rate

Abstract

This paper analyzes 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 proved that the regret vanishes at the approximate rate of $O(1/\sqrt{t})$, 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.