Tighter Regret Lower Bound for Gaussian Process Bandits with Squared Exponential Kernel in Hypersphere

TL;DR

This paper establishes tighter lower bounds on the regret for Gaussian process bandits with squared exponential kernels in hyperspherical domains, showing the bounds are nearly optimal and clarifying the role of dimension-dependent factors.

Contribution

It provides the first dimension-dependent lower bounds for GP bandits with SE kernels on hyperspheres, narrowing the gap with existing upper bounds and confirming the near-optimality of current algorithms.

Findings

Lower bound on cumulative regret: (\u221a{T} (\u2318; T)^d (\u2318; ; T)^{-d})

Lower bound on simple regret: (;  ; 1/)^d (; ; 1/)^{-d})

Upper bound on maximum information gain: O((; T)^{d+1}(; ; T)^{-d})

Abstract

We study an algorithm-independent, worst-case lower bound for the Gaussian process (GP) bandit problem in the frequentist setting, where the reward function is fixed and has a bounded norm in the known reproducing kernel Hilbert space (RKHS). Specifically, we focus on the squared exponential (SE) kernel, one of the most widely used kernel functions in GP bandits. One of the remaining open questions for this problem is the gap in the \emph{dimension-dependent} logarithmic factors between upper and lower bounds. This paper partially resolves this open question under a hyperspherical input domain. We show that any algorithm suffers $\Omega(\sqrt{T (\ln T)^{d} (\ln \ln T)^{-d}})$ cumulative regret, where $T$ and $d$ represent the total number of steps and the dimension of the hyperspherical domain, respectively. Regarding the simple regret, we show that any algorithm requires $\Omega(\epsilon^{-2}(\ln \frac{1}{\epsilon})^d (\ln \ln \frac{1}{\epsilon})^{-d})$ time steps to find an $\epsilon$-optimal point. We also provide the improved $O((\ln T)^{d+1}(\ln \ln T)^{-d})$ upper bound on the maximum information gain for the SE kernel. Our results guarantee the optimality of the existing best algorithm up to \emph{dimension-independent} logarithmic factors under a hyperspherical input domain.