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

TL;DR

The paper investigates algorithm-independent worst-case lower bounds for Gaussian process bandits with the squared exponential kernel on a hyperspherical input domain. By constructing a new hard function class via Mercer's representation and spherical harmonics, it establishes tighter regret lower bounds: a simple regret lower bound of Ω(ε^{-2}(ln(1/ε))^{d}(ln ln(1/ε))^{-d}) and a cumulative regret lower bound of Ω(√{Tσ^2}(ln(B^2T/σ^2))^{d}(ln ln(B^2T/σ^2))^{-d}) with high probability, thereby narrowing the dimension-dependent logarithmic gap. In parallel, the authors prove an improved maximum information gain upper bound of O((ln T)^{d+1}(ln ln T)^{-d}), which translates into tighter upper bounds for GP-based bandit algorithms up to dimension-independent logs. While the analysis is specialized to the hypersphere, the MIG bound is general to compact domains in ℝ^d, suggesting a path toward extending the tighter lower bounds more broadly. Overall, the work advances fundamental understanding of SE GP bandits by resolving a substantial portion of the log-gap and introducing new Fukushima-style harmonic analysis tools into lower-bound proofs.

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 $Ω(\sqrt{T (\ln T)^{d} (\ln \ln T)^{-d}})$ 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 $Ω(ε^{-2}(\ln \frac{1}ε)^d (\ln \ln \frac{1}ε)^{-d})$ time steps to find an $ε$-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.

Figures (2)

• Figure 1: Illustrative example of $\mathcal{F}_{\epsilon}$ in one dimension scarlett2017lower.
• Figure 2: Visualization of the function $f_{\epsilon, N, \bm{z}}$ with $\epsilon = 0.5$ and $d = 1$ (i.e., the function on the 2D circle $\mathbb{S}^1$). Note that the above plot takes the geodesic distance $\rho(\bm{x}, \bm{z}) \coloneqq \arccos(\bm{x}^{\top}\bm{z})$ from some center point $\bm{z} \in \mathbb{S}^1$ as the horizontal axis. Intuitively, the number $N$ controls the sharpness of the behavior around the peak at $\rho(\bm{x}, \bm{z}) = 0$.

Theorems & Definitions (48)

• Lemma 2.1: Properties of $g_{\epsilon}$
• Definition 2.2: Finite function class construction based on $g_{\epsilon}$
• Lemma 2.3: Relating two instances, Lemma 1 in cai2021on
• Lemma 2.4: Lemma 5 in scarlett2017lower
• Corollary 2.5: Simple regret lower bound on the hypersphere, extended from cai2021on
• Corollary 2.6: Cumulative regret lower bound on the hypersphere, extended from cai2021on
• Theorem 2.7: Mercer representation, e.g., Theorem 4.2 in kanagawa2018gaussian
• Definition 2.8: Spherical harmonics, e.g., Chapter 2.1.3 in atkinson2012spherical or Chapter 4.2 in efthimiou2014spherical
• Lemma 2.9: Theorem 2 in minh2006mercer
• Theorem 3.1: Improved simple regret lower bound for the SE kernel on the hypersphere