[Paper Review] Batched Kernelized Bandits: Refinements and Extensions

Chenkai Ma, Keqin Chen+1 more|arXiv (Cornell University)|Mar 13, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

This paper refines and extends batched kernelized bandits by optimizing batch counts, establishing lower bounds for adaptive batches, and introducing a robust variant that matches non-robust regret bounds while improving simple regret.

ABSTRACT

In this paper, we consider the problem of black-box optimization with noisy feedback revealed in batches, where the unknown function to optimize has a bounded norm in some Reproducing Kernel Hilbert Space (RKHS). We refer to this as the Batched Kernelized Bandits problem, and refine and extend existing results on regret bounds. For algorithmic upper bounds, (Li and Scarlett, 2022) shows that $B=O(\log\log T)$ batches suffice to attain near-optimal regret, where $T$ is the time horizon and $B$ is the number of batches. We further refine this by (i) finding the optimal number of batches including constant factors (to within $1+o(1)$), and (ii) removing a factor of $B$ in the regret bound. For algorithm-independent lower bounds, noticing that existing results only apply when the batch sizes are fixed in advance, we present novel lower bounds when the batch sizes are chosen adaptively, and show that adaptive batches have essentially same minimax regret scaling as fixed batches. Furthermore, we consider a robust setting where the goal is to choose points for which the function value remains high even after an adversarial perturbation. We present the robust-BPE algorithm, and show that a suitably-defined cumulative regret notion incurs the same bound as the non-robust setting, and derive a simple regret bound significantly below that of previous work.

Motivation & Objective

• Investigate batched black-box optimization with noisy RKHS-bounded functions, focusing on reducing regret under batched feedback.
• Determine the optimal number of batches (with precise constants) and remove extraneous factors in regret bounds.
• Develop lower bounds for adaptive batching to assess minimax limits.
• Extend to a robust setting where perturbations are adversarially bounded while preserving regret performance.

Proposed method

• Analyze and refine the Batched Pure Exploration (BPE) algorithm to operate with generalized batch sizes.
• Derive upper bounds for growing batch counts using modified batch-size schedule N_i = min{ceil(T^{1-a^i}), T - sum_{j<i} N_j}, and prove near-optimal regret O*(sqrt(T gamma_T)).
• Establish lower bounds for adaptive batches via a change-measure argument tailored to infinite-arm kernelized bandits.
• Introduce robust-BPE by expanding exploration to perturbation-robust candidate sets and proving cumulative regret bounds.
• Compare with and extend prior results on information gain gamma_T for SE and Matérn kernels, including handling adaptive batching.
• Provide high-level discussion on simple regret implications and robustness to adversarial perturbations.

Figure 1 : Illustration of a class of hard-to-distinguish functions $\mathcal{F}$ , where any $x\in\mathcal{X}$ can be $\epsilon$ -optimal for at most one bump function. This is an “idealized” illustration, with the actual functions used having infinite support but steady decay to zero.

Experimental results

Research questions

• RQ1 What is the optimal (up to 1+o(1)) number of batches needed to achieve near-optimal regret in batched kernelized bandits?
• RQ2 Do adaptive batch sizes offer any minimax advantage over fixed batches in this setting?
• RQ3 Can the batched framework be extended to adversarially robust objectives without sacrificing regret performance?
• RQ4 How do refined batch-size schedules affect regret bounds for SE and Matérn kernels?
• RQ5 What are the algorithm-independent lower bounds when batch sizes are chosen adaptively?

Key findings

• Batched Pure Exploration with generalized batch sizes achieves near-optimal O*(sqrt(T gamma_T)) regret with B = O(log log T) batches.
• A refined batch-size schedule N_i = min{ceil(T^{1-a^i}), T - sum_{j<i} N_j} yields tighter batch counts and enables constants-precise B ≈ (log_{1/a} log T)(1+o(1)).
• For a in (1/2,1), the regret bound improves; for Matérn with a in (ν/(2ν+d), 1/2], bounds hold with well-behaved gamma_bar_t.
• Adaptive batching incurs only a mild inverse-polynomial dependence on B in lower bounds, implying adaptivity does not substantially improve minimax regret.
• A robust-BPE algorithm achieves cumulative regret matching the non-robust setting and improves simple regret over prior robust results.
• Lower bounds for adaptive batches show that to achieve near-optimal regret, B must scale at least as Ω(log_{1/η} log T) with η depending on kernel parameters.

(a) SE kernel.