Year: 2025
Type: article
Abstract: This paper studies batched bandit learning problems for nondegenerate functions. Over a compact doubling metric space (X,D), a functionf : X→ R is called nondegenerate if there existsL≥ λ > 0 andq≥ 1, such that λ (D(x, x*))q≤f(x) −f(x*) ≤L(D(x, x*))q, x ∈X, where x* = arg minz∈Xf(z) is the unique minimizer offoverX. In this paper, we introduce an algorithm that solves the batched bandit problem for nondegenerate functions nearoptimally. More specifically, we introduce an algorithm, called Geometric Narrowing (GN), whose regret bound is of orderO(Ad+√T), wheredis the doubling dimension of (X,D), andA+is a constant independent ofdand the time horizonT. In addition, GN only needsO(log logT) batches to achieve this regret. We also provide lower bound analysis for this problem. More specifically, we prove that over some (compact) doubling metric space of doubling dimensiond: 1. For any policy π, there exists a problem instance on which π admits a regret of order Ω (Ad−√T), whereA−is a constant independent ofdandT; 2. No policy can achieve a regret of orderAd−√Tover all problem instances, using less than Ω(log logT) rounds of communications. Our lower bound analysis shows that the GN algorithm achieves near optimal regret with minimal number of batches.
Authors: Yu Liu, Yunlu Shu, Tianyu Wang
Institution: Fudan University
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W4409311565