Contextual Linear Bandits with Delay as Payoff

TL;DR

This paper extends the delay-as-payoff model to contextual linear bandits, proposing an efficient phased elimination algorithm with regret bounds that handle delays proportional to payoffs, and demonstrates its effectiveness through experiments.

Contribution

It introduces a novel phased arm elimination algorithm for contextual linear bandits with delay-as-payoff, achieving near-optimal regret bounds and extending to varying action sets.

Findings

Regret overhead is at most DΔ_max log T compared to no-delay case.

Further improvements are shown for the loss setting, indicating a separation from reward.

Experimental results demonstrate the algorithm's effectiveness and superior performance.

Abstract

A recent work by Schlisselberg et al. (2024) studies a delay-as-payoff model for stochastic multi-armed bandits, where the payoff (either loss or reward) is delayed for a period that is proportional to the payoff itself. While this captures many real-world applications, the simple multi-armed bandit setting limits the practicality of their results. In this paper, we address this limitation by studying the delay-as-payoff model for contextual linear bandits. Specifically, we start from the case with a fixed action set and propose an efficient algorithm whose regret overhead compared to the standard no-delay case is at most $D\Delta_{\max}\log T$, where $T$ is the total horizon, $D$ is the maximum delay, and $\Delta_{\max}$ is the maximum suboptimality gap. When payoff is loss, we also show further improvement of the bound, demonstrating a separation between reward and loss similar to Schlisselberg et al. (2024). Contrary to standard linear bandit algorithms that construct least squares estimator and confidence ellipsoid, the main novelty of our algorithm is to apply a phased arm elimination procedure by only picking actions in a volumetric spanner of the action set, which addresses challenges arising from both payoff-dependent delays and large action sets. We further extend our results to the case with varying action sets by adopting the reduction from Hanna et al. (2023). Finally, we implement our algorithm and showcase its effectiveness and superior performance in experiments.