Linear Bandits with Non-i.i.d. Noise

TL;DR

This paper extends linear bandit analysis to settings with dependent, sub-Gaussian noise, developing new confidence sequences and algorithms that adapt to the noise's dependence structure, achieving regret bounds related to the decay of dependencies.

Contribution

It introduces confidence sequences and a bandit algorithm for dependent noise, relaxing the i.i.d. assumption and providing regret bounds based on dependence decay rates.

Findings

Regret bounds depend on the decay rate of noise dependence.

Bounds recover standard rates up to a mixing time factor.

Algorithm performs well under geometrically mixing noise.

Abstract

We study the linear stochastic bandit problem, relaxing the standard i.i.d. assumption on the observation noise. As an alternative to this restrictive assumption, we allow the noise terms across rounds to be sub-Gaussian but interdependent, with dependencies that decay over time. To address this setting, we develop new confidence sequences using a recently introduced reduction scheme to sequential probability assignment, and use these to derive a bandit algorithm based on the principle of optimism in the face of uncertainty. We provide regret bounds for the resulting algorithm, expressed in terms of the decay rate of the strength of dependence between observations. Among other results, we show that our bounds recover the standard rates up to a factor of the mixing time for geometrically mixing observation noise.