Lasso Bandit with Compatibility Condition on Optimal Arm

TL;DR

This paper introduces a new Lasso bandit algorithm that achieves logarithmic regret in the ambient dimension without requiring diversity conditions, relying only on a weaker compatibility condition on the optimal arm.

Contribution

It demonstrates that the compatibility condition alone suffices for logarithmic regret bounds in sparse linear bandits, relaxing previous assumptions and proposing an adaptive algorithm.

Findings

Achieves $O( ext{poly}\log dT)$ regret under the margin condition.

Requires weaker assumptions than existing Lasso bandit algorithms.

Numerical experiments confirm superior performance.

Abstract

We consider a stochastic sparse linear bandit problem where only a sparse subset of context features affects the expected reward function, i.e., the unknown reward parameter has a sparse structure. In the existing Lasso bandit literature, the compatibility conditions, together with additional diversity conditions on the context features are imposed to achieve regret bounds that only depend logarithmically on the ambient dimension $d$. In this paper, we demonstrate that even without the additional diversity assumptions, the \textit{compatibility condition on the optimal arm} is sufficient to derive a regret bound that depends logarithmically on $d$, and our assumption is strictly weaker than those used in the lasso bandit literature under the single-parameter setting. We propose an algorithm that adapts the forced-sampling technique and prove that the proposed algorithm achieves $O(\text{poly}\log dT)$ regret under the margin condition. To our knowledge, the proposed algorithm requires the weakest assumptions among Lasso bandit algorithms under the single-parameter setting that achieve $O(\text{poly}\log dT)$ regret. Through numerical experiments, we confirm the superior performance of our proposed algorithm.