New Classes of the Greedy-Applicable Arm Feature Distributions in the Sparse Linear Bandit Problem

TL;DR

This paper broadens the theoretical understanding of greedy algorithms in sparse linear bandit problems by introducing new classes of arm feature distributions, including asymmetric ones, that ensure sample diversity and applicability.

Contribution

It introduces new distribution classes, related to Gaussian mixtures and discrete distributions, expanding the range of arm features where greedy algorithms are theoretically justified.

Findings

Mixture distributions with greedy-applicable components are also greedy-applicable.

New distribution classes guarantee sample diversity for asymmetric supports.

Theoretical guarantees extend to a wider range of arm feature distributions.

Abstract

We consider the sparse contextual bandit problem where arm feature affects reward through the inner product of sparse parameters. Recent studies have developed sparsity-agnostic algorithms based on the greedy arm selection policy. However, the analysis of these algorithms requires strong assumptions on the arm feature distribution to ensure that the greedily selected samples are sufficiently diverse; One of the most common assumptions, relaxed symmetry, imposes approximate origin-symmetry on the distribution, which cannot allow distributions that has origin-asymmetric support. In this paper, we show that the greedy algorithm is applicable to a wider range of the arm feature distributions from two aspects. Firstly, we show that a mixture distribution that has a greedy-applicable component is also greedy-applicable. Second, we propose new distribution classes, related to Gaussian mixture, discrete, and radial distribution, for which the sample diversity is guaranteed. The proposed classes can describe distributions with origin-asymmetric support and, in conjunction with the first claim, provide theoretical guarantees of the greedy policy for a very wide range of the arm feature distributions.