Greedy-Applicable Arm Feature Distributions for Sparse Linear Bandits

Core Concepts

The greedy algorithm is applicable to a wider range of arm feature distributions than previously shown, including distributions with origin-asymmetric support, by considering mixture distributions and proposing new representational distribution classes.

Abstract

The paper considers the sparse linear contextual bandit problem, where the reward is affected by the inner product of the selected arm features and an unknown sparse parameter. Recent studies have developed sparsity-agnostic algorithms based on the greedy arm selection policy, but the analysis requires strong assumptions on the arm feature distribution to ensure sample diversity.
The authors show that the applicability of the greedy algorithm can be extended to a wider range of arm feature distributions in two ways:

Distributions having a mixture component of a greedy-applicable distribution are also greedy-applicable. This allows the greedy algorithm to be applied to a broader class of distributions.

The authors propose new representational distribution classes, related to Gaussian mixture, discrete, and radial distributions, that are greedy-applicable. These classes can describe distributions with origin-asymmetric support, which were not covered by previous assumptions.

The authors provide theoretical guarantees for the greedy policy under these new distribution classes and demonstrate the usefulness of their analysis by applying it to other cases, such as the thresholded lasso bandit, combinatorial setting, and non-sparse setting.

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Key Insights Distilled From

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

by Koji Ichikaw... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2312.12400.pdf

Deeper Inquiries

What are the key factors that determine the size of the compatibility constant ϕ0 in the proposed greedy-applicable distribution classes

The size of the compatibility constant ϕ0 in the proposed greedy-applicable distribution classes is determined by several key factors. Firstly, the shape and spread of the arm feature distribution play a significant role. Distributions that have more diverse and spread-out features are likely to result in a larger ϕ0. Secondly, the dimensionality of the arm features can impact the size of ϕ0. Higher-dimensional feature spaces may require a larger ϕ0 to ensure sample diversity. Additionally, the number of arms in the bandit problem can influence the size of ϕ0, with a larger number of arms potentially requiring a larger constant for sample diversity. Lastly, the specific characteristics of the chosen basis for the distribution, such as the spread of the Gaussian components or the truncation in radial distributions, can also affect the size of ϕ0.

How can the analysis be extended to the case where all arms are correlated, without the assumption of at least one independent arm

Extending the analysis to the case where all arms are correlated, without the assumption of at least one independent arm, presents a more complex scenario. In this case, the analysis would need to consider the interplay between the correlated arms and how their features collectively contribute to sample diversity. One approach could involve developing a framework that accounts for the correlation structure among all arms and how it influences the diversity of the chosen arm features. This may require more sophisticated mathematical modeling and potentially the introduction of new assumptions or conditions that capture the correlated nature of the arms while ensuring the applicability of the greedy algorithm.

Can the proposed distribution classes be further generalized or combined to capture an even wider range of arm feature distributions that are greedy-applicable

The proposed distribution classes can be further generalized or combined to capture an even wider range of arm feature distributions that are greedy-applicable. One approach could involve exploring hybrid distributions that combine elements from different basis functions. For example, a mixture of Gaussian and radial distributions could be considered to capture both smooth and truncated features. Additionally, incorporating adaptive or dynamic elements into the distribution classes, such as varying weights or parameters based on feedback or exploration, could enhance the flexibility and applicability of the distributions. By continuously refining and expanding the proposed classes, a more comprehensive framework for greedy-applicable distributions can be developed to accommodate diverse and complex arm feature scenarios.