Variance-Aware Regret Bounds for Stochastic Contextual Dueling Bandits

TL;DR

This paper introduces a variance-aware regret bound for stochastic contextual dueling bandits, improving decision-making models that incorporate uncertainty in pairwise comparisons, with theoretical guarantees and empirical validation.

Contribution

The paper proposes a new efficient SupLinUCB-type algorithm with a variance-aware regret bound for contextual dueling bandits, addressing a gap in existing research.

Findings

The regret bound scales with the sum of variances in pairwise comparisons.

The algorithm performs better than previous variance-agnostic methods on synthetic data.

The regret is minimized in deterministic comparison scenarios.

Abstract

Dueling bandits is a prominent framework for decision-making involving preferential feedback, a valuable feature that fits various applications involving human interaction, such as ranking, information retrieval, and recommendation systems. While substantial efforts have been made to minimize the cumulative regret in dueling bandits, a notable gap in the current research is the absence of regret bounds that account for the inherent uncertainty in pairwise comparisons between the dueling arms. Intuitively, greater uncertainty suggests a higher level of difficulty in the problem. To bridge this gap, this paper studies the problem of contextual dueling bandits, where the binary comparison of dueling arms is generated from a generalized linear model (GLM). We propose a new SupLinUCB-type algorithm that enjoys computational efficiency and a variance-aware regret bound $\tilde O\big(d\sqrt{\sum_{t=1}^T\sigma_t^2} + d\big)$, where $\sigma_t$ is the variance of the pairwise comparison in round $t$, $d$ is the dimension of the context vectors, and $T$ is the time horizon. Our regret bound naturally aligns with the intuitive expectation in scenarios where the comparison is deterministic, the algorithm only suffers from an $\tilde O(d)$ regret. We perform empirical experiments on synthetic data to confirm the advantage of our method over previous variance-agnostic algorithms.