UCB algorithms for multi-armed bandits: Precise regret and adaptive inference (2412.06126v1)

Abstract: Upper Confidence Bound (UCB) algorithms are a widely-used class of sequential algorithms for the $K$-armed bandit problem. Despite extensive research over the past decades aimed at understanding their asymptotic and (near) minimax optimality properties, a precise understanding of their regret behavior remains elusive. This gap has not only hindered the evaluation of their actual algorithmic efficiency, but also limited further developments in statistical inference in sequential data collection. This paper bridges these two fundamental aspects--precise regret analysis and adaptive statistical inference--through a deterministic characterization of the number of arm pulls for an UCB index algorithm [Lai87, Agr95, ACBF02]. Our resulting precise regret formula not only accurately captures the actual behavior of the UCB algorithm for finite time horizons and individual problem instances, but also provides significant new insights into the regimes in which the existing theory remains informative. In particular, we show that the classical Lai-Robbins regret formula is exact if and only if the sub-optimality gaps exceed the order $\sigma\sqrt{K\log T/T}$. We also show that its maximal regret deviates from the minimax regret by a logarithmic factor, and therefore settling its strict minimax optimality in the negative. The deterministic characterization of the number of arm pulls for the UCB algorithm also has major implications in adaptive statistical inference. Building on the seminal work of [Lai82], we show that the UCB algorithm satisfies certain stability properties that lead to quantitative central limit theorems in two settings including the empirical means of unknown rewards in the bandit setting. These results have an important practical implication: conventional confidence sets designed for i.i.d. data remain valid even when data are collected sequentially.

• The paper provides a refined, deterministic characterization of UCB algorithm arm pulls to precisely analyze regret and enable adaptive inference.
• It introduces a precise regret formula for UCB algorithms, showing classical formulas hold only when sub-optimality gaps exceed a specific threshold.
• The deterministic pull characterization allows for a novel adaptive inference approach, validating conventional confidence sets for sequentially collected data.

Overview of "UCB Algorithms for Multi-Armed Bandits: Precise Regret and Adaptive Inference"

The paper "UCB Algorithms for Multi-Armed Bandits: Precise Regret and Adaptive Inference" by Qiyang Han, Koulik Khamaru, and Cun-Hui Zhang presents significant advancements in understanding Upper Confidence Bound (UCB) algorithms, commonly used in the $K$-armed bandit problem. The paper targets two primary challenges: the accurate analysis of the regret behavior of UCB algorithms and their application in adaptive statistical inference.

Regret Analysis of UCB Algorithms

The multi-armed bandit problem, an essential topic in sequential decision-making, typically seeks to balance exploration and exploitation to minimize cumulative regret. Regret quantifies the performance of an algorithm by measuring the difference between the rewards collected by a selected strategy and the rewards from an optimal strategy over multiple rounds. The paper specifically focuses on UCB strategies, which decide the arm to pull based on a combination of empirical means and an exploration term derived from confidence bounds.

Despite the widespread use of UCB strategies, their regret behavior has been inadequately understood. Traditional analyses provided asymptotic or near-minimax optimal bounds; however, these did not precisely capture regret behavior for finite-time horizons or for specific instances. The authors address this by offering a refined and deterministic characterization of the pulls of each arm using a fixed-point equation, which more accurately predicts UCB performance over finite periods.

Key Findings

1. Precise Regret Formula: The paper introduces a regret formula that better reflects the actual performance of UCB algorithms. It is shown that the classical Lai-Robbins regret formula holds exactly only when the sub-optimality gaps are larger than a threshold of $O(\sigma\sqrt{K\log T/T})$. This finding corrects and improves upon existing theoretical frameworks, suggesting that traditional analyses might be overly conservative under certain conditions.
2. Minimax Sub-Optimality: The research indicates that the maximal regret for UCB algorithms deviates from the minimax regret by a logarithmic factor, undermining the strict minimax optimality—suggesting a need for alternative strategies to achieve better worst-case guarantees.

Adaptive Inference

A novel contribution of the paper is illustrating how UCB algorithms can be leveraged for adaptive inference. The deterministic characterization of arm pulls leads to an innovative adaptive inference approach, which validates using conventional confidence sets even for data collected sequentially. This aspect addresses a significant challenge in bandit problems, where data-dependent sequential interventions complicate inferential statistics.

Implications and Future Directions

The implications of this research are twofold:

1. Practical Algorithm Design: By providing a more accurate estimation of regret, algorithm designers can fine-tune UCB strategies to work more efficiently within specific time constraints or problem environments.
2. Theoretical Insights: The paper challenges existing paradigms of UCB algorithm evaluation, suggesting that both practitioners and theorists need to reconsider how these strategies are analyzed in the context of finite-time horizons.

Future research directions could further refine the theoretical bounds for specific problem settings or consider other bandit algorithms within the same analytical framework proposed for UCBs. Additionally, exploring practical implementations of the adaptive inference methodology in real-world applications could validate the theoretical advancements presented.

In conclusion, this paper makes substantive contributions to understanding and improving UCB algorithms' performance in regret minimization and adaptive inference, reinforcing their importance and efficacy in both theoretical and applied statistics.