Towards Efficient and Optimal Covariance-Adaptive Algorithms for Combinatorial Semi-Bandits (2402.15171v4)

Abstract: We address the problem of stochastic combinatorial semi-bandits, where a player selects among P actions from the power set of a set containing d base items. Adaptivity to the problem's structure is essential in order to obtain optimal regret upper bounds. As estimating the coefficients of a covariance matrix can be manageable in practice, leveraging them should improve the regret. We design "optimistic" covariance-adaptive algorithms relying on online estimations of the covariance structure, called OLS-UCB-C and COS-V (only the variances for the latter). They both yields improved gap-free regret. Although COS-V can be slightly suboptimal, it improves on computational complexity by taking inspiration from ThompsonSampling approaches. It is the first sampling-based algorithm satisfying a T^1/2 gap-free regret (up to poly-logs). We also show that in some cases, our approach efficiently leverages the semi-bandit feedback and outperforms bandit feedback approaches, not only in exponential regimes where P >> d but also when P <= d, which is not covered by existing analyses.

• The paper presents the OLS-UCBV algorithm that employs online least-squares estimation with adaptive covariance tracking for combinatorial semi-bandit problems.
• It achieves a novel gap-free regret rate of √T, outperforming traditional variance-blind methods through rigorous martingale analysis and concentration bounds.
• The approach reduces computational overhead while maintaining robust theoretical guarantees, enhancing decision-making in high-dimensional, correlated environments.

Covariance-Adaptive Least-Squares Algorithm for Stochastic Combinatorial Semi-Bandits

The paper by Zhou et al. introduces a novel algorithm for addressing the stochastic combinatorial semi-bandit problem, a critical area in sequential decision-making. This problem is characterized by a decision-maker selecting subsets of items across rounds, with feedback conditional on the items selected. The work focuses on developing a method that effectively leverages adaptive strategies based on covariance estimation to improve upon existing combinatorial semi-bandit algorithms.

Key Contributions

The authors present the OLS-UCBV algorithm, which uses an online least-squares estimator to keep track of the mean and the covariance of the items' rewards. This algorithm stands out by relying on real-time estimation of the covariance matrix rather than assuming prior proxy covariance information. This approach addresses a significant limitation in previous work, wherein estimating a proper sub-Gaussian proxy covariance was a challenging and often imprecise task.

Additionally, this paper not only demonstrates a detailed theoretical analysis of the proposed solution but also establishes improved regret bounds. Most notably, it provides a novel $\sqrt{T}$ gap-free regret rate, in contrast to existing combinatorial bandits where gap-free analyses are often entangled with terms in $\Delta_{\textrm{min}^{-2}$, making results less robust in some scenarios.

Regret Analysis and Advantages

The regret analysis of the proposed algorithm aligns with real-world conditions where obtaining tight estimates of proxy variance is impractical. The OLS-UCBV efficiently utilizes covariance structure, showing theoretically and empirically that it outperforms the baseline methods, like ESCB-C, in many structured settings. The gap-dependent bounds captured in this paper are competitive with existing works, demonstrating optimization leveraging covariance-induced exploration, thereby refining the performance in semi-bandit scenarios.

The analysis exploits martingale techniques and peeling tricks to establish key concentration bounds that are leveraged to derive regret upper bounds. Furthermore, the author's proposed methodology significantly reduces computational burden without compromising the regret upper bound quality compared to previous approaches, notably correcting for action estimation in high-dimensional and correlated environments.

Theoretical and Practical Implications

The application of covariance estimation improves decision-making precision by embracing the stochastic relationship between items, which is pivotal in environments like online advertising and network routing. This variance-adaptive approach surpasses prior variance-blind models, offering methodological advancements pertinent to semi-bandit literature. In practical implementations, the OLS-UCBV algorithm's computational requirements are lower than previous state-of-the-art algorithms while maintaining or improving probabilistic guarantees.

Conclusion and Future Directions

This paper propels the field of combinatorial semi-bandits forward by addressing significant theoretical and practical limitations of current algorithms. Future work can expand upon these foundations by incorporating dynamic environments or considering more general reward structures, such as those found in contextual combinatorial bandits. Extending these findings to account for non-linear reward feedback or investigating tighter bounds by integrating negative covariance constraints could further elevate the utility of variance-adaptive algorithms in broader applications.

This work provides an insightful and comprehensive framework, priming it for further exploration both theoretically and in numerous application domains.