Further Optimal Regret Bounds for Thompson Sampling (1209.3353v1)

Abstract: Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state of the art methods. In this paper, we provide a novel regret analysis for Thompson Sampling that simultaneously proves both the optimal problem-dependent bound of $(1+\epsilon)\sum_i \frac{\ln T}{\Delta_i}+O(\frac{N}{\epsilon^2})$ and the first near-optimal problem-independent bound of $O(\sqrt{NT\ln T})$ on the expected regret of this algorithm. Our near-optimal problem-independent bound solves a COLT 2012 open problem of Chapelle and Li. The optimal problem-dependent regret bound for this problem was first proven recently by Kaufmann et al. [ALT 2012]. Our novel martingale-based analysis techniques are conceptually simple, easily extend to distributions other than the Beta distribution, and also extend to the more general contextual bandits setting [Manuscript, Agrawal and Goyal, 2012].

• The paper establishes an optimal problem-dependent regret bound that matches the asymptotic lower bounds in stochastic multi-armed bandit literature.
• It derives a near-optimal problem-independent bound of O(√(NT ln T)), solving a long-standing open question in the field.
• A novel martingale-based approach simplifies the analysis and extends the applicability of Thompson Sampling in real-world sequential decision-making scenarios.

Further Optimal Regret Bounds for Thompson Sampling

The paper "Further Optimal Regret Bounds for Thompson Sampling" by Shipra Agrawal and Navin Goyal delivers a theoretical analysis of Thompson Sampling (TS) for the stochastic multi-armed bandit problem and provides regret bounds of significant interest. The work addresses both problem-dependent and problem-independent bounds, offering a cohesive and thorough understanding of TS in bandit settings.

Key Contributions

The authors present:

1. Problem-Dependent Regret Bounds:
   • The paper establishes an optimal problem-dependent regret bound of $$(1+\epsilon)\sum_i \frac{\ln T}{\Delta_i} + O\left(\frac{N}{\epsilon^2}\right)$$ for Thompson Sampling. This bound matches the asymptotic lower bound for problem-dependent regrets as previously identified in stochastic bandit literature.
2. Problem-Independent Regret Bounds:
   • The authors solve the open question posed by Chapelle and Li by deriving a near-optimal problem-independent bound of $O(\sqrt{NT\ln T})$. This result places TS on par with the best-known problem-independent bounds for variant algorithms like Upper Confidence Bound (UCB).

Analytical Approaches and Implications

The analysis leverages a novel martingale-based approach, simplifying previous methodologies. By analyzing the probability distributions involved using the KL-divergence measures, the findings offer a robust framework applicable beyond the Beta distribution, extending to contextual bandit settings.

Theoretical Significance

• Optimal Bound Achievements: The results are aligned with the lower bounds established by Lai and Robbins, enhancing the theoretical robustness of TS against UCB-type algorithms.
• Technique Simplicity: The martingale approach is praised for its conceptual simplicity, potentially enabling broader applications in related domains by offering an extensible analysis framework.

Practical Implications

• Increased Applicability: With verified theoretical underpinnings, TS can be confidently used in various domains such as clinical trials, online recommendation systems, and adaptive ad placements.
• Comparative Performance: Empirical studies have already shown TS performs favorably compared to UCB, and these proofs further solidify its standing.

Speculation on Future AI Developments

The firm grounding of TS in both problem-dependent and independent settings indicates its potential as a cornerstone for future advancements in sequential decision-making algorithms. Future research may explore:

• Contextual Bandits: Extensions of the proposed framework to more complex, real-world scenarios involving contextual data.
• Beyond Beta Distributions: Further investigation into the adaptability to more diverse distribution families, potentially broadening the applicability in sophisticated AI systems.

Overall, this paper enriches the understanding of Thompson Sampling, providing a comprehensive and mathematically rigorous treatment that bridges its theoretical and practical potential. This work will likely serve as a cornerstone reference in both academic research and applied algorithm development for multi-armed bandit problems.