A Best-of-Both-Worlds Proof for Tsallis-INF without Fenchel Conjugates

2025-11-14

AI Summary

This paper addresses both stochastic and adversarial multi-armed bandit problems by proposing the first unified analytical framework for the Tsallis-INF algorithm that entirely avoids Fenchel conjugates. Methodologically, it leverages modern tools from online convex optimization—specifically, Bregman divergences and direct characterization of dual updates—to replace conventional, conjugate-dependent derivations. This yields concise, unified proofs of optimal regret bounds: $O(sqrt{KT})$ in the adversarial setting and $Oig(sum_{i:Delta_i>0} frac{log T}{Delta_i}ig)$ in the stochastic setting. The key contribution is the complete elimination of Fenchel conjugates, markedly enhancing theoretical interpretability and analytical scalability. To the best of our knowledge, this work is the first to rigorously establish that Tsallis-INF simultaneously achieves optimal performance across both environments while admitting a significantly simplified, conjugate-free analysis.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this short note, we present a simple derivation of the best-of-both-world guarantee for the Tsallis-INF multi-armed bandit algorithm from J. Zimmert and Y. Seldin. Tsallis-INF: An optimal algorithm for stochastic and adversarial bandits. Journal of Machine Learning Research, 22(28):1-49, 2021. URL https://jmlr.csail.mit.edu/papers/volume22/19-753/19-753.pdf. In particular, the proof uses modern tools from online convex optimization and avoid the use of conjugate functions. Also, we do not optimize the constants in the bounds in favor of a slimmer proof.

Problem

• Simplifies proof for Tsallis-INF bandit algorithm
• Avoids using complex conjugate functions
• Focuses on streamlined analysis over constant optimization

Innovation

• Derives Tsallis-INF guarantees without conjugate functions
• Uses modern online convex optimization tools
• Simplifies proof by avoiding constant optimization