Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures
This work addresses the problem of efficient decision-making in sequential settings like hyperparameter tuning for machine learning practitioners, but it is incremental as it builds on existing optimism-based methods with tighter confidence bounds.
The paper tackles the stochastic linear bandit problem by developing improved algorithms with worst-case regret guarantees, using novel tail bounds for adaptive martingale mixtures to construct tighter confidence sequences, which lead to competitive regret and better performance in hyperparameter tuning tasks.
We present improved algorithms with worst-case regret guarantees for the stochastic linear bandit problem. The widely used "optimism in the face of uncertainty" principle reduces a stochastic bandit problem to the construction of a confidence sequence for the unknown reward function. The performance of the resulting bandit algorithm depends on the size of the confidence sequence, with smaller confidence sets yielding better empirical performance and stronger regret guarantees. In this work, we use a novel tail bound for adaptive martingale mixtures to construct confidence sequences which are suitable for stochastic bandits. These confidence sequences allow for efficient action selection via convex programming. We prove that a linear bandit algorithm based on our confidence sequences is guaranteed to achieve competitive worst-case regret. We show that our confidence sequences are tighter than competitors, both empirically and theoretically. Finally, we demonstrate that our tighter confidence sequences give improved performance in several hyperparameter tuning tasks.