How can we make use of information parallelism in online decision making problems while efficiently balancing the exploration-exploitation trade-off? In this paper, we introduce a batch Thompson Sampling framework for two canonical online decision making problems, namely, stochastic multi-arm bandit and linear contextual bandit with finitely many arms. Over a time horizon T, our batch Thompson Sampling policy achieves the same (asymptotic) regret bound of a fully sequential one while carrying out only O(log T) batch queries. To achieve this exponential reduction, i.e., reducing the number of interactions from T to O(log T), our batch policy dynamically determines the duration of each batch in order to balance the exploration-exploitation trade-off. We also demonstrate experimentally that dynamic batch allocation dramatically outperforms natural baselines such as static batch allocations.

Unlike classical control theory, such as Linear Quadratic Control (LQC), real-world control problems are highly complex. These problems often involve adversarial perturbations, bandit feedback models, and non-quadratic, adversarially chosen cost functions. A fundamental yet unresolved question is whether optimal regret can be achieved for these general control problems. The standard approach to addressing this problem involves a reduction to bandit convex optimization with memory. In the bandit setting, constructing a gradient estimator with low variance is challenging due to the memory structure and non-quadratic loss functions.In this paper, we provide an affirmative answer to this question. Our main contribution is an algorithm that achieves an $\tilde{O}(\sqrt{T})$ optimal regret for bandit non-stochastic control with strongly-convex and smooth cost functions in the presence of adversarial perturbations, improving the previously known $\tilde{O}(T^{2/3})$ regret bound from \citep{cassel2020bandit}. Our algorithm overcomes the memory issue by reducing the problem to Bandit Convex Optimization (BCO) without memory and addresses general strongly-convex costs using recent advancements in BCO from \citep{suggala2024second}. Along the way, we develop an improved algorithm for BCO with memory, which may be of independent interest.

As one of the most mainstream paradigms for sequential decision-making, RL has extensive applications in many real-world problems (Kober et al., 2013; Mnih et al.,

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In this paper, we provide an affirmative answer to this question.

Various results suggest that achieving optimal regret in the fully adversarial sleeping experts problem is computationally hard.