Bayesian Optimization (BO) is an effective framework for black-box optimization.

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An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint.

We consider the problem of regret minimization in non-parametric stochastic bandits. When the rewards are known to be bounded from above, there exists asymptotically optimal algorithms, with asymptotic regret depending on an infi-mum of Kullback-Leibler divergences (KL).

In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy function to establish an $O(K^{-2/3})$ bound on its average regret after $K$ iterations. This quantitatively verifies the algorithm's better behavior than the simultaneous version of mirror descent algorithm, which is known to diverge and yields an $O(K^{-1/2})$ average regret bound. In the special case of an unconstrained setting, our results recover the behavior of alternating gradient descent algorithm for zero-sum games which was studied in (Bailey et al., COLT 2020).

We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $λ$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

We consider the problem of regret minimization in non-parametric stochastic bandits. When the rewards are known to be bounded from above, there exists asymptotically optimal algorithms, with asymptotic regret depending on an infi-mum of Kullback-Leibler divergences (KL).