We develop a general theory to optimize the frequentist regret for sequential learning problems, where efficient bandit and reinforcement learning algorithms can be derived from unified Bayesian principles. We propose a novel optimization approach to generate "algorithmic beliefs" at each round, and use Bayesian posteriors to make decisions. The optimization objective to create "algorithmic beliefs," which we term "Algorithmic Information Ratio," represents an intrinsic complexity measure that effectively characterizes the frequentist regret of any algorithm. To the best of our knowledge, this is the first systematical approach to make Bayesian-type algorithms prior-free and applicable to adversarial settings, in a generic and optimal manner. Moreover, the algorithms are simple and often efficient to implement. As a major application, we present a novel algorithm for multi-armed bandits that achieves the "best-of-all-worlds" empirical performance in the stochastic, adversarial, and non-stationary environments. And we illustrate how these principles can be used in linear bandits, bandit convex optimization, and reinforcement learning.

翻译：我们发展了一套通用理论，用于优化序贯学习问题中的频率学派遗憾值，其中高效的赌博机与强化学习算法可统一源于贝叶斯原理。我们提出了一种新型优化方法，在每个轮次生成“算法信念”，并利用贝叶斯后验进行决策。用于生成“算法信念”的优化目标——我们称之为“算法信息比”——代表了刻画任意算法频率学派遗憾值的内在复杂度度量。据我们所知，这是首个以通用且最优的方式，使贝叶斯类算法摆脱先验依赖并适用于对抗性环境的系统化方法。此外，这些算法简单且通常能高效实现。作为重要应用，我们提出了一种新型多臂赌博机算法，在随机、对抗及非平稳环境中实现了“全能型”实证性能。同时，我们阐释了这些原理如何应用于线性赌博机、赌博机凸优化以及强化学习领域。