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bandit和强化学习算法可源自统一的贝叶斯原理。我们提出了一种新颖的优化方法，在每一轮生成"算法信念"，并利用贝叶斯后验进行决策。用于创建"算法信念"的优化目标——我们称之为"算法信息比"——代表了一种内在复杂度度量，能够有效刻画任何算法的频率学派遗憾值。据我们所知，这是首个以通用且最优的方式，系统性地使贝叶斯类型算法摆脱先验依赖并适用于对抗性环境的方法。此外，这些算法实现简单且通常高效。作为一项重要应用，我们提出了一种多臂赌博机算法，在随机、对抗和非平稳环境中实现了"最佳全能"经验性能。我们还阐述了这些原理在线性赌博机、赌博机凸优化及强化学习中的应用。