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来优化序贯学习问题中的频率派遗憾值，使得高效的赌博机与强化学习算法可从统一的贝叶斯原理中推导得出。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。用于生成"算法信念"的优化目标（我们称之为"算法信息比"）代表了一种内在的复杂度度量，能够有效表征任意算法的频率派遗憾值。据我们所知，这是首个以通用且最优的方式使贝叶斯类型算法实现无先验假设并适用于对抗性设置的系统性方法。此外，该算法实现简单且通常计算高效。作为重要应用，我们提出了一种针对多臂赌博机的新算法，在随机、对抗和非平稳环境中实现了"所有世界最优"的实证性能。同时，我们展示了这些原理如何在线性赌博机、赌博机凸优化和强化学习中应用。