将Stein方法应用于在线线性优化：基于中心极限定理的最优权衡
Operationalizing Stein's Method for Online Linear Optimization: CLT-Based Optimal Tradeoffs
AI总结：
利用Stein方法改进在线线性优化算法，实现基于中心极限定理的最优权衡与更精确的性能保证。
摘要：
对抗性在线线性优化（OLO）本质上是关于在未知对手难度之间做出性能权衡。在单维固定时间OLO的有界域设置中，自Cover（1966）以来，已观察到可实现的权衡由概率不等式所支配，这些描述性结果可通过动态规划转换为算法，但该方法在计算上并不高效。我们通过展示Stein方法——一个用于概率极限定理证明的古典框架——可以被转化为计算高效的OLO算法，从而解决这一限制。关联的遗憾和总损失上界是“加法锐利”的，意味着它们超越了传统的大O最优性，通过加法的低阶项匹配基于正态近似下的下界。我们的构造受到Röllin（2018）对Wasserstein鞅中心极限定理（CLT）惊人清晰证明的启发。几种具体的益处可以从这一通用技术中获得。首先，与OGD和MWU的总损失上界相比，所提出的算法在相同计算复杂度下改进了总损失上界。其次，我们的算法可以实现总损失和比较器最大遗憾之间的连续最优两点权衡，优于参数自由在线学习的先前工作。第三，通过允许对手在无界支持上随机化，我们实现了具有噪声反馈的OLO的尖锐期望性能保证。
英文摘要：
Adversarial online linear optimization (OLO) is essentially about making performance tradeoffs with respect to the unknown difficulty of the adversary. In the setting of one-dimensional fixed-time OLO on a bounded domain, it has been observed since Cover (1966) that achievable tradeoffs are governed by probabilistic inequalities, and these descriptive results can be converted into algorithms via dynamic programming, which, however, is not computationally efficient. We address this limitation by showing that Stein's method, a classical framework underlying the proofs of probabilistic limit theorems, can be operationalized as computationally efficient OLO algorithms. The associated regret and total loss upper bounds are "additively sharp", meaning that they surpass the conventional big-O optimality and match normal-approximation-based lower bounds by additive lower order terms. Our construction is inspired by the remarkably clean proof of a Wasserstein martingale central limit theorem (CLT) due to Röllin (2018). Several concrete benefits can be obtained from this general technique. First, with the same computational complexity, the proposed algorithm improves upon the total loss upper bounds of online gradient descent (OGD) and multiplicative weight update (MWU). Second, our algorithm can realize a continuum of optimal two-point tradeoffs between the total loss and the maximum regret over comparators, improving upon prior works in parameter-free online learning. Third, by allowing the adversary to randomize on an unbounded support, we achieve sharp in-expectation performance guarantees for OLO with noisy feedback.
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