Understanding the generalization properties of heavy-tailed stochastic optimization algorithms has attracted increasing attention over the past years. While illuminating interesting aspects of stochastic optimizers by using heavy-tailed stochastic differential equations as proxies, prior works either provided expected generalization bounds, or introduced non-computable information theoretic terms. Addressing these drawbacks, in this work, we prove high-probability generalization bounds for heavy-tailed SDEs which do not contain any nontrivial information theoretic terms. To achieve this goal, we develop new proof techniques based on estimating the entropy flows associated with the so-called fractional Fokker-Planck equation (a partial differential equation that governs the evolution of the distribution of the corresponding heavy-tailed SDE). In addition to obtaining high-probability bounds, we show that our bounds have a better dependence on the dimension of parameters as compared to prior art. Our results further identify a phase transition phenomenon, which suggests that heavy tails can be either beneficial or harmful depending on the problem structure. We support our theory with experiments conducted in a variety of settings.

翻译：理解重尾随机优化算法的泛化特性在过去几年中受到越来越多的关注。尽管先前研究通过使用重尾随机微分方程作为代理模型揭示了随机优化器的一些有趣特性，但这些工作要么提供了期望泛化界，要么引入了不可计算的信息论项。为克服这些缺陷，本研究证明了不包含任何非平凡信息论项的重尾随机微分方程的高概率泛化界。为实现这一目标，我们开发了基于分数阶Fokker-Planck方程（该偏微分方程控制着相应重尾随机微分方程分布演化）熵流估计的新证明技术。除了获得高概率界外，我们还证明相较于现有研究，我们的界对参数维度具有更好的依赖性。研究结果进一步揭示了相变现象，表明重尾特性根据问题结构可能有益或有害。我们在多种设置下进行的实验为理论提供了支持。