New SGD Bounds for Markovian Noise Achieve Optimal Mixing

Researchers have developed new theoretical guarantees for Stochastic Gradient Descent (SGD) when the data samples used to compute gradients exhibit Markovian dependencies, a common scenario in many real-world applications. Specifically, the work focuses on objectives satisfying the Polyak-Łojasiewicz (PL) condition, which is a weaker but still powerful assumption for convergence. The study closes a significant gap in the understanding of SGD's performance under Markovian noise. Previous high-probability bounds for light-tailed settings were less optimistic than expectation bounds, scaling quadratically with mixing time. This new research establishes uniform high-probability guarantees that scale linearly with mixing time, proving this linear dependence is optimal. Furthermore, the framework is extended to handle heavy-tailed Markovian gradients, which are prevalent in financial data or network traffic. A novel clipped block method is introduced to mitigate Markovian bias, achieving optimal high-probability stochastic error bounds. This work provides a tight characterization of optimal mixing time and effective-sample-size dependence for robust SGD in these complex settings.