Large-scale nonconvex optimization problems are ubiquitous in modern machine learning, and among practitioners interested in solving them, Stochastic Gradient Descent (SGD) reigns supreme. We revisit the analysis of SGD in the nonconvex setting and propose a new variant of the recently introduced expected smoothness assumption which governs the behaviour of the second moment of the stochastic gradient. We show that our assumption is both more general and more reasonable than assumptions made in all prior work. Moreover, our results yield the optimal $\mathcal{O}(\varepsilon^{-4})$ rate for finding a stationary point of nonconvex smooth functions, and recover the optimal $\mathcal{O}(\varepsilon^{-1})$ rate for finding a global solution if the Polyak-{\L}ojasiewicz condition is satisfied. We compare against convergence rates under convexity and prove a theorem on the convergence of SGD under Quadratic Functional Growth and convexity, which might be of independent interest. Moreover, we perform our analysis in a framework which allows for a detailed study of the effects of a wide array of sampling strategies and minibatch sizes for finite-sum optimization problems. We corroborate our theoretical results with experiments on real and synthetic data.

本篇论文使用类似于期望光滑性假设的新方法来研究随机梯度下降法在非凸优化中的收敛率，并在考虑多种采样策略和小批量大小的情况下，探讨有限和优化问题的影响。

非凸世界中SGD的更好理论