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We prove that various stochastic gradient descent methods, including thestochastic gradient descent (SGD), stochastic heavy-ball (SHB), and stochasticNesterov's accelerated gradient (SNAG) methods, almost surely avoid any strictsaddle manifold. To the best of our knowledge, this is the first time suchresults are obtained for SHB and SNAG methods. Moreover, our analysis expandsupon previous studies on SGD by removing the need for bounded gradients of theobjective function and uniformly bounded noise. Instead, we introduce a morepractical local boundedness assumption for the noisy gradient, which isnaturally satisfied in empirical risk minimization problems typically seen intraining of neural networks.

Almost Sure Saddle Avoidance of Stochastic Gradient Methods without the Bounded Gradient Assumption