Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance

• URL: http://arxiv.org/abs/2012.04002v2
• Date: Fri, 16 Apr 2021 14:43:43 GMT
• Title: Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance
• Authors: A. Barakat, P. Bianchi, W. Hachem, and Sh. Schechtman
• Abstract summary: In this paper, a general optimization procedure is studied, unifying several variants of the gradient descent such as, among others, the heavy ball method, the Nesterov Accelerated Gradient (S-NAG), and the widely used Adam method. The avoidance is studied as a noisy discretization of a non-autonomous ordinary differential equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, a general stochastic optimization procedure is studied, unifying several variants of the stochastic gradient descent such as, among others, the stochastic heavy ball method, the Stochastic Nesterov Accelerated Gradient algorithm (S-NAG), and the widely used Adam algorithm. The algorithm is seen as a noisy Euler discretization of a non-autonomous ordinary differential equation, recently introduced by Belotto da Silva and Gazeau, which is analyzed in depth. Assuming that the objective function is non-convex and differentiable, the stability and the almost sure convergence of the iterates to the set of critical points are established. A noteworthy special case is the convergence proof of S-NAG in a non-convex setting. Under some assumptions, the convergence rate is provided under the form of a Central Limit Theorem. Finally, the non-convergence of the algorithm to undesired critical points, such as local maxima or saddle points, is established. Here, the main ingredient is a new avoidance of traps result for non-autonomous settings, which is of independent interest.

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