NAG-GS: semi-implicit, accelerated and robust stochastic optimizer.

Valentin Leplat, Daniil Merkulov, Aleksandr Katrutsa, Daniel Bershatsky, Ivan Oseledets

Keywords: Accelerated gradient methods, stochastic optimization, stochastic differential equations, semi-implicit solver, convergence analysis, deep neural networks

Abstract: Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal step size (or learning rate) in terms of rate of convergence while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. We show that NAG-GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, and Transformers in the frame of the GLUE benchmark.

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