On momentum methods and acceleration in stochastic optimization

  • Praneeth Netrapalli | Microsoft Research

It is well known that momentum gradient methods (e.g., Polyak’s heavy ball, Nesterov’s acceleration) yield significant improvements over vanilla gradient descent in deterministic optimization (i.e., where we have access to exact gradient of the function to be minimized). However, there is widespread sentiment that these momentum methods are not effective for the purposes of stochastic optimization due to their instability and error accumulation. Numerous works have attempted to quantify these instabilities in the face of either statistical or non-statistical errors (Paige, 1971; Proakis, 1974; Polyak, 1987; Greenbaum, 1989; Roy and Shynk, 1990; Sharma et al., 1998; d’Aspremont, 2008; Devolder et al., 2013, 2014; Yuan et al., 2016) but a precise understanding is lacking. This work considers these issues for the special case of stochastic approximation for the linear least squares regression problem, and shows that: 1. classical momentum methods (heavy ball and Nesterov’s acceleration) indeed do not offer any improvement over stochastic gradient descent, and 2. introduces an accelerated stochatic gradient method that provably achieves the minimax optimal statistical risk faster than stochastic gradient descent (and classical momentum methods). Critical to the analysis is a sharp characterization of accelerated stochastic gradient descent as a stochastic process. While the results are rigorously established for the special case of linear least squares regression, experiments suggest that the conclusions hold for the training of deep neural networks.

Joint work with Prateek Jain, Sham M. Kakade, Rahul Kidambi and Aaron Sidford.

Speaker Details

Praneeth Netrapalli is a POST DOC RESEARCHER at Microsoft Research. Earlier he was a PhD student in the ECE department at UT Austin, where he was advised by Prof. Sujay Sanghavi. His research focuses on designing and understanding practical algorithms for solving machine learning problems. He obtained MS from UT Austin, and B. Tech from IIT Bombay, both in Electrical Engineering. His prior experience includes two years as a quantitative analyst at Goldman Sachs where he worked on pricing and evaluating risk on complex financial products.

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