A Non-asymptotic comparison of SVRG and SGD: tradeoffs between compute and speed

Qingru Zhang, Yuhuai Wu, Fartash Faghri, Tianzong Zhang, Jimmy Ba

TL;DR: Non-asymptotic analysis of SGD and SVRG, showing the strength of each algorithm in convergence speed and computational cost, in both under-parametrized and over-parametrized settings.

Abstract: Stochastic gradient descent (SGD), which trades off noisy gradient updates for computational efficiency, is the de-facto optimization algorithm to solve large-scale machine learning problems. SGD can make rapid learning progress by performing updates using subsampled training data, but the noisy updates also lead to slow asymptotic convergence. Several variance reduction algorithms, such as SVRG, introduce control variates to obtain a lower variance gradient estimate and faster convergence. Despite their appealing asymptotic guarantees, SVRG-like algorithms have not been widely adopted in deep learning. The traditional asymptotic analysis in stochastic optimization provides limited insight into training deep learning models under a fixed number of epochs. In this paper, we present a non-asymptotic analysis of SVRG under a noisy least squares regression problem. Our primary focus is to compare the exact loss of SVRG to that of SGD at each iteration t. We show that the learning dynamics of our regression model closely matches with that of neural networks on MNIST and CIFAR-10 for both the underparameterized and the overparameterized models. Our analysis and experimental results suggest there is a trade-off between the computational cost and the convergence speed in underparametrized neural networks. SVRG outperforms SGD after a few epochs in this regime. However, SGD is shown to always outperform SVRG in the overparameterized regime.

Keywords: variance reduction, non-asymptotic analysis, trade-off, computational cost, convergence speed