Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm

TL;DR

This paper improves the theoretical understanding of stochastic gradient descent (SGD) convergence rates for smooth, strongly convex functions, highlighting the importance of importance sampling and connecting SGD with the randomized Kaczmarz algorithm.

Contribution

It provides a tighter finite-sample convergence guarantee for SGD, introduces importance sampling for further improvements, and establishes a novel link between SGD and the randomized Kaczmarz algorithm.

Findings

Linear convergence rate with dependence on L/μ

Importance sampling enhances convergence

Connection between SGD and Kaczmarz algorithm

Abstract

We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning $(L/\mu)^2$ (where $L$ is a bound on the smoothness and $\mu$ on the strong convexity) to a linear dependence on $L/\mu$. Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate.