Stochastic Optimization with Importance Sampling

TL;DR

This paper explores importance sampling in stochastic optimization algorithms like prox-SGD and prox-SDCA, demonstrating that it reduces variance and improves convergence rates through theoretical analysis and experiments.

Contribution

It introduces importance sampling schemes for prox-SGD and prox-SDCA, significantly enhancing their convergence performance over uniform sampling.

Findings

Importance sampling reduces variance in stochastic gradients.

Theoretical convergence rates are improved with importance sampling.

Experimental results verify the theoretical benefits of the proposed methods.

Abstract

Uniform sampling of training data has been commonly used in traditional stochastic optimization algorithms such as Proximal Stochastic Gradient Descent (prox-SGD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although uniform sampling can guarantee that the sampled stochastic quantity is an unbiased estimate of the corresponding true quantity, the resulting estimator may have a rather high variance, which negatively affects the convergence of the underlying optimization procedure. In this paper we study stochastic optimization with importance sampling, which improves the convergence rate by reducing the stochastic variance. Specifically, we study prox-SGD (actually, stochastic mirror descent) with importance sampling and prox-SDCA with importance sampling. For prox-SGD, instead of adopting uniform sampling throughout the training process, the proposed algorithm employs importance sampling to minimize the variance of the stochastic gradient. For prox-SDCA, the proposed importance sampling scheme aims to achieve higher expected dual value at each dual coordinate ascent step. We provide extensive theoretical analysis to show that the convergence rates with the proposed importance sampling methods can be significantly improved under suitable conditions both for prox-SGD and for prox-SDCA. Experiments are provided to verify the theoretical analysis.