# Convergence of Distributed Stochastic Variance Reduced Methods without Sampling Extra Data

## TL;DR

This paper analyzes the convergence of distributed stochastic variance reduced methods without extra data sampling, providing theoretical guarantees and insights into their behavior with data heterogeneity and nonconvexity.

## Contribution

It offers a comprehensive convergence analysis of distributed variance reduced algorithms, including accelerated and recursive variants, under data heterogeneity and nonconvex settings.

## Key findings

- Linear convergence for strongly convex losses.
- Convergence behavior depends on data homogeneity and regularization.
- Extension to nonconvex loss functions.

## Abstract

Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduced methods. In this paper, we consider a general framework that directly distributes popular stochastic variance reduced methods in the master/slave model, by assigning outer loops to the parameter server, and inner loops to worker machines. This framework is natural and friendly to implement, but its theoretical convergence is not well understood. We obtain a comprehensive understanding of algorithmic convergence with respect to data homogeneity by measuring the smoothness of the discrepancy between the local and global loss functions. We establish the linear convergence of distributed versions of a family of stochastic variance reduced algorithms, including those using accelerated and recursive gradient updates, for minimizing strongly convex losses. Our theory captures how the convergence of distributed algorithms behaves as the number of machines and the size of local data vary. Furthermore, we show that when the data are less balanced, regularization can be used to ensure convergence at a slower rate. We also demonstrate that our analysis can be further extended to handle nonconvex loss functions.