# Parallelizing Stochastic Gradient Descent for Least Squares Regression: mini-batching, averaging, and model misspecification

## TL;DR

This paper provides a detailed analysis of mini-batching and tail-averaging in stochastic gradient descent for least squares regression, demonstrating near-linear parallelization speedups and insights into noise effects on convergence.

## Contribution

It offers non-asymptotic excess risk bounds for averaging schemes, characterizes parallelization speedups, and analyzes the impact of noise on stepsize choices in SGD.

## Key findings

- Mini-batching reduces variance and enables near-linear parallel speedups.
- Tail-averaging decreases variance in the final iterate of SGD.
- The analysis shows how noise properties influence optimal stepsize choices.

## Abstract

This work characterizes the benefits of averaging schemes widely used in conjunction with stochastic gradient descent (SGD). In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD to decrease the variance in SGD's final iterate. This work presents non-asymptotic excess risk bounds for these schemes for the stochastic approximation problem of least squares regression. Furthermore, this work establishes a precise problem-dependent extent to which mini-batch SGD yields provable near-linear parallelization speedups over SGD with batch size one. This allows for understanding learning rate versus batch size tradeoffs for the final iterate of an SGD method. These results are then utilized in providing a highly parallelizable SGD method that obtains the minimax risk with nearly the same number of serial updates as batch gradient descent, improving significantly over existing SGD methods. A non-asymptotic analysis of communication efficient parallelization schemes such as model-averaging/parameter mixing methods is then provided. Finally, this work sheds light on some fundamental differences in SGD's behavior when dealing with agnostic noise in the (non-realizable) least squares regression problem. In particular, the work shows that the stepsizes that ensure minimax risk for the agnostic case must be a function of the noise properties. This paper builds on the operator view of analyzing SGD methods, introduced by Defossez and Bach (2015), followed by developing a novel analysis in bounding these operators to characterize the excess risk. These techniques are of broader interest in analyzing computational aspects of stochastic approximation.