Implicit regularisation in stochastic gradient descent: from single-objective to two-player games

TL;DR

This paper introduces a novel approach using backward error analysis to construct gradient-compatible continuous-time flows, revealing new implicit regularisation effects in stochastic gradient descent and two-player games.

Contribution

It develops a new method to utilize BEA for constructing gradient-representable flows, enabling discovery of previously unknown implicit regularisers in complex settings.

Findings

Identified implicit regularisation effects from multiple SGD steps with exact data batches.

Extended analysis to differentiable two-player games.

Provided a method to construct flows with gradient vector fields using BEA.

Abstract

Recent years have seen many insights on deep learning optimisation being brought forward by finding implicit regularisation effects of commonly used gradient-based optimisers. Understanding implicit regularisation can not only shed light on optimisation dynamics, but it can also be used to improve performance and stability across problem domains, from supervised learning to two-player games such as Generative Adversarial Networks. An avenue for finding such implicit regularisation effects has been quantifying the discretisation errors of discrete optimisers via continuous-time flows constructed by backward error analysis (BEA). The current usage of BEA is not without limitations, since not all the vector fields of continuous-time flows obtained using BEA can be written as a gradient, hindering the construction of modified losses revealing implicit regularisers. In this work, we provide a novel approach to use BEA, and show how our approach can be used to construct continuous-time flows with vector fields that can be written as gradients. We then use this to find previously unknown implicit regularisation effects, such as those induced by multiple stochastic gradient descent steps while accounting for the exact data batches used in the updates, and in generally differentiable two-player games.