Implicit regularisation in stochastic gradient descent: from single-objective to two-player games
This work addresses a methodological bottleneck for researchers studying optimisation dynamics in deep learning and game theory, offering incremental improvements in analytical tools.
The paper tackles the limitation of backward error analysis (BEA) in revealing implicit regularisation effects in gradient-based optimisers by introducing a novel approach that constructs continuous-time flows with gradient vector fields, enabling the discovery of previously unknown regularisation effects in stochastic gradient descent and two-player games.
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.