A Hessian-Aware Stochastic Differential Equation for Modelling SGD

• URL: http://arxiv.org/abs/2405.18373v2
• Date: Mon, 5 Aug 2024 22:25:10 GMT
• Title: A Hessian-Aware Stochastic Differential Equation for Modelling SGD
• Authors: Xiang Li, Zebang Shen, Liang Zhang, Niao He,
• Abstract summary: Hessian-Aware Modified Equation (HA-SME) is an approximation SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. For quadratic objectives, HA-SME is proved to be the first SDE model that recovers exactly the SGD dynamics in the distributional sense.
• Score: 28.974147174627102
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME matches the order-best approximation error guarantee among existing SDE models in the literature, while achieving a significantly reduced dependence on the smoothness parameter of the objective. Further, for quadratic objectives, under mild conditions, HA-SME is proved to be the first SDE model that recovers exactly the SGD dynamics in the distributional sense. Consequently, when the local landscape near a stationary point can be approximated by quadratics, HA-SME is expected to accurately predict the local escaping behaviors of SGD.

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