Two Facets of SDE Under an Information-Theoretic Lens: Generalization of SGD via Training Trajectories and via Terminal States
This provides theoretical insights into generalization for machine learning practitioners, but it is incremental as it builds on existing information-theoretic frameworks.
The authors tackled the problem of understanding SGD's generalization behavior by approximating it with stochastic differential equations (SDEs), resulting in trajectory-based and terminal-state-based generalization bounds that outperform prior work and show fast decay rates.
Stochastic differential equations (SDEs) have been shown recently to characterize well the dynamics of training machine learning models with SGD. When the generalization error of the SDE approximation closely aligns with that of SGD in expectation, it provides two opportunities for understanding better the generalization behaviour of SGD through its SDE approximation. Firstly, viewing SGD as full-batch gradient descent with Gaussian gradient noise allows us to obtain trajectory-based generalization bound using the information-theoretic bound from Xu and Raginsky [2017]. Secondly, assuming mild conditions, we estimate the steady-state weight distribution of SDE and use information-theoretic bounds from Xu and Raginsky [2017] and Negrea et al. [2019] to establish terminal-state-based generalization bounds. Our proposed bounds have some advantages, notably the trajectory-based bound outperforms results in Wang and Mao [2022], and the terminal-state-based bound exhibits a fast decay rate comparable to stability-based bounds.