Implicit regularization refers to the tendency of local search algorithms to converge to low-dimensional solutions, even when such structures are not explicitly enforced. Despite its ubiquity, the mechanism underlying this behavior remains poorly understood, particularly in over-parameterized settings. We analyze gradient descent dynamics and identify three conditions under which it converges to second-order stationary points within an implicit low-dimensional region: (i) suitable initialization, (ii) efficient escape from saddle points, and (iii) sustained proximity to the region. We show that these can be achieved through infinitesimal perturbations and a small deviation rate. Building on this, we introduce Infinitesimally Perturbed Gradient Descent (IPGD), which satisfies these conditions under mild assumptions. We provide theoretical guarantees for IPGD in over-parameterized matrix sensing and empirical evidence of its broader applicability.
@article{arxiv.2505.17304,
title = {Understanding the Implicit Regularization of Gradient Descent in Over-parameterized Models},
author = {Jianhao Ma and Geyu Liang and Salar Fattahi},
journal= {arXiv preprint arXiv:2505.17304},
year = {2025}
}