# Understanding the Implicit Regularization of Gradient Descent in Over-parameterized Models

arXiv: https://arxiv.org/abs/2505.17304 · HTML: https://mlsanity.com/paper/2505.17304/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 22 May 2025.

## Abstract (arXiv)

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.