# Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process

arXiv: https://arxiv.org/abs/1904.09080 · HTML: https://mlsanity.com/paper/1904.09080/

## Key facts

- Cited by 123 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2019.

## Built on by (most cited first, 30 of 123)

- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [Towards Understanding Sharpness-Aware Minimization](https://mlsanity.com/paper/2206.06232/) (2022)
- [Label Noise SGD Provably Prefers Flat Global Minimizers](https://mlsanity.com/paper/2106.06530/) (2021)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [What Happens after SGD Reaches Zero Loss? --A Mathematical Framework](https://mlsanity.com/paper/2110.06914/) (2021)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction](https://mlsanity.com/paper/2206.07085/) (2022)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Same Pre-training Loss, Better Downstream: Implicit Bias Matters for Language Models](https://mlsanity.com/paper/2210.14199/) (2022)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Transformers as Support Vector Machines](https://mlsanity.com/paper/2308.16898/) (2023)
- [Why Do We Need Weight Decay in Modern Deep Learning?](https://mlsanity.com/paper/2310.04415/) (2023)
- [SGD with Large Step Sizes Learns Sparse Features](https://mlsanity.com/paper/2210.05337/) (2022)
- [DR3: Value-Based Deep Reinforcement Learning Requires Explicit Regularization](https://mlsanity.com/paper/2112.04716/) (2021)
- [How Does Sharpness-Aware Minimization Minimize Sharpness?](https://mlsanity.com/paper/2211.05729/) (2022)
- [The alignment property of SGD noise and how it helps select flat minima: A stability analysis](https://mlsanity.com/paper/2207.02628/) (2022)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Max-Margin Token Selection in Attention Mechanism](https://mlsanity.com/paper/2306.13596/) (2023)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [The Implicit Regularization of Dynamical Stability in Stochastic Gradient Descent](https://mlsanity.com/paper/2305.17490/) (2023)
- [Beyond the Quadratic Approximation: the Multiscale Structure of Neural Network Loss Landscapes](https://mlsanity.com/paper/2204.11326/) (2022)
- [Anticorrelated Noise Injection for Improved Generalization](https://mlsanity.com/paper/2202.02831/) (2022)
- [Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing](https://mlsanity.com/paper/2301.11500/) (2023)
- [Strength of Minibatch Noise in SGD](https://mlsanity.com/paper/2102.05375/) (2021)
- [Implicit Bias of Gradient Descent on Reparametrized Models: On Equivalence to Mirror Descent](https://mlsanity.com/paper/2207.04036/) (2022)
- [Why neural networks find simple solutions: the many regularizers of geometric complexity](https://mlsanity.com/paper/2209.13083/) (2022)
- [Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks](https://mlsanity.com/paper/2306.04251/) (2023)
- [Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalization](https://mlsanity.com/paper/2307.11007/) (2023)
- [Sharp Bounds for Federated Averaging (Local SGD) and Continuous Perspective](https://mlsanity.com/paper/2111.03741/) (2021)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)

## Abstract (arXiv)

We consider networks, trained via stochastic gradient descent to minimize ℓ2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the data points, of the squared ℓ2 norm of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width, depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models.