# Limitation of Characterizing Implicit Regularization by Data-independent Functions

arXiv: https://arxiv.org/abs/2201.12198 · HTML: https://mlsanity.com/paper/2201.12198/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 28 Jan 2022.

## Builds on (most cited first, 27 of 27)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [Regularization for Deep Learning: A Taxonomy](https://mlsanity.com/paper/1710.10686/) (2017)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [Phase diagram for two-layer ReLU neural networks at infinite-width limit](https://mlsanity.com/paper/2007.07497/) (2020)
- [The Implicit Bias of Depth: How Incremental Learning Drives Generalization](https://mlsanity.com/paper/1909.12051/) (2019)
- [Implicit Regularization in ReLU Networks with the Square Loss](https://mlsanity.com/paper/2012.05156/) (2020)
- [Gradient Descent for Deep Matrix Factorization: Dynamics and Implicit Bias towards Low Rank](https://mlsanity.com/paper/2011.13772/) (2020)
- [A type of generalization error induced by initialization in deep neural networks](https://mlsanity.com/paper/1905.07777/) (2019)
- [Deep frequency principle towards understanding why deeper learning is faster](https://mlsanity.com/paper/2007.14313/) (2020)
- [Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness](https://mlsanity.com/paper/1905.11427/) (2019)
- [Can Implicit Bias Explain Generalization? Stochastic Convex Optimization as a Case Study](https://mlsanity.com/paper/2003.06152/) (2020)
- [On the exact computation of linear frequency principle dynamics and its generalization](https://mlsanity.com/paper/2010.08153/) (2020)
- [Linear Frequency Principle Model to Understand the Absence of Overfitting in Neural Networks](https://mlsanity.com/paper/2102.00200/) (2021)

## Abstract (arXiv)

In recent years, understanding the implicit regularization of neural networks (NNs) has become a central task in deep learning theory. However, implicit regularization is itself not completely defined and well understood. In this work, we attempt to mathematically define and study implicit regularization. Importantly, we explore the limitations of a common approach to characterizing implicit regularization using data-independent functions. We propose two dynamical mechanisms, i.e., Two-point and One-point Overlapping mechanisms, based on which we provide two recipes for producing classes of one-hidden-neuron NNs that provably cannot be fully characterized by a type of or all data-independent functions. Following the previous works, our results further emphasize the profound data dependency of implicit regularization in general, inspiring us to study in detail the data dependency of NN implicit regularization in the future.