# Weight fluctuations in (deep) linear neural networks and a derivation of the inverse-variance flatness relation

arXiv: https://arxiv.org/abs/2311.14120 · HTML: https://mlsanity.com/paper/2311.14120/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 80 of its 130 references that are in the map.
- First posted on arXiv 23 Nov 2023.

## Builds on (most cited first, 30 of 80)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [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)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)

## Built on by (most cited first, 1 of 1)

- [Stochastic Gradient Descent in the Saddle-to-Saddle Regime of Deep Linear Networks](https://mlsanity.com/paper/2604.06366/) (2026)

## Abstract (arXiv)

We investigate the stationary (late-time) training regime of single- and two-layer underparameterized linear neural networks within the continuum limit of stochastic gradient descent (SGD) for synthetic Gaussian data. In the case of a single-layer network in the weakly underparameterized regime, the spectrum of the noise covariance matrix deviates notably from the Hessian, which can be attributed to the broken detailed balance of SGD dynamics. The weight fluctuations are in this case generally anisotropic, but effectively experience an isotropic loss. For an underparameterized two-layer network, we describe the stochastic dynamics of the weights in each layer and analyze the associated stationary covariances. We identify the inter-layer coupling as a distinct source of anisotropy for the weight fluctuations. In contrast to the single-layer case, the weight fluctuations are effectively subject to an anisotropic loss, the flatness of which is inversely related to the fluctuation variance. We thereby provide an analytical derivation of the recently observed inverse variance-flatness relation in a model of a deep linear neural network.