# Rigorous dynamical mean field theory for stochastic gradient descent methods

arXiv: https://arxiv.org/abs/2210.06591 · HTML: https://mlsanity.com/paper/2210.06591/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2022.

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

- [Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer](https://mlsanity.com/paper/2605.07870/) (2026)

## Abstract (arXiv)

We prove closed-form equations for the exact high-dimensional asymptotics of a family of first order gradient-based methods, learning an estimator (e.g. M-estimator, shallow neural network, ...) from observations on Gaussian data with empirical risk minimization. This includes widely used algorithms such as stochastic gradient descent (SGD) or Nesterov acceleration. The obtained equations match those resulting from the discretization of dynamical mean-field theory (DMFT) equations from statistical physics when applied to gradient flow. Our proof method allows us to give an explicit description of how memory kernels build up in the effective dynamics, and to include non-separable update functions, allowing datasets with non-identity covariance matrices. Finally, we provide numerical implementations of the equations for SGD with generic extensive batch-size and with constant learning rates.