Rigorous dynamical mean field theory for stochastic gradient descent methods

Autor:	Gerbelot, Cedric, Troiani, Emanuele, Mignacco, Francesca, Krzakala, Florent, Zdeborova, Lenka
Rok vydání:	2022
Předmět:	Mathematical Physics Computer Science - Information Theory Computer Science - Machine Learning Statistics - Machine Learning
Zdroj:	SIAM Journal on Mathematics of Data Science 6.2 (2024): 400-427
Druh dokumentu:	Working Paper
DOI:	10.1137/23M159438
Popis:	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. Comment: 40 pages, 4 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2210.06591 Zobrazit plný text záznamu View this record from Arxiv