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We prove closed-form equations for the exact high-dimensional asymptotics ofa family of first order gradient-based methods, learning an estimator (e.g.M-estimator, shallow neural network, ...) from observations on Gaussian datawith empirical risk minimization. This includes widely used algorithms such asstochastic gradient descent (SGD) or Nesterov acceleration. The obtainedequations match those resulting from the discretization of dynamical mean-fieldtheory (DMFT) equations from statistical physics when applied to gradient flow.Our proof method allows us to give an explicit description of how memorykernels build up in the effective dynamics, and to include non-separable updatefunctions, allowing datasets with non-identity covariance matrices. Finally, weprovide numerical implementations of the equations for SGD with genericextensive batch-size and with constant learning rates.

Rigorous dynamical mean field theory for stochastic gradient descent methods