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We study a class of deterministic flows in ${\mathbb R}^{d\times k}$,parametrized by a random matrix ${\boldsymbol X}\in {\mathbb R}^{n\times d}$with i.i.d. centered subgaussian entries. We characterize the asymptoticbehavior of these flows over bounded time horizons, in the high-dimensionallimit in which $n,d\to\infty$ with $k$ fixed and converging aspect ratios$n/d\to\delta$. The asymptotic characterization we prove is in terms of asystem of a nonlinear stochastic process in $k$ dimensions, whose parametersare determined by a fixed point condition. This type of characterization isknown in physics as dynamical mean field theory. Rigorous results of this typehave been obtained in the past for a few spin glass models. Our proof is basedon time discretization and a reduction to certain iterative schemes known asapproximate message passing (AMP) algorithms, as opposed to earlier work thatwas based on large deviations theory and stochastic processes theory. The newapproach allows for a more elementary proof and implies that thehigh-dimensional behavior of the flow is universal with respect to thedistribution of the entries of ${\boldsymbol X}$. As specific applications, weobtain high-dimensional characterizations of gradient flow in some classicalmodels from statistics and machine learning, under a random design assumption.

The high-dimensional asymptotics of first order methods with random data