The High Line: Exact Risk and Learning Rate Curves of Stochastic Adaptive Learning Rate Algorithms

We develop a framework for analyzing the training and learning rate dynamics on a large class of high-dimensional optimization problems, which we call the high line, trained using one-pass stochastic gradient descent (SGD) with adaptive learning rates. We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs. We then investigate in detail two adaptive learning rates -- an idealized exact line search and AdaGrad-Norm -- on the least squares problem. When the data covariance matrix has strictly positive eigenvalues, this idealized exact line search strategy can exhibit arbitrarily slower convergence when compared to the optimal fixed learning rate with SGD. Moreover we exactly characterize the limiting learning rate (as time goes to infinity) for line search in the setting where the data covariance has only two distinct eigenvalues. For noiseless targets, we further demonstrate that the AdaGrad-Norm learning rate converges to a deterministic constant inversely proportional to the average eigenvalue of the data covariance matrix, and identify a phase transition when the covariance density of eigenvalues follows a power law distribution.

Hitting the High-Dimensional Notes: An ODE for SGD learning dynamics on GLMs and multi-index models

We analyze the dynamics of streaming stochastic gradient descent (SGD) in the high-dimensional limit when applied to generalized linear models and multi-index models (e.g. logistic regression, phase retrieval) with general data-covariance. In particular, we demonstrate a deterministic equivalent of SGD in the form of a system of ordinary differential equations that describes a wide class of statistics, such as the risk and other measures of sub-optimality. This equivalence holds with overwhelming probability when the model parameter count grows proportionally to the number of data. This framework allows us to obtain learning rate thresholds for stability of SGD as well as convergence guarantees. In addition to the deterministic equivalent, we introduce an SDE with a simplified diffusion coefficient (homogenized SGD) which allows us to analyze the dynamics of general statistics of SGD iterates. Finally, we illustrate this theory on some standard examples and show numerical simulations which give an excellent match to the theory.