Reza Gheissari

Gerard Ben Arous, Reza Gheissari, Aukosh Jagannath

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step-size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.

Gerard Ben Arous, Reza Gheissari, Jiaoyang Huang et al.

We rigorously study the relation between the training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, both the SGD trajectory and emergent outlier eigenspaces of the Hessian and gradient matrices align with a common low-dimensional subspace. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

Reza Gheissari, Aukosh Jagannath

We consider statistical tasks in high dimensions whose loss depends on the data only through its projection into a fixed-dimensional subspace spanned by the parameter vectors and certain ground truth vectors. This includes classifying mixture distributions with cross-entropy loss with one and two-layer networks, and learning single and multi-index models with one and two-layer networks. When the data is drawn from an isotropic Gaussian mixture distribution, it is known that the evolution of a finite family of summary statistics under stochastic gradient descent converges to an autonomous ordinary differential equation (ODE), as the dimension and sample size go to $\infty$ and the step size goes to $0$ commensurately. Our main result is that these ODE limits are universal in that this limit is the same whenever the data is drawn from mixtures of arbitrary product distributions whose first two moments match the corresponding Gaussian distribution, provided the initialization and ground truth vectors are coordinate-delocalized. We complement this by proving two corresponding non-universality results. We provide a simple example where the ODE limits are non-universal if the initialization is coordinate aligned. We also show that the stochastic differential equation limits arising as fluctuations of the summary statistics around their ODE's fixed points are not universal.

Gerard Ben Arous, Reza Gheissari, Aukosh Jagannath

Stochastic gradient descent (SGD) is a popular algorithm for optimization problems arising in high-dimensional inference tasks. Here one produces an estimator of an unknown parameter from independent samples of data by iteratively optimizing a loss function. This loss function is random and often non-convex. We study the performance of the simplest version of SGD, namely online SGD, from a random start in the setting where the parameter space is high-dimensional. We develop nearly sharp thresholds for the number of samples needed for consistent estimation as one varies the dimension. Our thresholds depend only on an intrinsic property of the population loss which we call the information exponent. In particular, our results do not assume uniform control on the loss itself, such as convexity or uniform derivative bounds. The thresholds we obtain are polynomial in the dimension and the precise exponent depends explicitly on the information exponent. As a consequence of our results, we find that except for the simplest tasks, almost all of the data is used simply in the initial search phase to obtain non-trivial correlation with the ground truth. Upon attaining non-trivial correlation, the descent is rapid and exhibits law of large numbers type behavior. We illustrate our approach by applying it to a wide set of inference tasks such as phase retrieval, and parameter estimation for generalized linear models, online PCA, and spiked tensor models, as well as to supervised learning for single-layer networks with general activation functions.