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Begoña García Malaxechebarría

Alumni

Publications

High-dimensional Limit of SGD for Diagonal Linear Networks
Begoña García Malaxechebarría
Courtney Paquette
Maryam Fazel
Dmitriy Drusvyatskiy
Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diago… (voir plus)nal linear networks as a simplified yet expressive setting for analyzing the optimization and generalization properties of neural models. In this work, we show that in the high-dimensional regime, stochastic gradient descent on diagonal linear networks is well-approximated by continuous dynamics governed by a stochastic differential equation (SDE), which explicitly decouples the drift from the gradient noise. We further derive a deterministic partial differential equation whose solution propagates the relevant state of the iterates and characterizes the time evolution of a broad class of observable statistics, including the risk, curvature, and other metrics for optimality. Finally, we show that, under a suitable parametrization, the stochastic dynamics are globally well posed and converge exponentially fast to zero risk with high probability, yielding a fully explicit non-asymptotic description of their long-time behavior. Numerical simulations corroborate our theoretical findings.
2026-05-15
arXiv (prépublication)
doi.org
arxiv.org
The High Line: Exact Risk and Learning Rate Curves of Stochastic Adaptive Learning Rate Algorithms
Elizabeth Collins-Woodfin
INBAR SEROUSSI
Begoña García Malaxechebarría
Andrew W. Mackenzie
Elliot Paquette
Courtney Paquette
We develop a framework for analyzing the training and learning rate dynamics on a large class of high-dimensional optimization problems, whi… (voir plus)ch 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. We provide our code for evaluation at https://github.com/amackenzie1/highline2024.
2024-12-10
Conference on Neural Information Processing Systems (Accept (poster))
doi.org
openreview.net