Scaling Limit: Exact and Tractable Analysis of Online Learning Algorithms with Applications to Regularized Regression and PCA

TL;DR

This paper introduces a framework for analyzing the exact dynamics of online learning algorithms in high-dimensional settings, enabling precise performance predictions and insights into their asymptotic behavior.

Contribution

It provides a novel PDE-based approach to characterize the dynamics of online algorithms like regularized regression and PCA in the high-dimensional limit.

Findings

Empirical measures converge to a deterministic PDE in high dimensions

Performance metrics can be accurately predicted from PDE solutions

Asymptotic decoupling simplifies analysis of complex algorithms

Abstract

We present a framework for analyzing the exact dynamics of a class of online learning algorithms in the high-dimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression and principal component analysis. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measures of the target feature vector and its estimates provided by the algorithms will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE can be efficiently obtained. These solutions lead to precise predictions of the performance of the algorithms, as many practical performance metrics are linear functionals of the joint empirical measures. In addition to characterizing the dynamic performance of online learning algorithms, our asymptotic analysis also provides useful insights. In particular, in the high-dimensional limit, and due to exchangeability, the original coupled dynamics associated with the algorithms will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight for nonconvex optimization problems may prove an interesting line of future research.