Stochastic Differential Equations models for Least-Squares Stochastic Gradient Descent

2025-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

A study published in 2026, titled "Stochastic Differential Equations models for Least-Squares Stochastic Gradient Descent" by Adrien Schertzer and Loucas Pillaud-Vivien, analyzes continuous-time stochastic differential equations (SDEs) that model the dynamics of stochastic gradient descent (SGD) for the least-squares problem. The research covers both the training loss scenario with finite samples and the online setting for population loss. A key finding is the existence of a perfect interpolator for the data, irrespective of sample size. The authors provide precise, non-asymptotic rates of convergence to the stationary distribution and describe this asymptotic distribution, including estimates of its mean, deviations, and a proof of heavy-tail emergence related to the step-size magnitude. Numerical simulations support these findings.

Key takeaway

For AI Scientists optimizing machine learning models, understanding the theoretical underpinnings of SGD is crucial. This analysis provides a robust SDE framework to predict SGD behavior in least-squares problems, offering insights into convergence rates and the emergence of heavy-tailed distributions. You should consider these dynamics when selecting step-sizes and evaluating model stability, particularly in scenarios where precise control over asymptotic behavior is critical for performance and generalization.

Key insights

SDEs model SGD dynamics for least-squares, revealing convergence rates, asymptotic distribution, and heavy-tail emergence.

Principles

• SGD dynamics for least-squares can be modeled by SDEs.
• A perfect interpolator exists regardless of sample size.
• Heavy-tails in SGD relate to step-size magnitude.

Method

The paper analyzes SDEs modeling SGD for least-squares, covering both finite sample training loss and online population loss scenarios to derive convergence and asymptotic distribution properties.

In practice

• Understand SGD convergence in least-squares.
• Analyze asymptotic distribution characteristics.
• Relate step-size to heavy-tail phenomena.

Topics

• Stochastic Gradient Descent
• Stochastic Differential Equations
• Least-Squares Problem
• Convergence Rates
• Asymptotic Distribution
• Heavy-Tailed Distributions

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.