Rigorous Asymptotics for First-Order Algorithms Through the Dynamical Cavity Method

2026-03-15

AI Summary

This work addresses the lack of rigorous theoretical foundations for first-order optimization algorithms in high-dimensional stochastic systems by providing, for the first time, a mathematically precise formulation of the dynamical cavity method commonly used in physics. Building on this formulation, the authors establish a dynamical mean-field theory (DMFT) framework that characterizes the asymptotic dynamics of generalized first-order algorithms—such as gradient descent and approximate message passing—in the high-dimensional limit. By integrating tools from high-dimensional probability, limit theory of stochastic processes, and the dynamical cavity method, this study bridges the gap between physics-inspired heuristics and mathematical rigor. It delivers the first provably correct DMFT framework for analyzing the dynamic behavior of high-dimensional statistical and machine learning models, thereby confirming both the validity and universality of the approach.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Dynamical Mean Field Theory (DMFT) provides an asymptotic description of the dynamics of macroscopic observables in certain disordered systems. Originally pioneered in the context of spin glasses by Sompolinsky and Zippelius (1982), it has since been used to derive asymptotic dynamical equations for a wide range of models in physics, high-dimensional statistics and machine learning. One of the main tools used by physicists to obtain these equations is the dynamical cavity method, which has remained largely non-rigorous. In contrast, existing mathematical formalizations have relied on alternative approaches, including Gaussian conditioning, large deviations over paths, or Fourier analysis. In this work, we formalize the dynamical cavity method and use it to give a new proof of the DMFT equations for General First Order Methods, a broad class of dynamics encompassing algorithms such as Gradient Descent and Approximate Message Passing.

Problem

• dynamical cavity method
• Dynamical Mean Field Theory
• rigorous asymptotics
• first-order algorithms

Innovation

• dynamical cavity method
• Dynamical Mean Field Theory
• rigorous asymptotics
• first-order algorithms
• disordered systems