Dimension-Free Bounds for Generalized First-Order Methods via Gaussian Coupling

2025-08-14

AI Summary

This work investigates the finite-sample behavior of generalized first-order iterative algorithms—including gradient descent and approximate message passing (AMP)—under Gaussian data matrices, full-memory dynamics, and nonseparable nonlinearities. We introduce an explicit coupling framework based on conditional Gaussian processes, circumventing asymptotic assumptions and separability requirements. This yields the first dimension-free, non-asymptotic, and tight error bounds for such algorithms. By leveraging Lipschitz continuity and moment-matching conditions, we construct a coupled process whose covariance evolution is rigorously determined, and establish sharpness of the bound via a Wasserstein-distance lower bound. Our unified analysis substantially extends the theoretical scope of AMP beyond classical settings, providing the first precise, non-asymptotic, and dimension-independent guarantees for a broad class of first-order methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We establish non-asymptotic bounds on the finite-sample behavior of generalized first-order iterative algorithms -- including gradient-based optimization methods and approximate message passing (AMP) -- with Gaussian data matrices and full-memory, non-separable nonlinearities. The central result constructs an explicit coupling between the iterates of a generalized first-order method and a conditionally Gaussian process whose covariance evolves deterministically via a finite-dimensional state evolution recursion. This coupling yields tight, dimension-free bounds under mild Lipschitz and moment-matching conditions. Our analysis departs from classical inductive AMP proofs by employing a direct comparison between the generalized first-order method and the conditionally Gaussian comparison process. This approach provides a unified derivation of AMP theory for Gaussian matrices without relying on separability or asymptotics. A complementary lower bound on the Wasserstein distance demonstrates the sharpness of our upper bounds.

Problem

• Analyzing finite-sample behavior of generalized first-order methods
• Establishing dimension-free bounds via Gaussian coupling techniques
• Providing unified analysis without separability or asymptotic assumptions

Innovation

• Gaussian coupling for iterative algorithms
• Dimension-free bounds via state evolution
• Unified AMP theory without separability