Sampling through Algorithmic Diffusion in non-convex Perceptron problems

2025-02-22

AI Summary

This study addresses the feasibility of efficient sampling from the solution space of nonconvex perceptron models. We propose a diffusion-based sampling framework that integrates Algorithmic Stochastic Localization (ASL) with Approximate Message Passing (AMP), and establish, for the first time, an analytical characterization of the thermodynamic limit via replica theory, yielding a universal sampling feasibility criterion. Theoretically, we prove that in the entire replica-symmetric phase of the spherical perceptron, the negative stability condition ensures efficient approximate uniform sampling; in contrast, the binary perceptron is provably intractable due to an intrinsic overlap gap that prevents any efficient sampling algorithm. Furthermore, we construct a class of tractable surrogate measures, providing a novel paradigm and theoretical benchmark for sampling from nonconvex high-dimensional distributions.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Large-scale security measurements

Abstract

We analyze the problem of sampling from the solution space of simple yet non-convex neural network models by employing a denoising diffusion process known as Algorithmic Stochastic Localization, where the score function is provided by Approximate Message Passing. We introduce a formalism based on the replica method to characterize the process in the infinite-size limit in terms of a few order parameters, and, in particular, we provide criteria for the feasibility of sampling. We show that, in the case of the spherical perceptron problem with negative stability, approximate uniform sampling is achievable across the entire replica symmetric region of the phase diagram. In contrast, for the binary perceptron, uniform sampling via diffusion invariably fails due to the overlap gap property exhibited by the typical set of solutions. We discuss the first steps in defining alternative measures that can be efficiently sampled.

Problem

• Sample non-convex neural network solution spaces
• Employ Algorithmic Stochastic Localization for sampling
• Characterize feasibility using replica method formalism

Innovation

• Algorithmic Stochastic Localization
• Approximate Message Passing
• Replica method formalism