Path Integral Optimiser: Global Optimisation via Neural Schr""odinger-F""ollmer Diffusion

2025-06-07

AI Summary

This work addresses the limited global exploration capability of neural diffusion methods in high-dimensional parameter spaces (e.g., 15.9k dimensions). We propose a novel optimization framework grounded in Schrödinger bridge theory: the optimization problem is formulated as an optimal transport task under a Boltzmann distribution, transformed into a stochastic control problem via Girsanov’s theorem, and solved approximately using a Fourier-feature-mapped MLP to estimate the path integral. To our knowledge, this is the first application of neural Schrödinger–Föllmer diffusion to optimization, providing theoretical convergence guarantees and establishing a new conceptual link between optimization and quantum-inspired stochastic differential equations. Empirical evaluation on synthetic benchmarks spanning 2–1,247 dimensions demonstrates superior single-step optimization performance; however, exploration degrades in ultra-high dimensions, highlighting the need for adaptive diffusion mechanisms.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We present an early investigation into the use of neural diffusion processes for global optimisation, focusing on Zhang et al.'s Path Integral Sampler. One can use the Boltzmann distribution to formulate optimization as solving a Schr""odinger bridge sampling problem, then apply Girsanov's theorem with a simple (single-point) prior to frame it in stochastic control terms, and compute the solution's integral terms via a neural approximation (a Fourier MLP). We provide theoretical bounds for this optimiser, results on toy optimisation tasks, and a summary of the stochastic theory motivating the model. Ultimately, we found the optimiser to display promising per-step performance at optimisation tasks between 2 and 1,247 dimensions, but struggle to explore higher-dimensional spaces when faced with a 15.9k parameter model, indicating a need for work on adaptation in such environments.

Problem

• Global optimization via neural diffusion processes
• Formulating optimization as Schrödinger bridge sampling
• Challenges in high-dimensional space exploration

Innovation

• Neural diffusion for global optimization
• Schrödinger bridge sampling via Girsanov
• Fourier MLP approximates integral terms