Thompson Sampling in Function Spaces via Neural Operators

2025-06-27

AI Summary

This work addresses efficient optimization in function spaces: minimizing a known functional whose output is generated by an unknown operator (e.g., a PDE solver), where functional evaluations are inexpensive but operator queries (e.g., high-fidelity simulations) are prohibitively costly. To tackle this, we introduce the first Thompson sampling framework for infinite-dimensional function spaces—termed “sample-then-optimize”: it leverages pretrained neural operators as approximate samples from an infinite-dimensional Gaussian process, circumventing explicit uncertainty quantification. We further establish the first convergence theory for Thompson sampling in function spaces. Experiments on PDE-constrained and nonlinear-operator-driven functional optimization demonstrate substantial improvements in sample efficiency, consistently outperforming state-of-the-art baselines.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We propose an extension of Thompson sampling to optimization problems over function spaces where the objective is a known functional of an unknown operator's output. We assume that functional evaluations are inexpensive, while queries to the operator (such as running a high-fidelity simulator) are costly. Our algorithm employs a sample-then-optimize approach using neural operator surrogates. This strategy avoids explicit uncertainty quantification by treating trained neural operators as approximate samples from a Gaussian process. We provide novel theoretical convergence guarantees, based on Gaussian processes in the infinite-dimensional setting, under minimal assumptions. We benchmark our method against existing baselines on functional optimization tasks involving partial differential equations and other nonlinear operator-driven phenomena, demonstrating improved sample efficiency and competitive performance.

Problem

• Extends Thompson sampling to functional optimization problems
• Optimizes expensive operator queries using neural surrogates
• Provides convergence guarantees for infinite-dimensional Gaussian processes

Innovation

• Extends Thompson sampling to functional spaces
• Uses neural operator surrogates for optimization
• Avoids explicit uncertainty quantification via Gaussian processes