Learning Operators with Stochastic Gradient Descent in General Hilbert Spaces

2024-02-07

arXiv.org

AI Summary

This work investigates the convergence and learnability of stochastic gradient descent (SGD) for learning linear and nonlinear operators in general Hilbert spaces. To accommodate structural properties of target operators, we introduce weak and strong regularity conditions—functional analogues of classical smoothness and decay assumptions. Our analysis constitutes the first systematic extension of SGD convergence theory to infinite-dimensional operator learning. We establish that SGD converges to the optimal linear approximation of a nonlinear operator; derive tight non-asymptotic upper bounds on the convergence rate; and construct matching minimax lower bounds, thereby characterizing fundamental statistical limits. The results hold uniformly across both vector-valued and scalar-valued reproducing kernel Hilbert spaces (RKHS). By transcending the conventional restrictions of SGD analysis—namely, finite-dimensional parameterizations or linear models—this work provides the first theoretical framework for operator learning that incorporates functional regularity characterizations and delivers quantitative, provable solvability guarantees.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• 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: Social networks and social learning

Abstract

This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and complexity. Under these conditions, we establish upper bounds for convergence rates of the SGD algorithm and conduct a minimax lower bound analysis, further illustrating that our convergence analysis and regularity conditions quantitatively characterize the tractability of solving operator learning problems using the SGD algorithm. It is crucial to highlight that our convergence analysis is still valid for nonlinear operator learning. We show that the SGD estimator will converge to the best linear approximation of the nonlinear target operator. Moreover, applying our analysis to operator learning problems based on vector-valued and real-valued reproducing kernel Hilbert spaces yields new convergence results, thereby refining the conclusions of existing literature.

Problem

• Random Gradient Descent
• Mathematical Space
• Nonlinear Operator Learning

Innovation

• Stochastic Gradient Descent
• Operator Learning
• Nonlinear Operators