Generalization bound of globally optimal non-convex neural network training: Transportation map estimation by infinite dimensional Langevin dynamics
This provides a theoretical framework for analyzing neural networks of any width, addressing limitations of existing approaches like mean field theory and neural tangent kernel theory.
The authors tackled the problem of analyzing deep learning optimization and generalization error by formulating parameter training as transportation map estimation and proving global convergence via infinite dimensional Langevin dynamics. They achieved fast learning rates, including exponential convergence for classification and minimax optimal rates for regression.
We introduce a new theoretical framework to analyze deep learning optimization with connection to its generalization error. Existing frameworks such as mean field theory and neural tangent kernel theory for neural network optimization analysis typically require taking limit of infinite width of the network to show its global convergence. This potentially makes it difficult to directly deal with finite width network; especially in the neural tangent kernel regime, we cannot reveal favorable properties of neural networks beyond kernel methods. To realize more natural analysis, we consider a completely different approach in which we formulate the parameter training as a transportation map estimation and show its global convergence via the theory of the infinite dimensional Langevin dynamics. This enables us to analyze narrow and wide networks in a unifying manner. Moreover, we give generalization gap and excess risk bounds for the solution obtained by the dynamics. The excess risk bound achieves the so-called fast learning rate. In particular, we show an exponential convergence for a classification problem and a minimax optimal rate for a regression problem.