Measuring Generalization with Optimal Transport

TL;DR

This paper introduces a new theoretical framework for understanding neural network generalization using optimal transport costs, providing bounds that align well with empirical observations on large datasets.

Contribution

It develops margin-based generalization bounds normalized with optimal transport costs, linking feature space structure to generalization performance.

Findings

Optimal transport costs generalize variance and capture feature space structure.

The bounds accurately predict generalization error on large datasets.

Feature concentration and separation are key factors in generalization.

Abstract

Understanding the generalization of deep neural networks is one of the most important tasks in deep learning. Although much progress has been made, theoretical error bounds still often behave disparately from empirical observations. In this work, we develop margin-based generalization bounds, where the margins are normalized with optimal transport costs between independent random subsets sampled from the training distribution. In particular, the optimal transport cost can be interpreted as a generalization of variance which captures the structural properties of the learned feature space. Our bounds robustly predict the generalization error, given training data and network parameters, on large scale datasets. Theoretically, we demonstrate that the concentration and separation of features play crucial roles in generalization, supporting empirical results in the literature. The code is available at \url{https://github.com/chingyaoc/kV-Margin}.