Instance-Dependent Generalization Bounds via Optimal Transport

Existing generalization bounds fail to explain crucial factors that drive generalization of modern neural networks. Since such bounds often hold uniformly over all parameters, they suffer from over-parametrization, and fail to account for the fact that the set of parameters, considered during initialization and training, is much more restricted than the entire parameter space. As an alternative, we propose a novel optimal transport interpretation of the generalization problem. This allows us to derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function} in the data space. Therefore, our bounds are agnostic to the parametrization of the model and work well when the number of training samples is much smaller than the number of parameters. With small modifications, our approach yields accelerated rates for data on low-dimensional manifolds, and guarantees under distribution shifts. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.