Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data

TL;DR

This paper develops a method to compute meaningful, nonvacuous generalization bounds for deep stochastic neural networks with many parameters, explaining why they generalize well despite overparameterization.

Contribution

It extends PAC-Bayes bounds to deep neural networks, providing the first nonvacuous bounds for models with millions of parameters trained on limited data.

Findings

Achieves nonvacuous generalization bounds for deep stochastic neural networks.

Connects bounds to flat minima and MDL explanations.

Demonstrates bounds for networks trained on tens of thousands of examples.

Abstract

One of the defining properties of deep learning is that models are chosen to have many more parameters than available training data. In light of this capacity for overfitting, it is remarkable that simple algorithms like SGD reliably return solutions with low test error. One roadblock to explaining these phenomena in terms of implicit regularization, structural properties of the solution, and/or easiness of the data is that many learning bounds are quantitatively vacuous when applied to networks learned by SGD in this "deep learning" regime. Logically, in order to explain generalization, we need nonvacuous bounds. We return to an idea by Langford and Caruana (2001), who used PAC-Bayes bounds to compute nonvacuous numerical bounds on generalization error for stochastic two-layer two-hidden-unit neural networks via a sensitivity analysis. By optimizing the PAC-Bayes bound directly, we are able to extend their approach and obtain nonvacuous generalization bounds for deep stochastic neural network classifiers with millions of parameters trained on only tens of thousands of examples. We connect our findings to recent and old work on flat minima and MDL-based explanations of generalization.