A path-norm toolkit for modern networks: consequences, promises and challenges

TL;DR

This paper develops a comprehensive path-norm toolkit for modern neural networks, enabling tighter generalization bounds and practical evaluation on complex architectures like ResNets.

Contribution

It introduces the first versatile path-norm toolkit applicable to general DAG ReLU networks with various operations, improving theoretical bounds and practical assessments.

Findings

Extended path-norms recover or surpass existing bounds.

The toolkit is easy to compute and invariant under network symmetries.

Numerical evaluation on ResNets challenges existing generalization promises.

Abstract

This work introduces the first toolkit around path-norms that fully encompasses general DAG ReLU networks with biases, skip connections and any operation based on the extraction of order statistics: max pooling, GroupSort etc. This toolkit notably allows us to establish generalization bounds for modern neural networks that are not only the most widely applicable path-norm based ones, but also recover or beat the sharpest known bounds of this type. These extended path-norms further enjoy the usual benefits of path-norms: ease of computation, invariance under the symmetries of the network, and improved sharpness on layered fully-connected networks compared to the product of operator norms, another complexity measure most commonly used. The versatility of the toolkit and its ease of implementation allow us to challenge the concrete promises of path-norm-based generalization bounds, by numerically evaluating the sharpest known bounds for ResNets on ImageNet.