The Nonlinearity Coefficient - Predicting Generalization in Deep Neural Networks

George Philipp, Jaime G. Carbonell

Abstract:

For a long time, designing neural architectures that exhibit high performance was considered a dark art that required expert hand-tuning. One of the few well-known guidelines for architecture design is the avoidance of exploding or vanishing gradients. However, even this guideline has remained relatively vague and circumstantial, because there exists no well-defined, gradient-based metric that can be computed {\it before} training begins and can robustly predict the performance of the network {\it after} training is complete.

We introduce what is, to the best of our knowledge, the first such metric: the nonlinearity coefficient (NLC). Via an extensive empirical study, we show that the NLC, computed in the network's randomly initialized state, is a powerful predictor of test error and that attaining a right-sized NLC is essential for attaining an optimal test error, at least in fully-connected feedforward networks. The NLC is also conceptually simple, cheap to compute, and is robust to a range of confounders and architectural design choices that comparable metrics are not necessarily robust to. Hence, we argue the NLC is an important tool for architecture search and design, as it can robustly predict poor training outcomes before training even begins.

Keywords: deep learning, neural networks, nonlinearity, activation functions, exploding gradients, vanishing gradients, neural architecture search

TL;DR: We introduce the NLC, a metric that is cheap to compute in the networks randomly initialized state and is highly predictive of generalization, at least in fully-connected networks.