The Nonlinearity Coefficient -- A Practical Guide to Neural Architecture Design

In essence, a neural network is an arbitrary differentiable, parametrized function. Choosing a neural network architecture for any task is as complex as searching the space of those functions. For the last few years, 'neural architecture design' has been largely synonymous with 'neural architecture search' (NAS), i.e. brute-force, large-scale search. NAS has yielded significant gains on practical tasks. However, NAS methods end up searching for a local optimum in architecture space in a small neighborhood around architectures that often go back decades, based on CNN or LSTM. In this work, we present a different and complementary approach to architecture design, which we term 'zero-shot architecture design' (ZSAD). We develop methods that can predict, without any training, whether an architecture will achieve a relatively high test or training error on a task after training. We then go on to explain the error in terms of the architecture definition itself and develop tools for modifying the architecture based on this explanation. This confers an unprecedented level of control on the deep learning practitioner. They can make informed design decisions before the first line of code is written, even for tasks for which no prior art exists. Our first major contribution is to show that the 'degree of nonlinearity' of a neural architecture is a key causal driver behind its performance, and a primary aspect of the architecture's model complexity. We introduce the 'nonlinearity coefficient' (NLC), a scalar metric for measuring nonlinearity. Via extensive empirical study, we show that the value of the NLC in the architecture's randomly initialized state before training is a powerful predictor of test error after training and that attaining a right-sized NLC is essential for attaining an optimal test error. The NLC is also conceptually simple, well-defined for any feedforward network, easy and cheap to compute, has extensive theoretical, empirical and conceptual grounding, follows instructively from the architecture definition, and can be easily controlled via our 'nonlinearity normalization' algorithm. We argue that the NLC is the most powerful scalar statistic for architecture design specifically and neural network analysis in general. Our analysis is fueled by mean field theory, which we use to uncover the 'meta-distribution' of layers. Beyond the NLC, we uncover and flesh out a range of metrics and properties that have a significant explanatory influence on test and training error. We go on to explain the majority of the error variation across a wide range of randomly generated architectures with these metrics and properties. We compile our insights into a practical guide for architecture designers, which we argue can significantly shorten the trial-and-error phase of deep learning deployment. Our results are grounded in an experimental protocol that exceeds that of the vast majority of other deep learning studies in terms of carefulness and rigor. We study the impact of e.g. dataset, learning rate, floating-point precision, loss function, statistical estimation error and batch inter-dependency on performance and other key properties. We promote research practices that we believe can significantly accelerate progress in architecture design research.