Spotting Expressivity Bottlenecks and Fixing Them Optimally

Manon Verbockhaven, Guillaume Charpiat

Keywords: neural network, expressivity, optimization, quadratic programming, layer growth, tangent space, gradient

TL;DR: We quantify the concept of lack of expressivity in neural networks and propose an algorithm to fix them by appropriately adding neurons.

Abstract: Machine learning tasks are generally formulated as optimization problems, where one searches for an optimal function within a certain functional space. In practice, parameterized functional spaces are considered, in order to be able to perform gradient descent. Typically, a neural network architecture is chosen and fixed, and its parameters (connection weights) are optimized, yielding an architecture-dependent result. This way of proceeding however forces the evolution of the function during training to lie within the realm of what is expressible with the chosen architecture, and prevents any optimization across possible architectures. Costly architectural hyper-parameter optimization is often performed to compensate for this. Instead, we propose to adapt the architecture on the fly during training. We show that the information about desirable architectural changes, due to expressivity bottlenecks when attempting to follow the functional gradient, can be extracted from the back-propagation. To do this, we propose a new mathematically well-grounded method to detect expressivity bottlenecks on the fly and solve them by adding suitable neurons when and where needed. Thus, while the standard approach requires large networks, in terms of number of neurons per layer, for expressivity and optimization reasons, we are able to start with very small neural networks and let them grow appropriately. As a proof of concept, we show convincing results on the MNIST dataset, matching large neural network accuracy, with competitive training time, while removing the need for standard architectural hyper-parameter optimization.

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