Growing Tiny Networks: Spotting Expressivity Bottlenecks and Fixing Them Optimally

TL;DR

This paper introduces a method to dynamically adapt neural network architectures during training by detecting and fixing expressivity bottlenecks, enabling smaller networks to achieve large-model performance efficiently.

Contribution

It proposes a mathematical framework to identify and address expressivity bottlenecks in neural networks during training, allowing for architecture adaptation and reducing the need for large, fixed networks.

Findings

Achieves CIFAR accuracy comparable to large networks with smaller architectures.

Reduces reliance on extensive hyper-parameter tuning.

Demonstrates efficient training with adaptive architecture modifications.

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 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 backpropagation. To do this, we propose a mathematical definition of expressivity bottlenecks, which enables us to detect, quantify and solve them while training, by adding suitable neurons. Thus, while the standard approach requires large networks, in terms of number of neurons per layer, for expressivity and optimization reasons, we provide tools and properties to develop an architecture starting with a very small number of neurons. As a proof of concept, we show results~on the CIFAR dataset, matching large neural network accuracy, with competitive training time, while removing the need for standard architectural hyper-parameter search.