Conceptual capacity and effective complexity of neural networks

TL;DR

This paper introduces a new complexity measure for neural networks based on tangent space diversity, which correlates with the network's generalization ability and captures its effective complexity rather than just theoretical capacity.

Contribution

It proposes an entropy-based complexity measure using tangent spaces, revealing the gap between theoretical and effective capacity of neural networks.

Findings

The maximal capacity of a ReLU network equals its number of neurons.

Actual capacity is often much smaller due to neuron activity correlations.

The measure correlates with the network's generalization performance.

Abstract

We propose a complexity measure of a neural network mapping function based on the diversity of the set of tangent spaces from different inputs. Treating each tangent space as a linear PAC concept we use an entropy-based measure of the bundle of concepts in order to estimate the conceptual capacity of the network. The theoretical maximal capacity of a ReLU network is equivalent to the number of its neurons. In practice however, due to correlations between neuron activities within the network, the actual capacity can be remarkably small, even for very big networks. Empirical evaluations show that this new measure is correlated with the complexity of the mapping function and thus the generalisation capabilities of the corresponding network. It captures the effective, as oppose to the theoretical, complexity of the network function. We also showcase some uses of the proposed measure for analysis and comparison of trained neural network models.