Deep ReLU Networks Have Surprisingly Few Activation Patterns

TL;DR

This paper demonstrates that the average number of activation patterns in deep ReLU networks at initialization is surprisingly limited and does not grow with depth, challenging assumptions about their practical expressivity.

Contribution

It provides a theoretical bound on activation patterns at initialization and shows empirically that this bound remains tight during training, indicating limited practical expressivity.

Findings

Average activation patterns are bounded by total neurons to input dimension

Bound remains tight at initialization and during training

Practical expressivity of deep networks may be limited

Abstract

The success of deep networks has been attributed in part to their expressivity: per parameter, deep networks can approximate a richer class of functions than shallow networks. In ReLU networks, the number of activation patterns is one measure of expressivity; and the maximum number of patterns grows exponentially with the depth. However, recent work has showed that the practical expressivity of deep networks - the functions they can learn rather than express - is often far from the theoretical maximum. In this paper, we show that the average number of activation patterns for ReLU networks at initialization is bounded by the total number of neurons raised to the input dimension. We show empirically that this bound, which is independent of the depth, is tight both at initialization and during training, even on memorization tasks that should maximize the number of activation patterns. Our work suggests that realizing the full expressivity of deep networks may not be possible in practice, at least with current methods.