Size and Depth Separation in Approximating Benign Functions with Neural Networks
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by
Gal Vardi, Daniel Reichman, Toniann Pitassi, Ohad Shamir
2021
Abstract
When studying the expressive power of neural networks, a main challenge is to
understand how the size and depth of the network affect its ability to
approximate real functions. However, not all functions are interesting from a
practical viewpoint: functions of interest usually have a polynomially-bounded
Lipschitz constant, and can be computed efficiently. We call functions that
satisfy these conditions "benign", and explore the benefits of size and depth
for approximation of benign functions with ReLU networks. As we show, this
problem is more challenging than the corresponding problem for non-benign
functions. We give barriers to showing depth-lower-bounds: Proving existence of
a benign function that cannot be approximated by polynomial-size networks of
depth 4 would settle longstanding open problems in computational complexity.
It implies that beyond depth 4 there is a barrier to showing depth-separation
for benign functions, even between networks of constant depth and networks of
nonconstant depth. We also study size-separation, namely, whether there are
benign functions that can be approximated with networks of size O(s(d)), but
not with networks of size O(s'(d)). We show a complexity-theoretic barrier to
proving such results beyond size O(dlog^2(d)), but also show an explicit
benign function, that can be approximated with networks of size O(d) and not
with networks of size o(d/log d). For approximation in L_∞ we achieve
such separation already between size O(d) and size o(d). Moreover, we show
superpolynomial size lower bounds and barriers to such lower bounds, depending
on the assumptions on the function. Our size-separation results rely on an
analysis of size lower bounds for Boolean functions, which is of independent
interest: We show linear size lower bounds for computing explicit Boolean
functions with neural networks and threshold circuits.
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