Size and Depth Separation in Approximating Natural Functions with Neural Networks

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 "natural", and explore the benefits of size and depth for approximation of natural functions with ReLU networks. As we show, this problem is more challenging than the corresponding problem for non-natural functions. We give barriers to showing depth-lower-bounds: Proving existence of a natural 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 natural functions, even between networks of constant depth and networks of nonconstant depth. We also study size-separation, namely, whether there are natural 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 $O(d\log^2(d))$, but also show an explicit natural function, that can be approximated with networks of size $O(d)$ and not with networks of size $o(d/\log d)$. For approximation in $L_\infty$ 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.