Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuits

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for depth-two linear threshold circuits with arbitrary weights, and depth-three majority circuits computing an explicit function. (1) We prove that for all ε ≪ √log(n)/n, the linear-time computable Andreev’s function cannot be computed on a (1/2+ε)-fraction of n-bit inputs by depth-two circuits of o(ε3 n3/2/log3 n) gates, nor can it be computed with o(ε3 n5/2/log7/2 n) wires. This establishes an average-case “size hierarchy” for threshold circuits, as Andreev’s function is computable by uniform depth-two circuits of o(n3) linear threshold gates, and by uniform depth-three circuits of O(n) majority gates. (2) We present a new function in P based on small-biased sets, which we prove cannot be computed by a majority vote of depth-two threshold circuits of o(n3/2/log3 n) gates, nor with o(n5/2/log7/2n) wires. (3) We give tight average-case (gate and wire) complexity results for computing PARITY with depth-two threshold circuits; the answer turns out to be the same as for depth-two majority circuits. The key is a new method for analyzing random restrictions to linear threshold functions. Our main analytical tool is the Littlewood-Offord Lemma from additive combinatorics.