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.

\bullet We prove that for all \epsilon\gg \sqrt{\log(n)/n}, the linear-time computable Andreev's function cannot be computed on a (1/2+\epsilon)-fraction of n-bit inputs by depth-two linear threshold circuits of o(\epsilon^3 n^{3/2}/\log^3 n) gates, nor can it be computed with o(\epsilon^{3} n^{5/2}/\log^{7/2} 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(n^3) linear threshold gates, and by uniform depth-three circuits of O(n) majority gates.

\bullet 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 linear threshold circuits with o(n^{3/2}/\log^3 n) gates, nor with o(n^{5/2}/\log^{7/2}n) wires.

\bullet 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 random restriction lemma for linear threshold functions. Our main analytical tool is the Littlewood-Offord Lemma from additive combinatorics.