  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.
