Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is ed > 0 such that Parity has correlation at most 1/nΩ(1) with depth-d threshold circuits which have at most n1+ed wires, and the Generalized Andreev Function has correlation at most 1/2nΩ(1) with depth-d threshold circuits which have at most n1+ed wires. Previously, only worst-case lower bounds in this setting were known [22]. We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth-d threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC0 circuits with no(1) general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC0 with no(1) threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas of any depth. Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.