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

We show average-case lower bounds for explicit Boolean functions against bounded-depth thresh-old circuits with a superlinear number of wires. We show that for each integer d > 1, there isεd > 0 such that Parity has correlation at most 1/nΩ(1) with depth-d threshold circuits whichhave at most n1+εd wires, and the Generalized Andreev Function has correlation at most 1/2nwith depth-d threshold circuits which have at most n1+εd wires. Previously, only worst-case lowerbounds in this setting were known [22].We use our ideas to make progress on several related questions. We give satisfiability algo-rithms beating brute force search for depth-d threshold circuits with a superlinear number ofwires. These are the first such algorithms for depth greater than 2. We also show that Paritycannot be computed by polynomial-size AC0 circuits with no(1) general threshold gates. Previ-ously no lower bound for Parity in this setting could handle more than log(n) gates. This resultalso implies subexponential-time learning algorithms for AC0 with no(1) threshold gates underthe uniform distribution. In addition, we give almost optimal bounds for the number of gates ina depth-d threshold circuit computing Parity on average, and show average-case lower boundsfor threshold formulas of any depth.Our techniques include adaptive random restrictions, anti-concentration and the structuraltheory of linear threshold functions, and bounded-read Chernoff bounds. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes