Depth-d Threshold Circuits vs. Depth-(d+1) AND-OR Trees

For any n is an element of N and d = o (log log n), we prove that there is a Boolean function F on n bits and a value gamma = 2(-Theta(d)) such that F can be computed by a uniform depth-(3 + 1) AC(0) circuit with O(n) wires, but F cannot be computed by any depth-d TC0 circuit with n(1+gamma) wires. This bound matches the current state-of-the-art lower bounds for computing explicit functions by threshold circuits of depth d > 2, which were previously known only for functions outside AC(0) such as the parity function. Furthermore, in our result, the AC(0) circuit computing F is a monotone read-once formula (i.e., an AND-OR tree), and the lower bound holds even in the average-case setting with respect to advantage n(-gamma). At a high level, our proof strategy combines two prominent approaches in circuit complexity from the last decade: The celebrated random projections method of Hastad, Rossman, Servedio, and Tan (J. ACM 2017), which was previously used to show a tight average-case depth hierarchy for AC(0); and the line of works analyzing the effect of random restrictions on threshold circuits. We show that under a modified version of Hastad, Rossman, Servedio, and Tan's projection procedure, any depth-3 threshold circuit with n(1+gamma) wires simplifies to a near-trivial function, whereas an appropriately parameterized AND-OR tree of depth d + 1 maintains structure.