Quantified Derandomization of Linear Threshold Circuits

Abstract: One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC<sup>0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for $TC<sup>0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC<sup>0$ circuits of depth $d&gt;2$. Our first main result is a quantified derandomization algorithm for $TC<sup>0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC<sup>0$ circuit $C$ over $n$ input bits with depth $d$ and $n<sup>{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2<sup>{n<sup>{1-1/5d}}$ inputs and the case that $C$ accepts at most $2<sup>{n<sup>{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC<sup>0$ circuit (i.e., $C$ is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of $TC<sup>0$, and would consequently imply that $NEXP\not\subseteq TC<sup>0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC<sup>0$ circuit with depth $d$ and $n<sup>{1+O(1/d)}$ wires (rather than $n<sup>{1+\exp(-d)}$ wires), runs in time at most $2<sup>{n<sup>{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2<sup>{n<sup>{1-1/5d}}$ inputs and the case that $C$ accepts at most $2<sup>{n<sup>{1-1/5d}}$ inputs, then there exists an algorithm with running time $2<sup>{n<sup>{1-\Omega(1)}}$ for standard derandomization of $TC<sup>0$.