Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits

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

Circuit Complexity of Visual Search

We study computational hardness of feature and conjunction search through the lens of circuit complexity. Let $x = (x_1, ... , x_n)$ (resp., $y = (y_1, ... , y_n)$) be Boolean variables each of which takes the value one if and only if a neuron at place $i$ detects a feature (resp., another feature). We then simply formulate the feature and conjunction search as Boolean functions ${\rm FTR}_n(x) = \bigvee_{i=1}^n x_i$ and ${\rm CONJ}_n(x, y) = \bigvee_{i=1}^n x_i \wedge y_i$, respectively. We employ a threshold circuit or a discretized circuit (such as a sigmoid circuit or a ReLU circuit with discretization) as our models of neural networks, and consider the following four computational resources: [i] the number of neurons (size), [ii] the number of levels (depth), [iii] the number of active neurons outputting non-zero values (energy), and [iv] synaptic weight resolution (weight). We first prove that any threshold circuit $C$ of size $s$, depth $d$, energy $e$ and weight $w$ satisfies $\log rk(M_C) \le ed (\log s + \log w + \log n)$, where $rk(M_C)$ is the rank of the communication matrix $M_C$ of a $2n$-variable Boolean function that $C$ computes. Since ${\rm CONJ}_n$ has rank $2^n$, we have $n \le ed (\log s + \log w + \log n)$. Thus, an exponential lower bound on the size of even sublinear-depth threshold circuits exists if the energy and weight are sufficiently small. Since ${\rm FTR}_n$ is computable independently of $n$, our result suggests that computational capacity for the feature and conjunction search are different. We also show that the inequality is tight up to a constant factor if $ed = o(n/ \log n)$. We next show that a similar inequality holds for any discretized circuit. Thus, if we regard the number of gates outputting non-zero values as a measure for sparse activity, our results suggest that larger depth helps neural networks to acquire sparse activity

Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, 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 $TC^0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC^0$ circuits of depth $d>2$. Our first main result is a quantified derandomization algorithm for $TC^0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC^0$ circuit $C$ over $n$ input bits with depth $d$ and $n^{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC^0$ 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 $TC^0$, and would consequently imply that $NEXP\not\subseteq TC^0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC^0$ circuit with depth $d$ and $n^{1+O(1/d)}$ wires (rather than $n^{1+\exp(-d)}$ wires), runs in time at most $2^{n^{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs, then there exists an algorithm with running time $2^{n^{1-\Omega(1)}}$ for standard derandomization of $TC^0$.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction