Deterministic Approximate Counting for Juntas of Degree-2 Polynomial Threshold Functions
A. De and I. Diakonikolas and R. Servedio.
In 29th Conference on Computational Complexity (CCC), 2014, pp. 229--240.


Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[ \Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] \] to within an additive $\pm \eps$. For any constant $\eps > 0$ and $k \geq 1$ the running time of our algorithm is a fixed polynomial in $n$ (in fact this is true even for some not-too-small $\eps = o_n(1)$ and not-too-large $k = \omega_n(1)$). This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits.

Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function $\sign(q(x)),$ corresponding to the $k=1$ case of our result. Note that even in the $k=1$ case it is NP-hard to determine whether $\Pr_{x \sim \{-1,1\}^n}[\sign(q(x))=1]$ is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms.

Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the $k=1$ case in \cite{DDS13:deg2count}. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for $k$-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution, i.e., an algorithm for estimating \[ \Pr_{x \sim N(0,1)^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1]. \] Finally, a third new ingredient is a ``regularity lemma'' for \emph{$k$-tuples} of degree-$d$ polynomial threshold functions. This generalizes both the regularity lemmas of \cite{DSTW:10,HKM:09} (which apply to a single degree-$d$ polynomial threshold function) and the regularity lemma of Gopalan et al \cite{GOWZ10} (which applies to a $k$-tuples of \emph{linear} threshold functions, i.e., the case $d=1$). Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.

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