Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$

Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations

Consider a database of $n$ people, each represented by a bit-string of length $d$ corresponding to the setting of $d$ binary attributes. A $k$-way marginal query is specified by a subset $S$ of $k$ attributes, and a $|S|$-dimensional binary vector $\beta$ specifying their values. The result for this query is a count of the number of people in the database whose attribute vector restricted to $S$ agrees with $\beta$. Privately releasing approximate answers to a set of $k$-way marginal queries is one of the most important and well-motivated problems in differential privacy. Information theoretically, the error complexity of marginal queries is well-understood: the per-query additive error is known to be at least $\Omega(\min\{\sqrt{n},d^{\frac{k}{2}}\})$ and at most $\tilde{O}(\min\{\sqrt{n} d^{1/4},d^{\frac{k}{2}}\})$. However, no polynomial time algorithm with error complexity as low as the information theoretic upper bound is known for small $n$. In this work we present a polynomial time algorithm that, for any distribution on marginal queries, achieves average error at most $\tilde{O}(\sqrt{n} d^{\frac{\lceil k/2 \rceil}{4}})$. This error bound is as good as the best known information theoretic upper bounds for $k=2$. This bound is an improvement over previous work on efficiently releasing marginals when $k$ is small and when error $o(n)$ is desirable. Using private boosting we are also able to give nearly matching worst-case error bounds. Our algorithms are based on the geometric techniques of Nikolov, Talwar, and Zhang. The main new ingredients are convex relaxations and careful use of the Frank-Wolfe algorithm for constrained convex minimization. To design our relaxations, we rely on the Grothendieck inequality from functional analysis