Agnostic Learning of Disjunctions on Symmetric Distributions

We consider the problem of approximating and learning disjunctions (or equivalently, conjunctions) on symmetric distributions over $\{0,1\}^n$. Symmetric distributions are distributions whose PDF is invariant under any permutation of the variables. We give a simple proof that for every symmetric distribution $\mathcal{D}$, there exists a set of $n^{O(\log{(1/\epsilon)})}$ functions $\mathcal{S}$, such that for every disjunction $c$, there is function $p$, expressible as a linear combination of functions in $\mathcal{S}$, such that $p$ $\epsilon$-approximates $c$ in $\ell_1$ distance on $\mathcal{D}$ or $\mathbf{E}_{x \sim \mathcal{D}}[ |c(x)-p(x)|] \leq \epsilon$. This directly gives an agnostic learning algorithm for disjunctions on symmetric distributions that runs in time $n^{O( \log{(1/\epsilon)})}$. The best known previous bound is $n^{O(1/\epsilon^4)}$ and follows from approximation of the more general class of halfspaces (Wimmer, 2010). We also show that there exists a symmetric distribution $\mathcal{D}$, such that the minimum degree of a polynomial that $1/3$-approximates the disjunction of all $n$ variables is $\ell_1$ distance on $\mathcal{D}$ is $\Omega( \sqrt{n})$. Therefore the learning result above cannot be achieved via $\ell_1$-regression with a polynomial basis used in most other agnostic learning algorithms. Our technique also gives a simple proof that for any product distribution $\mathcal{D}$ and every disjunction $c$, there exists a polynomial $p$ of degree $O(\log{(1/\epsilon)})$ such that $p$ $\epsilon$-approximates $c$ in $\ell_1$ distance on $\mathcal{D}$. This was first proved by Blais et al. (2008) via a more involved argument

Learning Coverage Functions and Private Release of Marginals

We study the problem of approximating and learning coverage functions. A function $c: 2^{[n]} \rightarrow \mathbf{R}^{+}$ is a coverage function, if there exists a universe $U$ with non-negative weights $w(u)$ for each $u \in U$ and subsets $A_1, A_2, \ldots, A_n$ of $U$ such that $c(S) = \sum_{u \in \cup_{i \in S} A_i} w(u)$. Alternatively, coverage functions can be described as non-negative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any $\gamma,\delta>0$, given random and uniform examples of an unknown coverage function $c$, finds a function $h$ that approximates $c$ within factor $1+\gamma$ on all but $\delta$-fraction of the points in time $poly(n,1/\gamma,1/\delta)$. This is the first fully-polynomial algorithm for learning an interesting class of functions in the demanding PMAC model of Balcan and Harvey (2011). Our algorithms are based on several new structural properties of coverage functions. Using the results in (Feldman and Kothari, 2014), we also show that coverage functions are learnable agnostically with excess $\ell_1$-error $\epsilon$ over all product and symmetric distributions in time $n^{\log(1/\epsilon)}$. In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomial-size disjoint DNF formulas, a class of functions for which the best known algorithm runs in time $2^{\tilde{O}(n^{1/3})}$ (Klivans and Servedio, 2004). As an application of our learning results, we give simple differentially-private algorithms for releasing monotone conjunction counting queries with low average error. In particular, for any $k \leq n$, we obtain private release of $k$-way marginals with average error $\bar{\alpha}$ in time $n^{O(\log(1/\bar{\alpha}))}$

Quantum entanglement, sum of squares, and the log rank conjecture

For every $\epsilon>0$, we give an $\exp(\tilde{O}(\sqrt{n}/\epsilon^2))$-time algorithm for the $1$ vs $1-\epsilon$ \emph{Best Separable State (BSS)} problem of distinguishing, given an $n^2\times n^2$ matrix $\mathcal{M}$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $\rho$ that $\mathcal{M}$ accepts with probability $1$, and the case that every separable state is accepted with probability at most $1-\epsilon$. Equivalently, our algorithm takes the description of a subspace $\mathcal{W} \subseteq \mathbb{F}^{n^2}$ (where $\mathbb{F}$ can be either the real or complex field) and distinguishes between the case that $\mathcal{W}$ contains a rank one matrix, and the case that every rank one matrix is at least $\epsilon$ far (in $\ell_2$ distance) from $\mathcal{W}$. To the best of our knowledge, this is the first improvement over the brute-force $\exp(n)$-time algorithm for this problem. Our algorithm is based on the \emph{sum-of-squares} hierarchy and its analysis is inspired by Lovett's proof (STOC '14, JACM '16) that the communication complexity of every rank-$n$ Boolean matrix is bounded by $\tilde{O}(\sqrt{n})$.Comment: 23 pages + 1 title-page + 1 table-of-content