Learning pseudo-Boolean k-DNF and Submodular Functions

We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f:{0,1}^n -> {0,1,...,k}. Our algorithm runs in time polynomial in n, k^{O(k \log k / \epsilon)}, 1/\epsilon and log(1/\delta) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi, Klivans, Kothari, Lee (SODA '12)] implies only n^{O(k)} query complexity for learning submodular functions in this setting, for fixed epsilon and delta. Our learning algorithm implies a property tester for submodularity of functions f:{0,1}^n -> {0, ..., k} with query complexity polynomial in n for k=O((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon

Fast Private Data Release Algorithms for Sparse Queries

We revisit the problem of accurately answering large classes of statistical queries while preserving differential privacy. Previous approaches to this problem have either been very general but have not had run-time polynomial in the size of the database, have applied only to very limited classes of queries, or have relaxed the notion of worst-case error guarantees. In this paper we consider the large class of sparse queries, which take non-zero values on only polynomially many universe elements. We give efficient query release algorithms for this class, in both the interactive and the non-interactive setting. Our algorithms also achieve better accuracy bounds than previous general techniques do when applied to sparse queries: our bounds are independent of the universe size. In fact, even the runtime of our interactive mechanism is independent of the universe size, and so can be implemented in the "infinite universe" model in which no finite universe need be specified by the data curator

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

Nearly Optimal Private Convolution

We study computing the convolution of a private input $x$ with a public input $h$, while satisfying the guarantees of $(\epsilon, \delta)$-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolution then captures important primitives including linear filtering, which is an essential tool in time series analysis, and aggregation queries on projections of the data. We give a nearly optimal algorithm for computing convolutions while satisfying $(\epsilon, \delta)$-differential privacy. Surprisingly, we follow the simple strategy of adding independent Laplacian noise to each Fourier coefficient and bounding the privacy loss using the composition theorem of Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal noise to add to each Fourier coefficient using convex programming duality. Our algorithm is very efficient -- it is essentially no more computationally expensive than a Fast Fourier Transform. To prove near optimality, we use the recent discrepancy lowerbounds of Muthukrishnan and Nikolov and derive a spectral lower bound using a characterization of discrepancy in terms of determinants

Differentially Private Data Releasing for Smooth Queries with Synthetic Database Output

We consider accurately answering smooth queries while preserving differential privacy. A query is said to be $K$-smooth if it is specified by a function defined on $[-1,1]^d$ whose partial derivatives up to order $K$ are all bounded. We develop an $\epsilon$-differentially private mechanism for the class of $K$-smooth queries. The major advantage of the algorithm is that it outputs a synthetic database. In real applications, a synthetic database output is appealing. Our mechanism achieves an accuracy of $O (n^{-\frac{K}{2d+K}}/\epsilon )$, and runs in polynomial time. We also generalize the mechanism to preserve $(\epsilon, \delta)$-differential privacy with slightly improved accuracy. Extensive experiments on benchmark datasets demonstrate that the mechanisms have good accuracy and are efficient

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 $n^{O(\log(1/\bar{\alpha}))}$