Optimal Testing for Planted Satisfiability Problems

We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas

Reweighted belief propagation and quiet planting for random K-SAT

We study the random K-satisfiability problem using a partition function where each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several new results on the properties of random K-satisfiability problem. In particular the reweighting allows to introduce a planted ensemble that generates instances that are, in some region of parameters, equivalent to random instances. We are hence able to generate at the same time a typical random SAT instance and one of its solutions. We study the relation between clustering and belief propagation fixed points and we give a direct evidence for the existence of purely entropic (rather than energetic) barriers between clusters in some region of parameters in the random K-satisfiability problem. We exhibit, in some large planted instances, solutions with a non-trivial whitening core; such solutions were known to exist but were so far never found on very large instances. Finally, we discuss algorithmic hardness of such planted instances and we determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression correcte

Hide and Seek: Scaling Machine Learning for Combinatorial Optimization via the Probabilistic Method

Applying deep learning to solve real-life instances of hard combinatorial problems has tremendous potential. Research in this direction has focused on the Boolean satisfiability (SAT) problem, both because of its theoretical centrality and practical importance. A major roadblock faced, though, is that training sets are restricted to random formulas of size several orders of magnitude smaller than formulas of practical interest, raising serious concerns about generalization. This is because labeling random formulas of increasing size rapidly becomes intractable. By exploiting the probabilistic method in a fundamental way, we remove this roadblock entirely: we show how to generate correctly labeled random formulas of any desired size, without having to solve the underlying decision problem. Moreover, the difficulty of the classification task for the formulas produced by our generator is tunable by varying a simple scalar parameter. This opens up an entirely new level of sophistication for the machine learning methods that can be brought to bear on Satisfiability. Using our generator, we train existing state-of-the-art models for the task of predicting satisfiability on formulas with 10,000 variables. We find that they do no better than random guessing. As a first indication of what can be achieved with the new generator, we present a novel classifier that performs significantly better than random guessing 99% on the same datasets, for most difficulty levels. Crucially, unlike past approaches that learn based on syntactic features of a formula, our classifier performs its learning on a short prefix of a solver's computation, an approach that we expect to be of independent interest

Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's

We present an algorithm for recovering planted solutions in two well-known models, the stochastic block model and planted constraint satisfaction problems, via a common generalization in terms of random bipartite graphs. Our algorithm matches up to a constant factor the best-known bounds for the number of edges (or constraints) needed for perfect recovery and its running time is linear in the number of edges used. The time complexity is significantly better than both spectral and SDP-based approaches. The main contribution of the algorithm is in the case of unequal sizes in the bipartition (corresponding to odd uniformity in the CSP). Here our algorithm succeeds at a significantly lower density than the spectral approaches, surpassing a barrier based on the spectral norm of a random matrix. Other significant features of the algorithm and analysis include (i) the critical use of power iteration with subsampling, which might be of independent interest; its analysis requires keeping track of multiple norms of an evolving solution (ii) it can be implemented statistically, i.e., with very limited access to the input distribution (iii) the algorithm is extremely simple to implement and runs in linear time, and thus is practical even for very large instances

NegDL: Privacy-Preserving Deep Learning Based on Negative Database

In the era of big data, deep learning has become an increasingly popular topic. It has outstanding achievements in the fields of image recognition, object detection, and natural language processing et al. The first priority of deep learning is exploiting valuable information from a large amount of data, which will inevitably induce privacy issues that are worthy of attention. Presently, several privacy-preserving deep learning methods have been proposed, but most of them suffer from a non-negligible degradation of either efficiency or accuracy. Negative database (\textit{NDB}) is a new type of data representation which can protect data privacy by storing and utilizing the complementary form of original data. In this paper, we propose a privacy-preserving deep learning method named NegDL based on \textit{NDB}. Specifically, private data are first converted to \textit{NDB} as the input of deep learning models by a generation algorithm called \textit{QK}-hidden algorithm, and then the sketches of \textit{NDB} are extracted for training and inference. We demonstrate that the computational complexity of NegDL is the same as the original deep learning model without privacy protection. Experimental results on Breast Cancer, MNIST, and CIFAR-10 benchmark datasets demonstrate that the accuracy of NegDL could be comparable to the original deep learning model in most cases, and it performs better than the method based on differential privacy

Proceedings of SAT Competition 2016 : Solver and Benchmark Descriptions

Peer reviewe

On the Complexity of Random Satisfiability Problems with Planted Solutions

The problem of identifying a planted assignment given a random $k$-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution becomes unique and can be identified given a formula with $O(n\log n)$ clauses, there are distributions over clauses for which the best known efficient algorithms require $n^{k/2}$ clauses. We propose and study a unified model for planted $k$-SAT, which captures well-known special cases. An instance is described by a planted assignment $\sigma$ and a distribution on clauses with $k$ literals. We define its distribution complexity as the largest $r$ for which the distribution is not $r$-wise independent ($1 \le r \le k$ for any distribution with a planted assignment). Our main result is an unconditional lower bound, tight up to logarithmic factors, for statistical (query) algorithms [Kearns 1998, Feldman et. al 2012], matching known upper bounds, which, as we show, can be implemented using a statistical algorithm. Since known approaches for problems over distributions have statistical analogues (spectral, MCMC, gradient-based, convex optimization etc.), this lower bound provides a rigorous explanation of the observed algorithmic gap. The proof introduces a new general technique for the analysis of statistical query algorithms. It also points to a geometric paring phenomenon in the space of all planted assignments. We describe consequences of our lower bounds to Feige's refutation hypothesis [Feige 2002] and to lower bounds on general convex programs that solve planted $k$-SAT. Our bounds also extend to other planted $k$-CSP models, and, in particular, provide concrete evidence for the security of Goldreich's one-way function and the associated pseudorandom generator when used with a sufficiently hard predicate [Goldreich 2000].Comment: Extended abstract appeared in STOC 201