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This paper presents a randomized scheduler for finding concurrency bugs. Like current stress-testing methods, it repeatedly runs a given test program with supplied inputs. However, it improves on stress-testing by finding buggy schedules more effectively and by quantifying the probability of missing concurrency bugs. Key to its design is the… (More)

In this paper, we consider the problem of preserving privacy in the online learning setting. Online learning involves learning from the data in real-time, so that the learned model as well as its outputs are also continuously changing. This makes preserving privacy of each data point significantly more challenging as its effect on the learned model can be… (More)

We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any… (More)

The problem of finding large cliques in random graphs and its " planted" variant, where one wants to recover a clique of size ω log (n) added to an Erd˝ os-Rényi graph G ∼ G(n, 1 2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω(√ n). By contrast, information theoretically, one… (More)

Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However, these algorithms are typically not practical. In 1976, Wolfe [21] proposed an algorithm to find the minimum Euclidean… (More)

What we'd like: f 2 NP f / 2 P/poly such that. What we'd like: f 2 NP f / 2 P/poly such that. Explicit circuit lower bounds known for very few circuit classes. What we'd like: f 2 NP f / 2 P/poly such that. Explicit circuit lower bounds known for very few circuit classes. Even for low depth circuits of AND, OR, NOT and mod-m gates, a lower bound eluded us… (More)

We show that all non-negative submodular functions have high noise-stability. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on {−1, 1} n (for any constant accuracy parameter). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular functions required… (More)

We prove that for every ε>0 and predicate P:{0,1}<sup>k</sup>-> {0,1} that supports a pairwise independent distribution, there exists an instance I of the Max P constraint satisfaction problem on n variables such that no assignment can satisfy more than a ~(|P<sup>-1</sup>(1)|)/(2<sup>k</sup>)+ε fraction of I's constraints but the degree… (More)

We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1} n. Our main result is the following structural theorem: any submodular function is-close in 2 to a real-valued decision tree (DT) of depth O(1// 2). This immediately implies that any submodular function… (More)

We study the problem of approximating and learning coverage functions. A function c : 2 [n] → R + is a coverage function, if there exists a universe U with non-negative weights w(u) for each u ∈ U and subsets A1, A2,. .. , An of U such that c(S) = u∈∪ i∈S A i w(u). Alternatively, coverage functions can be described as non-negative linear combinations of… (More)