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- research-articleJune 2017
Area-convexity, l∞ regularization, and undirected multicommodity flow
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 452–460https://doi.org/10.1145/3055399.3055501We show the strong-convexity assumption of regularization-based methods for solving bilinear saddle point problems may be relaxed to a weaker notion of area-convexity with respect to an alternating bilinear form. This allows bypassing the infamous '' ...
- research-articleJune 2017
Quantum entanglement, sum of squares, and the log rank conjecture
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 975–988https://doi.org/10.1145/3055399.3055488For every constant ε>0, we give an exp(Õ(∞n))-time algorithm for the 1 vs 1 - ε Best Separable State (BSS) problem of distinguishing, given an n2 x n2 matrix ℳ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-...
- research-articleJune 2017
Sum of squares lower bounds for refuting any CSP
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 132–145https://doi.org/10.1145/3055399.3055485Let P:{0,1}k → {0,1} be a nontrivial k-ary predicate. Consider a random instance of the constraint satisfaction problem (P) on n variables with Δ n constraints, each being P applied to k randomly chosen literals. Provided the constraint density satisfies ...
- research-articleJune 2017
A strongly polynomial algorithm for bimodular integer linear programming
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 1206–1219https://doi.org/10.1145/3055399.3055473We present a strongly polynomial algorithm to solve integer programs of the form max{cT x: Ax≤ b, xεℤn }, for AεℤmXn with rank(A)=n, bε≤m, cε≤n, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, ...
- research-articleJune 2017
A generalization of permanent inequalities and applications in counting and optimization
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 384–396https://doi.org/10.1145/3055399.3055469A polynomial pΕℝ[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This ...
- research-articleJune 2017
Finding approximate local minima faster than gradient descent
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 1195–1199https://doi.org/10.1145/3055399.3055464We design a non-convex second-order optimization algorithm that is guaranteed to return an approximate local minimum in time which scales linearly in the underlying dimension and the number of training examples. The time complexity of our algorithm to ...
- research-articleJune 2017
Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via operator scaling
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 397–409https://doi.org/10.1145/3055399.3055458The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous in- equalities in analysis, convex geometry and information theory, with many ...
- research-articleJune 2017
Real stable polynomials and matroids: optimization and counting
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 370–383https://doi.org/10.1145/3055399.3055457Several fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an oracle access to an m-variate real ...
- research-articleJune 2017
How well do local algorithms solve semidefinite programs?
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 604–614https://doi.org/10.1145/3055399.3055451Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing and yet poorly understood dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite ...
- research-articleJune 2017
Katyusha: the first direct acceleration of stochastic gradient methods
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 1200–1205https://doi.org/10.1145/3055399.3055448Nesterov's momentum trick is famously known for accelerating gradient descent, and has been proven useful in building fast iterative algorithms. However, in the stochastic setting, counterexamples exist and prevent Nesterov's momentum from providing ...
- research-articleJune 2017
Complexity of short Presburger arithmetic
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 812–820https://doi.org/10.1145/3055399.3055435We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by “short” we mean sentences with a bounded number of variables, quantifers, inequalities and Boolean operations; the input consists only of the integers involved in the ...
- research-articleJune 2017
Subquadratic submodular function minimization
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 1220–1231https://doi.org/10.1145/3055399.3055419Submodular function minimization (SFM) is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in polynomial time. Owing to applications in computer vision and ...
- research-articleJune 2017
Geodesic walks in polytopes
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 927–940https://doi.org/10.1145/3055399.3055416We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of polytopes in ℝn specified by m inequalities. The walk is a discrete-time simulation of a stochastic ...
- research-articleJune 2017
The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 564–575https://doi.org/10.1145/3055399.3055413We prove that the integrality gap of the Goemans-Linial semidefinite programming relaxation for the Sparsest Cut Problem is Ω(√logn) on inputs with n vertices, thus matching the previously best known upper bound (logn)1/2+o(1) up to lower-order factors. ...
- research-articleJune 2017
Kernel-based methods for bandit convex optimization
STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of ComputingPages 72–85https://doi.org/10.1145/3055399.3055403We consider the adversarial convex bandit problem and we build the first poly(T)-time algorithm with poly(n) √T-regret for this problem. To do so we introduce three new ideas in the derivative-free optimization
literature: (i) kernel methods, (ii) a ...