http://dl.acm.org/citation.cfm?id=3446831 http://dl.acm.org/citation.cfm?id=3446831 Fri, 30 Apr 2021 00:00:00 GMT 3446831 http://dl.acm.org/citation.cfm?id=3456290
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Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial.]]> Wed, 24 Feb 2021 00:00:00 GMT 3418526 http://dl.acm.org/citation.cfm?id=3446383
We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Ω(D log n/D + log2 n) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O(D log n log log n/log D + logO(1) n)-time broadcasting algorithm.]]> Thu, 21 Jan 2021 00:00:00 GMT 3446383 http://dl.acm.org/citation.cfm?id=3431922
Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF(2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC2; the latter is a first-order theory corresponding to the complexity class NC2 consisting of problems solvable by uniform families of polynomial-size circuits and O(log2 n)-depth.]]> Wed, 13 Jan 2021 00:00:00 GMT 3431922 http://dl.acm.org/citation.cfm?id=3431725
In this article, we study the geometry of units and ideals of cyclotomic rings and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, given an ideal lattice of the cyclotomic ring of conductor m, the algorithm finds an approximation of the shortest vector by a factor exp (Õ(√ m)). This result exposes an unexpected hardness gap between these structured lattices and general lattices: The best known polynomial time generic lattice algorithms can only reach an approximation factor exp (Õ(m)). Following a recent series of attacks, these results call into question the hardness of various problems over structured lattices, such as Ideal-SVP and Ring-LWE, upon which relies the security of a number of cryptographic schemes.]]> Wed, 06 Jan 2021 00:00:00 GMT 3431725 http://dl.acm.org/citation.cfm?id=3433160
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for MAXIMUM CLIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2Õ(n2/3) for MAXIMUM CLIQUE on disk and unit ball graphs.]]> Wed, 06 Jan 2021 00:00:00 GMT 3433160 http://dl.acm.org/citation.cfm?id=3440988
We provide a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories.]]> Wed, 06 Jan 2021 00:00:00 GMT 3440988