This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.
Second, the Lazy Flipper is a deterministic alternative to the randomized search for tighter bounds proposed and analyzed in 2009 by @cite_22 . Exactly as in @cite_22 , sets of variables that are connected via potentials in the graphical model are considered and variables flipped if these flips lead to a smaller upper bound on the sum of potentials. In contrast to @cite_22 , unique representatives of these sets are visited in a deterministic order. Both algorithms maintain a current best assignment of values to the variables and are thus related with the Swendsen-Wang algorithm @cite_24 @cite_23 and Wolff algorithm @cite_33 .
@cite_24 @cite_33 @cite_22 @cite_23
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Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We present a deterministic (1+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o(1))-approximation O(n1 2D1 4+D)-time algorithm of Nanongkai [STOC 2014] by a factor of as large as n1 8, and (ii) the O(є−1logє−1)-approximation factor of Lenzen and Patt-Shamir’s O(n1 2+є+D)-time algorithm [STOC 2013] within the same running time. Our running time matches the known time lower bound of Ω(n1 2 logn + D) [Das STOC 2011] modulo some lower-order terms, thus essentially settling the status of this problem which was raised at least a decade ago [Elkin SIGACT News 2004]. It also implies a (2+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for approximating a network’s weighted diameter which almost matches the lower bound by [PODC 2012]. In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the “hitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (no(1), o(1))-hop set of size O(n1+o(1)). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [ STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+o(1))-approximation O(no(1))-time algorithm on congested cliques, and (ii) a (1+o(1))-approximation O(no(1)logW)-pass O(n1+o(1)logW)-space streaming algorithm, when edge weights are in 1, 2, …, W . The first result answers an open problem in [Nanongkai, STOC 2014]. The second result partially answers an open problem raised by McGregor in 2006 [ sublinear.info , Problem 14]. In this paper, we theoretically study the problem of binary classification in the presence of random classification noise—the learner, instead of seeing the true labels, sees labels that have independently been flipped with some small probability. Moreover, random label noise is class-conditional— the flip probability depends on the class. We provide two approaches to suitably modify any given surrogate loss function. First, we provide a simple unbiased estimator of any loss, and obtain performance bounds for empirical risk minimization in the presence of iid data with noisy labels. If the loss function satisfies a simple symmetry condition, we show that the method leads to an efficient algorithm for empirical minimization. Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain strong empirical risk bounds. This approach has a very remarkable consequence — methods used in practice such as biased SVM and weighted logistic regression are provably noise-tolerant. On a synthetic non-separable dataset, our methods achieve over 88 accuracy even when 40 of the labels are corrupted, and are competitive with respect to recently proposed methods for dealing with label noise in several benchmark datasets. The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in (2001), (1998), and (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by (2006) for a general label set. We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints objective function offered by QP and SOCP, the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCP-MS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field. Using some examples we show that the new SOCP relaxations strictly dominate the previous approaches.