27,993 research outputs found

    Nonlinear response and fluctuation dissipation relations

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    A unified derivation of the off equilibrium fluctuation dissipation relations (FDR) is given for Ising and continous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDR allows to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second one, the important problem of the definition of an off equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.Comment: 24 pages, 6 figure

    Improved Approximation Algorithms for Stochastic Matching

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    In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node vv upper-bounds the number of edges incident to vv that can be probed. The goal is to maximize the expected profit of the constructed matching. For this problem Bansal et al. (Algorithmica 2012) provided a 33-approximation algorithm for bipartite graphs, and a 44-approximation for general graphs. In this work we improve the approximation factors to 2.8452.845 and 3.7093.709, respectively. We also consider an online version of the bipartite case, where one side of the partition arrives node by node, and each time a node bb arrives we have to decide which edges incident to bb we want to probe, and in which order. Here we present a 4.074.07-approximation, improving on the 7.927.92-approximation of Bansal et al. The main technical ingredient in our result is a novel way of probing edges according to a random but non-uniform permutation. Patching this method with an algorithm that works best for large probability edges (plus some additional ideas) leads to our improved approximation factors

    Stochastic determination of matrix determinants

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    Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly but are only represented indirectly in form of a computer routine. Such a routine implements the transformation a data vector undergoes under matrix multiplication. While efficient probing routines to estimate a matrix's diagonal or trace, based solely on such computationally affordable matrix-vector multiplications, are well known and frequently used in signal inference, there is no stochastic estimate for its determinant. We introduce a probing method for the logarithm of a determinant of a linear operator. Our method rests upon a reformulation of the log-determinant by an integral representation and the transformation of the involved terms into stochastic expressions. This stochastic determinant determination enables large-size applications in Bayesian inference, in particular evidence calculations, model comparison, and posterior determination.Comment: 8 pages, 5 figure

    Decision making under uncertainty

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    Almost all important decision problems are inevitably subject to some level of uncertainty either about data measurements, the parameters, or predictions describing future evolution. The significance of handling uncertainty is further amplified by the large volume of uncertain data automatically generated by modern data gathering or integration systems. Various types of problems of decision making under uncertainty have been subject to extensive research in computer science, economics and social science. In this dissertation, I study three major problems in this context, ranking, utility maximization, and matching, all involving uncertain datasets. First, we consider the problem of ranking and top-k query processing over probabilistic datasets. By illustrating the diverse and conflicting behaviors of the prior proposals, we contend that a single, specific ranking function may not suffice for probabilistic datasets. Instead we propose the notion of parameterized ranking functions, that generalize or can approximate many of the previously proposed ranking functions. We present novel exact or approximate algorithms for efficiently ranking large datasets according to these ranking functions, even if the datasets exhibit complex correlations or the probability distributions are continuous. The second problem concerns with the stochastic versions of a broad class of combinatorial optimization problems. We observe that the expected value is inadequate in capturing different types of risk-averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function. We present a polynomial time approximation algorithm with additive error ε for any ε > 0, under certain conditions. Our result generalizes and improves several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. The third is the stochastic matching problem which finds interesting applications in online dating, kidney exchange and online ad assignment. In this problem, the existence of each edge is uncertain and can be only found out by probing the edge. The goal is to design a probing strategy to maximize the expected weight of the matching. We give linear programming based constant-factor approximation algorithms for weighted stochastic matching, which answer an open question raised in prior work

    Submodular Stochastic Probing on Matroids

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    In a stochastic probing problem we are given a universe EE, where each element eEe \in E is active independently with probability pep_e, and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements --- if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13), and provides a unified view of a number of problems. Thus far, all the results falling under this general framework pertain mainly to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a (11/e)/(kin+kout+1)(1 - 1/e)/(k_{in} + k_{out}+1)-approximation algorithm for the case in which we are given kink_{in} matroids as inner constraints and koutk_{out} matroids as outer constraints. Additionally, we obtain an improved 1/(kin+kout)1/(k_{in} + k_{out})-approximation algorithm for linear objective functions

    The Price of Information in Combinatorial Optimization

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    Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of nn independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.Comment: SODA 201

    Online Contention Resolution Schemes

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    We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider. We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.Comment: 33 pages. To appear in SODA 201

    DMFSGD: A Decentralized Matrix Factorization Algorithm for Network Distance Prediction

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    The knowledge of end-to-end network distances is essential to many Internet applications. As active probing of all pairwise distances is infeasible in large-scale networks, a natural idea is to measure a few pairs and to predict the other ones without actually measuring them. This paper formulates the distance prediction problem as matrix completion where unknown entries of an incomplete matrix of pairwise distances are to be predicted. The problem is solvable because strong correlations among network distances exist and cause the constructed distance matrix to be low rank. The new formulation circumvents the well-known drawbacks of existing approaches based on Euclidean embedding. A new algorithm, so-called Decentralized Matrix Factorization by Stochastic Gradient Descent (DMFSGD), is proposed to solve the network distance prediction problem. By letting network nodes exchange messages with each other, the algorithm is fully decentralized and only requires each node to collect and to process local measurements, with neither explicit matrix constructions nor special nodes such as landmarks and central servers. In addition, we compared comprehensively matrix factorization and Euclidean embedding to demonstrate the suitability of the former on network distance prediction. We further studied the incorporation of a robust loss function and of non-negativity constraints. Extensive experiments on various publicly-available datasets of network delays show not only the scalability and the accuracy of our approach but also its usability in real Internet applications.Comment: submitted to IEEE/ACM Transactions on Networking on Nov. 201
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