In the multislope ski rental problem, the user needs a certain resource for some unknown period of time. To use the resource, the user must subscribe to one of several options, each of which consists of a one-time set...
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In the multislope ski rental problem, the user needs a certain resource for some unknown period of time. To use the resource, the user must subscribe to one of several options, each of which consists of a one-time setup cost ("buying price") and cost proportional to the duration of the usage ("rental rate"). The larger the price, the smaller the rent. The actual usage time is determined by an adversary, and the goal of an algorithm is to minimize the cost by choosing the best alternative at any point in time. Multislope ski rental is a natural generalization of the classical ski rental problem (where there are only two available alternatives, namely pure rent and pure buy), which is one of the fundamental problems of online computation. The multislope ski rental problem is an abstraction of many problems, where online choices cannot be modeled by just two alternatives, e. g., power management in systems which can be shut down in parts. In this paper we study randomized algorithms for multislope ski rental. Our results include an algorithm that produces the best possible online randomized strategy for any additive instance, where the cost of switching from one alternative to another is the difference in their buying prices, and an e-competitive randomized strategy for any (not necessarily additive) instance.
Sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint. Employing convex optimization and random s...
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Sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint. Employing convex optimization and random sampling of parameter, these algorithms enable us to obtain a suboptimal solution within reasonable computational time. The suboptimal solution is feasible in a probabilistic sense and the suboptimal value belongs to an interval which contains the optimal value. The maximum of the interval is the optimal value of the robust convex optimization plus a specified tolerance. On the other hand, its minimum is the optimal value of the chance constrained optimization which is a probabilistic relaxation of the robust convex optimization, with high probability.
The multislope ski-rental problem is an extension of the classical ski-rental problem, where the player has several lease options in addition to the pure rent and buy options. For the additive general model, Lotker, P...
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The multislope ski-rental problem is an extension of the classical ski-rental problem, where the player has several lease options in addition to the pure rent and buy options. For the additive general model, Lotker, Patt-Shamir and Rawitz [in: SIAM J. Discr. Math. 26 (2012) 718-736] obtained a randomized algorithm with the competitive ratio bounded by e-r(k)/r(0) /e-1. However, obtaining a better bound on the competitive factor as a function of the slopes parameters remains an open problem in their paper. In this paper, we study randomized algorithm for the additive multislope ski rental problem, and extend the competitive ratio bound e-r(k)/r(0) /e-1 proposed by Lotker et al. to e/e-1+r(k)/r(0)
In this paper we study randomized algorithms with random input. We adapt to such algorithms the notion of probability of a false positive which is common in epidemiological studies. The probability of a false positive...
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In this paper we study randomized algorithms with random input. We adapt to such algorithms the notion of probability of a false positive which is common in epidemiological studies. The probability of a false positive takes into account both the (controlled) error of the randomization and the randomness of the input, which needs to be modeled. We illustrate our idea on two classes of problems: primality testing and fingerprinting in strings transmission. Although in both cases the randomization has low error, in the first one the probability of a false positive is very low, while in the second one it is not. We end the paper with a discussion of randomness illustrated in a textbook example. (C) 2000 Academic Press.
In the majority problem, we are given n balls coloured black or white and we are allowed to query whether two balls have the same colour or not. The goal is to find a ball of majority colour in the minimum number of q...
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In the majority problem, we are given n balls coloured black or white and we are allowed to query whether two balls have the same colour or not. The goal is to find a ball of majority colour in the minimum number of queries. The answer is known to be n - B(n) where B(n) is the number of 1's in the binary representation of n. In this paper we study randomized algorithms for determining majority, which are allowed to err with probability at most epsilon. We show that any such algorithm must have expected running time at least (2/3 - o(1)) n. Moreover, we provide a randomized algorithm which shows that this result is best possible. These extend a result of De Marco and Pelc [G. De Marco, A. Pelc, randomized algorithms for determining the majority on graphs, Combin. Probab. Comput. 15 (2006) 823-834]. (C) 2008 Elsevier B.V. All rights reserved.
Run time distributions or time-to-target plots display on the ordinate axis the probability that an algorithm will find a solution at least as good as a given target value within a given running time, shown on the abs...
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Run time distributions or time-to-target plots display on the ordinate axis the probability that an algorithm will find a solution at least as good as a given target value within a given running time, shown on the abscissa axis. Given a pair of different randomized algorithms and , we describe a numerical method that gives the probability that finds a solution at least as good as a given target value in a smaller computation time than , for the case where the runtimes of each of the two algorithms follow any runtime distribution. An illustrative example of a numerical application is also reported. We describe the perl program tttplots-compare, developed to compare time-to-target plots or general runtime distribution for measured CPU times of any two randomized heuristics. A listing of the perl program is given, and the program can also be downloaded from http://***/similar to celso/compare-tttplots..
algorithms and dynamics over networks often involve randomization and randomization can induce oscillating dynamics that fail to converge in a deterministic sense. Under assumptions of independence across time and lin...
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algorithms and dynamics over networks often involve randomization and randomization can induce oscillating dynamics that fail to converge in a deterministic sense. Under assumptions of independence across time and linearity of the updates, we show that the oscillations are ergodic if the expected dynamics is stable. We apply this result to three problems of network systems, namely, the estimation from relative measurements, the PageRank computation, and the dynamics of opinions in social networks. In these applications, the randomized dynamics is the asynchronous counterpart of a deterministic (stable) synchronous one. By ergodicity, the deterministic limit can be recovered via a time-averaging operation, which can be performed locally by each node of the network.
The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Recently there has been a great deal of interest is robust estimators, because of...
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The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Recently there has been a great deal of interest is robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator, One problem with robust estimators is that achieving high breakdown points (near 50%) has proved to be computationally demanding. In this paper we present the best known theoretical algorithm and a practical subquadratic algorithm for computing a 50% breakdown point line estimator, the Siegel or repeated median line estimator. We first present an O(n log n) randomized expected-time algorithm, where la is the number of given points. This algorithm relies, however, on sophisticated data structures. We also present a very simple O(n log(2) n) randomized algorithm for this problem, which uses no complex data structures. We provide empirical evidence that, for many realistic input distributions, the running time of this second algorithm is actually O (n log n) expected time.
Feature selection is a crucial problem in efficient machine learning,and it also greatly contributes to the explainability of machine-driven ***,like decision trees and Least Absolute Shrinkage and Selection Operator(...
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Feature selection is a crucial problem in efficient machine learning,and it also greatly contributes to the explainability of machine-driven ***,like decision trees and Least Absolute Shrinkage and Selection Operator(LASSO),can select features during ***,these embedded approaches can only be applied to a small subset of machine learning *** based methods can select features independently from machine learning models but they often suffer from a high computational *** enhance their efficiency,many randomized algorithms have been *** this paper,we propose automatic breadth searching and attention searching adjustment approaches to further speedup randomized wrapper based feature *** conduct theoretical computational complexity analysis and further explain our algorithms’generic *** conduct experiments on both synthetic and real datasets with different machine learning base *** show that,compared with existing approaches,our proposed techniques can locate a more meaningful set of features with a high efficiency.
The inference of a lexicographic rule from paired comparisons, ranking, or choice data is a discrete optimization problem that generalizes the linear ordering problem. We develop an approach to its solution using rand...
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The inference of a lexicographic rule from paired comparisons, ranking, or choice data is a discrete optimization problem that generalizes the linear ordering problem. We develop an approach to its solution using randomized algorithms. First, we show that maximizing the expected value of a randomized solution is equivalent to solving the lexicographic inference problem. As a result, the discrete problem is transformed into a continuous and unconstrained nonlinear program that can be solved, possibly only to a local optimum, using nonlinear optimization methods. Second, we show that a maximum likelihood procedure, which runs in polynomial time, can be used to implement the randomized algorithm. The maximum likelihood value determines a lower bound on the performance ratio of the randomized algorithm. We employ the proposed approach to infer lexicographic rules for individuals using data from a choice experiment for electronic tablets. These rules obtain substantially better fit and predictions than a previously described greedy algorithm, a local search algorithm, and a multinomial logit model.
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