Distributed conditional gradient online optimization with recursive variance reduction

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 20 August 2020Accepted: 01 September 2021Published online: 05 January 2022Keywordsdecentralized optimization, stochastic optimization, nonconvex optimization, variance reduction, gradient trackingAMS Subject Headings90C26, 90C15, 93A16Publication DataISSN (print): 1052-6234ISSN (online): 1095-7189Publisher: Society for Industrial and Applied MathematicsCODEN: sjope8

The object under study is a metric associated to each graph, called diameter constrained reliability. The exact evaluation of the diameter constrained reliability belongs to the class of NP-Hard problems, and becomes prohibitive in large graphs. In the literature, several estimation methods have been developed, inspired in statistics, combinatorics, algebra and other branches of knowledge. We are focused on the statistical evaluation of the diameter constrained reliability under rare event scenarios. Under these assumptions (highly reliable networks), Crude Monte Carlo method is not accurate. More sophisticated methods meet both accuracy and bounded relative error. We compare the performance of two variance reduction methods, to know, Approximate Zero Variance Importance Sampling (AZVIS) and Recursive Variance Reduction (RVR). These methods are compared to Crude Monte Carlo in terms of accuracy and computational effort. Numerical comparisons show the improvement in the global performance of these alternative statistical methods. The paper is closed with a discussion of novel hybrid methods to address network reliability analysis in robust networks, when failures represent a rare event.

In an era of data explosion and uncertain information, online optimization becomes a more and more powerful framework. And online DR-submodular maximization is an important subclass because its wide aplications in machine learning, statistics, etc., and significance for exploring general non-convex problems. In this paper, we focus on the online non-monotone DR-submodular maximizaition under general constraint set, and propose a meta-Frank-Wolfe online algorithm with appropriately choosing parameters. Based on the Lyapunov function approach in [8] and variance reduction technique in [16], we show that the proposed online algorithm attains sublinear regret against a 1/4 approximation ratio to the best fixed action in hindsight.

Variance reduction is popular in accelerating gradient descent and stochastic gradient descent for optimization problems defined on both euclidean space and Riemannian manifold. This paper further improves on existing variance reduction methods for non-convex Riemannian optimization, including R-SVRG and R-SRG/R-SPIDER by providing a unified framework for batch size adaptation. Such framework is more general than the existing works by considering retraction and vector transport and mini-batch stochastic gradients. We show that the adaptive-batch variance reduction methods require lower gradient complexities for both general non-convex and gradient dominated functions, under both finite-sum and online optimization settings. Moreover, under the new framework, we complete the analysis of R-SVRG and R-SRG, which is currently missing in the literature. We prove convergence of R-SVRG with much simpler analysis, which leads to curvature-free complexity bounds. We also show improved results for R-SRG under double-loop convergence, which match the optimal complexities as the R-SPIDER. In addition, we prove the first online complexity results for R-SVRG and R-SRG. Lastly, we discuss the potential of adapting batch size for non-smooth, constrained and second-order Riemannian optimizers. Extensive experiments on a variety of applications support the analysis and claims in the paper.

Exact evaluation of static network reliability parameters belongs to the NP-hard family, and Monte Carlo simulation is therefore a relevant tool to provide their estimations. The first goal of this work is to review a Recursive Variance Reduction (RVR) estimator, which approaches the unreliability by recursively reducing the graph from the random choice of the first working link on selected cuts. We show that the method does not verify the bounded relative error (BRE) property as reliability of individual links goes to one—that is, that the estimator is not robust in general to high reliability of links. We then propose to use the decomposition ideas of the RVR estimator in conjunction with the importance sampling technique. Two new estimators are presented: the first one—the Balanced Recursive Decomposition estimator—chooses the first working link on cuts uniformly, whereas the second—the Zero-Variance Approximation Recursive Decomposition estimator—tries to mimic the estimator with variance zero for this technique. We show that in both cases the BRE property is verified and, moreover, that a vanishing relative error (VRE) property can be obtained for the Zero-Variance Approximation RVR under specific sufficient conditions. A numerical illustration of the power of the methods is provided on several benchmark networks.

Linear models have been successfully used to establish the connections between sensor measurements and source variables in sensor networks. Sensor failures are a leading concern during the estimation of these source variables that cannot be measured directly. The reliability of a sensor system is a probabilistic evaluation of the ability of a system to withstand sensor failures. Finding the exact reliability of a linear sensor system is proven to be a #P problem. Consequently, for most practical systems, it is highly unlikely to obtain exact solutions to this problem within a reasonable timeframe. A viable alternative is to estimate the reliability using the crude Monte Carlo method. However, this method is known to be inefficient for highly reliable systems. An improved Monte Carlo approach called the Recursive Variance Reduction (RVR) method is commonly used in the literature to obtain better reliable estimates. However, the accuracy of this method banks heavily on the approach used in finding minimal cut sets of the linear sensor system. In this paper, we introduce two enhanced RVR methods in which mixed integer programming algorithms are deployed to find minimal cut sets that significantly improve the accuracy of the overall RVR technique. A case study over a wide range of test instances is conducted to establish the efficiency of the proposed methods.

A stochastic binary system (SBS) is a multicomponent on‐off system subject to random independent failures on its components. After potential failures, the state of the subsystem is ruled by a logical function (called structure function) that determines whether the system is operational or not. A SBS serves as a natural generalization of network reliability analysis, where the goal is to find the probability of correct operation of the system (in terms of connectivity, network diameter, or different measures of success). A particular subclass of interest is stochastic monotone binary systems (SMBS), which are characterized by nondecreasing structure. We explore the combinatorics of SBS, which provide building blocks for system reliability estimation, looking at minimal nonoperational subsystems, called mincuts. One key concept to understand the underlying combinatorics of SBS is duality. As methods for exact evaluation take exponential time, we discuss the use of Monte Carlo algorithms. In particular, we discuss the F‐Monte Carlo method for estimating the reliability polynomial for homogeneous SBS, the recursive variance reduction for SMBS, which builds upon the efficient determination of mincuts, and three additional methods that combine in different ways the well‐known techniques of permutation Monte Carlo and splitting. These last three methods are based on a stochastic process called the creation process, a temporal evolution of the SBS which is static by definition. All the methods are compared using different topologies, showing large efficiency gains over the basic Monte Carlo scheme.

A linear sensor system is defined as a sensor system in which the sensor measurements have a linear relationship to source variables that cannot be directly measured. Evaluation of the reliability of a general linear sensor system is a #P problem whose computational time increases exponentially with the increment of the number of sensors. To overcome the computational complexity, Monte Carlo methods are developed in this paper to approximate the sensor system's reliability. The crude Monte Carlo method is not efficient when the sensor system is highly reliable. A Monte Carlo method that has been improved for network reliability, known as the Recursive Variance Reduction (RVR) method, is further adapted for the reliability problem of linear sensor systems. To apply the RVR method, new methods are proposed to obtain minimal cut sets of the linear sensor system, particularly under the conditions where the states of some sensors are fixed as failed or functional. A case study in a multistage automotive assembly process is conducted to demonstrate the efficiency of the proposed methods.

A multi-component system is usually defined over a ground set S with m = |S| components that work (or fail) stochastically and independently, ruled by the probability vector p ∊ [0, 1]m, where p i is the probability that component i works. We study systems which can be either in “up” or “down” state, according to their ability to comply with their stated mission given the subset of components under operation, through a function φ :P(S) → {0,1}, called structure. A stochastic binary system (SBS) is the triad (S, p, φ), and the reliability r of an SBS is the probability that the system is up. The reliability evaluation of an arbitrary SBS belongs to the class of NP-Hard computational problems. Therefore, there is no polynomial time algorithm to find r for every SBS, unless P = NP. The goal of this paper is to study approximation algorithms to accurately estimate the reliability of a stochastic monotone binary system, or SMBS, where its structure is monotonous. First, two Monte Carlo approaches are discussed. Then, the Recursive Variance Reduction (RVR) method (designed originally for the particular case of network reliability) is generalized to estimate the reliability of an SBMS. The performance of these algorithms under different SMBS (inspired mainly in network design and k-out-of-m structures) is illustrated numerically. Hints and challenges for future work are discussed in the conclusions.

The RVR (recursive variance reduction) simulation technique has been used with success for the evaluation of the K-terminal reliability measure of networks where only links can fail. In this paper, we show how this technique can be adapted for computing the K-terminal residual connectedness reliability measure in the case of networks where nodes can fail. We prove that an RVR simulation of the residual connectedness reliability has a lower variance than standard Monte Carlo simulation, leading to better estimates. We study the worst-case computational complexity of the RVR method, and we discuss the influence of the node failure probability on the algorithm performance, which makes it more efficient and especially suited for very reliable networks.

In reliability analysis, the goal is to determine the probability of consistent operation of a system. We introduce the Hostile model, where the system under study is a network, and all the components may fail (both sites and links), except for a distinguished subset of sites, called terminals. The Hostile model includes the Classical Reliability model as a particular case. As a corollary, the exact reliability evaluation of a network in the Hostile model belongs to the list of N P-hard computational problems. Traditional methods for the classical reliability model such as Crude Monte Carlo, Importance Sampling and Recursive Variance Reduction are here adapted for the Hostile model. The performance of these methods is finally discussed using real-life networks.

Monte Carlo is a convenient method for statistics sampling theory of mathematical and physical. It is widely applied to the field of staff performance in geological, electric, medical, optical, water conservancy and so on. In order to effectively assess the stability and reliability of the network, this paper put forward an improved Monte Carlo evaluation method on the basis of the traditional Monte Carlo method. First of all, the paper elaborated the principle of improved Monte Carlo method and the specific method of how to calculate the stability of the network. The improved Monte Carlo method employed the recursive variance reduction method to get a smaller sample in the original sample, to ensure the estimate is unbiased. Secondly, the paper applied the ideology of permutation and combination to the network nodes and links in the model, and combined with the upper and lower boundary of network performance, to calculate the connected stability. Finally, the paper carried out experiments between the improved and traditional Monte Carlo method in Matlab simulation to contrast and analyze. The results show that the improved Monte Carlo method to evaluate the stability of the network has higher accuracy and smaller variance. So the experiment proved that it has the high feasibility and validity.

Monte Carlo simulation appears to be very useful in the evaluation of K-terminal-reliability of large communication systems because the exact algorithms are extremely time consuming. This paper shows that the well-known series-parallel reductions can be incorporated in the recursive variance reduction simulation method, leading to a more efficient estimator, as demonstrated by experimental results.