Stochastic Setting Research Articles

Toward a Better Understanding of Randomized Greedy Matching

There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze [ 9 ]. We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. [ 12 ]. We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al. [ 1 ] that is proved to achieve a (0.5+ε)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm named Random Decision Order (RDO) that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that the RDO algorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio of MRG . Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501-approximation ratio. This result solves the open question by Chan et al. [ 4 ] and Gamlath et al. [ 12 ] about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1-1/ e )-approximation for edge-weighted bipartite graphs, which generalizes the (1-1/ e )-approximation ratio of Gamlath et al. [ 12 ] for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.

An enhanced gradient-tracking bound for distributed online stochastic convex optimization

Gradient-tracking (GT) based decentralized methods have emerged as an effective and viable alternative method to decentralized (stochastic) gradient descent (DSGD) when solving distributed online stochastic optimization problems. Initial studies of GT methods implied that GT methods have worse network dependent rate than DSGD, contradicting experimental results. This dilemma has recently been resolved, and tighter rates for GT methods have been established, which improves upon DSGD.In this work, we establish more enhanced rates for GT methods under the online stochastic convex settings. We present an alternative approach for analyzing GT methods for convex problems and over static graphs. When compared to previous analyses, this approach allows us to establish enhanced network dependent rates.

Valuation of guaranteed minimum maturity benefits under mean reversion and jump models with surrender risk

We propose an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity (VA) contracts, where the price of underlying asset is modeled by a mean reversion process with jumps. We consider the case in which early surrender is permitted, and an intensity-based framework is used to capture surrender behavior and other asset price volatilities. To analyze the contract within this complex stochastic setting, we rely on the Fourier cosine series expansion method to approximate the expectation of the value function, which serves as an estimation of the value of VA benefits. We conduct numerical examples to demonstrate the efficiency and accuracy of the proposed method. Sensitivity analysis of involved parameters is given.

Anomalous Dissipation and Lack of Selection in the Obukhov–Corrsin Theory of Scalar Turbulence

The Obukhov–Corrsin theory of scalar turbulence [21, 54] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov’s K41 theory of fully developed turbulence [47]. The scaling analysis of Obukhov and Corrsin from 1949 to 1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation. Our approach relies on a fine quantitative analysis of the interaction between the spatial scale of the solution and the scale of the Brownian motion which represents diffusion in a stochastic Lagrangian setting. This provides a direct Lagrangian approach to anomalous dissipation which is fundamental in order to get detailed insight on the behavior of the solution. Exploiting further this approach, we also show that for a velocity field in Cα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\alpha $$\\end{document} of space and time (for an arbitrary 0≤α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le \\alpha < 1$$\\end{document}) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation. This is motivated by the fundamental open problem of the selection of solutions of the Euler equations as vanishing-viscosity limit of solutions of the Navier-Stokes equations and provides a complete negative answer in the case of passive advection.

AdaSAM: Boosting sharpness-aware minimization with adaptive learning rate and momentum for training deep neural networks

Sharpness aware minimization (SAM) optimizer has been extensively explored as it can generalize better for training deep neural networks via introducing extra perturbation steps to flatten the landscape of deep learning models. Integrating SAM with adaptive learning rate and momentum acceleration, dubbed AdaSAM, has already been explored empirically to train large-scale deep neural networks without theoretical guarantee due to the triple difficulties in analyzing the coupled perturbation step, adaptive learning rate and momentum step. In this paper, we try to analyze the convergence rate of AdaSAM in the stochastic non-convex setting. We theoretically show that AdaSAM admits a O(1/bT) convergence rate, which achieves linear speedup property with respect to mini-batch size b. Specifically, to decouple the stochastic gradient steps with the adaptive learning rate and perturbed gradient, we introduce the delayed second-order momentum term to decompose them to make them independent while taking an expectation during the analysis. Then we bound them by showing the adaptive learning rate has a limited range, which makes our analysis feasible. To the best of our knowledge, we are the first to provide the non-trivial convergence rate of SAM with an adaptive learning rate and momentum acceleration. At last, we conduct several experiments on several NLP tasks and the synthetic task, which show that AdaSAM could achieve superior performance compared with SGD, AMSGrad, and SAM optimizers.

Design-space dimensionality reduction in global optimization of functional surfaces: recent developments and way forward

ABSTRACT In shape optimization of complex industrial products (such as, but not limited to, hull forms, rudder and appendages, propellers), there exists an inherent similarity between global optimization (GO) and uncertainty quantification (UQ): they rely on an extensive exploration of the design and operational spaces, respectively; often, they need local refinements to ensure accurate identification of optimal solutions or probability density regions (such as distribution tails), respectively; they both are dramatically affected by the curse of dimensionality as GO and UQ algorithms' complexity and especially computational cost rapidly increase with the problem dimension. Therefore, there exists a natural ground for transferring dimensionality reduction methods from UQ to GO. These enable the efficient exploration of large design spaces in shape optimization, which, in turn, enable global optimization (possibly in a multidisciplinary and stochastic setting). The paper reviews and discusses recent techniques for design-space dimensionality reduction in shape optimization, based on the Karhunen-Loève expansion (equivalent to proper orthogonal decomposition and, at the discrete level, principal component analysis). An example is shown and discussed for the hydrodynamic optimization of a ship hull.

Penalized sieve estimation of zero‐inefficiency stochastic frontiers

SummaryStochastic frontier models for cross‐sectional data typically assume that the one‐sided distribution of firm‐level inefficiency is continuous. However, it may be reasonable to hypothesize that inefficiency is continuous except for a discrete mass at zero capturing fully efficient firms (zero‐inefficiency). We propose a sieve‐type density estimator for such a mixture distribution in a nonparametric stochastic frontier setting under a unimodality‐at‐zero assumption. Consistency, rates of convergence and asymptotic normality of the estimators are established, as well as a test of the zero‐inefficiency hypothesis. Simulations and two applications are provided to demonstrate the practicality of the method.

Z-relation-based multistage decision making

Multistage decision problems have attracted interest in various fields. Starting from the pioneering work of Bellman and Zadeh, a large class of comprehensive studies for fuzzy, stochastic and other settings were proposed. The main issue missed in the existing works is partial reliability of information. Indeed, partial reliability of information about states and actions is important for multistage decisions. Particularly, information about future state of a system resulting from current strategies is partially reliable due to uncertainty. To account for partial reliability aspect, Zadeh introduced the concept of a Z-number. In this paper, we use this concept to consider multistage decision making under partially reliable information. States, strategies, and goals are described by using Z-numbers. State transition matrix that describes evolution of a system in time is formed by using Z-number-based relations. This description leads to a problem of dynamic programming in Z-environment. We use Z-numbers to account for the fact that information related to state transitions is partially reliable – we are partially sure in a result of each transition. The reason behind partial reliability is complexity and uncertainty of real-world problems. A typical example and an application study are used to illustrate the approach.

Provably Good Region Partitioning for On-Time Last-Mile Delivery

Managing on-time delivery systems is challenging because of the underlying uncertainties and combinatorial nature of the routing decision. In practice, the efficiency of such systems also hinges on the driver’s familiarity with the local neighborhood. In “Provably Good Region Partitioning for On-Time Last-Mile Delivery,” Carlsson et al. study a region partitioning policy to minimize the expected delivery time of customer orders in a stochastic and dynamic setting. This policy assigns every driver to a subregion, ensuring that drivers are only dispatched to their territories. The authors characterize the structure of the optimal partitioning policy and show its expected on-time performance converges to that of the flexible dispatching policy in heavy traffic. The optimal characterization features two insightful conditions that are critical to the on-time performance of last-mile delivery systems. Furthermore, the paper develops partitioning algorithms with performance guarantees, leveraging ham sandwich cuts and three-partitions from discrete geometry.

Towards More General Loss and Setting in Unsupervised Domain Adaptation

In this paper, we present an analysis of unsupervised domain adaptation with a series of theoretical and algorithmic results. We derive a novel Rényi- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> divergence-based generalization bound, which is tailored to domain adaptation algorithms with arbitrary loss functions in a stochastic setting. Moreover, our theoretical results provide new insights into the assumptions for successful domain adaptation: the closeness between the conditional distributions of the domains and the Lipschitzness on the source domain. With these assumptions, we reveal the following: if their conditional generation distributions are close, the Lipschitzness property of the target domain can be transferred from the Lipschitzness on the source domain, without knowing the exact target distribution. Motivated by our analysis and assumptions, we further derive practical principles for deep domain adaptation: 1) Rényi-2 adversarial training for marginal distributions matching and 2) Lipschitz regularization for the classifier. Our experimental results on both synthetic and real-world datasets support our theoretical findings and the practical efficiency of the proposed principles.

Stochastic Analysis of Control Systems Subject to Communication and Computation Faults

Control theory allows one to design controllers that are robust to external disturbances, model simplification, and modelling inaccuracy. Researchers have investigated whether the robustness carries on to the controller’s digital implementation, mostly looking at how the controller reacts to either communication or computational problems. Communication problems are typically modelled using random variables (i.e., estimating the probability that a fault will occur during a transmission), while computational problems are modelled using deterministic guarantees on the number of deadlines that the control task has to meet. These fault models allow the engineer to both design robust controllers and assess the controllers’ behaviour in the presence of isolated faults. Despite being very relevant for the real-world implementations of control system, the question of what happens when these faults occur simultaneously does not yet have a proper answer. In this paper, we answer this question in the stochastic setting, using the theory of Markov Jump Linear Systems to provide stability contracts with almost sure guarantees of convergence. For linear time-invariant Markov jump linear systems, mean square stability implies almost sure convergence – a property that is central to our investigation. Our research primarily emphasises the validation of this property for closed-loop systems that are subject to packet losses and computational overruns, potentially occurring simultaneously. We apply our method to two case studies from the recent literature and show their robustness to a comprehensive set of faults. We employ closed-loop system simulations to empirically derive performance metrics that elucidate the quality of the controller implementation, such as the system settling time and the integral absolute error.

Mixing Renewable Energy with Pumped Hydropower Storage: Design Optimization under Uncertainty and Other Challenges

Hybrid renewable energy systems, complemented by pumped hydropower storage, have become increasingly popular amidst the increase in renewable energy penetration. Such configurations are even more prosperous in remote regions that are typically not connected to the mainland power grid, where the energy independence challenge intensifies. This research focuses on the design of such systems from the perspective of establishing an optimal mix of renewable sources that takes advantage of their complementarities and synergies, combined with the versatility of pumped hydropower storage. However, this design is subject to substantial complexities, due to the multiple objectives and constraints to fulfill, on the one hand, and the inherent uncertainties, on the other, which span over all the underlying processes, i.e., external and internal. In this vein, we utilize a proposed hybrid renewable energy system layout for the Aegean Island of Sifnos, Greece, to develop and evaluate a comprehensive simulation-optimization scheme in deterministic and, eventually, stochastic settings, revealing the design problem under the umbrella of uncertainty. In particular, we account for three major uncertain elements, namely, wind velocity (natural process), energy demand (anthropogenic process), and wind-to-power conversion (internal process, expressed in terms of a probabilistic power curve). Emphasis is also given to the decision-making procedure regarding the system’s key design parameters (reservoir size and solar power capacity), which is achieved by thoroughly interpreting the uncertainty-aware optimization outcomes. Finally, since the proposed pumped hydropower storage uses the sea as the lower reservoir, additional technical challenges are addressed.

Fractal–fractional order stochastic chaotic model: A synchronization study

This manuscript provides the existence and uniqueness of solution and synchronization of fractal dimensional fractional order stochastic chaotic systems in Rn space. Based on fractal–fractional analysis, Rn stochastic settings, fixed point rout, stability theory and suitable feedback controller, a valuable adequate useful sufficient conditions are determined for existence and uniqueness of solution and synchronization study of proposed model. We have addressed the nonlinear terms of the systems under consideration in this study by using local assumptions. The theory’s findings are then validated by a numerical simulation.

Quantifying Parameter Interdependence in Stochastic Discrete Models of Biochemical Systems.

Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to many reactions. As a result, their mathematical models depend on many parameters. In applications, some of the model parameters may be unknown, so their values need to be estimated from the experimental data. However, the problem of parameter value inference can be quite challenging, especially in the stochastic setting. To estimate accurately the values of a subset of parameters, the system should be sensitive with respect to variations in each of these parameters and they should not be correlated. In this paper, we propose a technique for detecting collinearity among models' parameters and we apply this method for selecting subsets of parameters that can be estimated from the available data. The analysis relies on finite-difference sensitivity estimations and the singular value decomposition of the sensitivity matrix. We illustrated the advantages of the proposed method by successfully testing it on several models of biochemical systems of practical interest.

A Derivative-Incorporated Adaptive Gradient Method for Federated Learning

As a new machine learning technique, federated learning has received more attention in recent years, which enables decentralized model training across data silos or edge intelligent devices in the Internet of Things without exchanging local raw data. All kinds of algorithms are proposed to solve the challenges in federated learning. However, most of these methods are based on stochastic gradient descent, which undergoes slow convergence and unstable performance during the training stage. In this paper, we propose a differential adaptive federated optimization method, which incorporates an adaptive learning rate and the gradient difference into the iteration rule of the global model. We further adopt the first-order moment estimation to compute the approximate value of the differential term so as to avoid amplifying the random noise from the input data sample. The theoretical convergence guarantee is established for our proposed method in a stochastic non-convex setting under full client participation and partial client participation cases. Experiments for the image classification task are performed on two standard datasets by training a neural network model, and experiment results on different baselines demonstrate the effectiveness of our proposed method.

Stable Matching for Crowdsourcing Last-Mile Delivery

This study investigates a crowdsourcing last-mile delivery problem considering orders with different destinations and time windows and crowdsourced drivers with preplanned trips. In this problem, customers announce requests on a crowdsourced delivery platform to deliver orders from a depot to their destinations and crowdsourced drivers are willing to make a detour to deliver orders in exchange for rewards provided by the platform. The crowdsourced drivers have preference lists over groups of orders based on their profits and meanwhile each order has a preference list over crowdsourced drivers based on the drivers’ arrival times at the depot. According to their preference lists, to maximize profits, crowdsourced drivers need to consider routing and scheduling decisions for delivering orders. This crowdsourced driver-order matching problem is considered to be a non-cooperative game between crowdsourced drivers. A Nash equilibrium for such a non-cooperative game is a stable matching between crowdsourced drivers and orders. We propose two exact algorithms to find stable matchings and develop a heuristic algorithm to find feasible matchings for large scale problem. The results from computational experiments demonstrate that the proposed approaches are highly efficient and effective, and the matching rate of stable matching is larger than that of feasible matching. Finally, we investigate two extensions to demonstrate the applicability of our methods and also extend to a stochastic setting with random release times of orders.

A viral co-infection model with general infection rate in deterministic and stochastic environments

In this paper, we present a model of infection in which a virus can simultaneously infect two types of target cells. The model has a general form of infection rates in deterministic and stochastic settings, incorporating bilinear infection rates, saturated incidences and half-saturated incidences. In the deterministic case, we investigate the existence and stability of infection-free steady state and infection steady state, respectively. Considering that the infection rate coefficient is affected by random noise, we built the corresponding stochastic model using the Ornstein–Uhlenbeck process. In the stochastic case, by constructing suitable Lyapunov functions, we established sufficient conditions for the stationary distribution and extinction of the model, respectively. In addition, the covariance matrix in the probability density function of the model near the infection steady state is determined. Finally, we consider the three most common forms of virus infection and perform comprehensive numerical simulations to support our theoretical results.

A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes

We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance.

No sensitivity to functional forms in the Rosenzweig–MacArthur model with strong environmental stochasticity

The classic Rosenzweig–MacArthur predator–prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig–MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator–prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues’ real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator–prey or other ecological systems.

Decentralized Riemannian Algorithm for Nonconvex Minimax Problems

The minimax optimization over Riemannian manifolds (possibly nonconvex constraints) has been actively applied to solve many problems, such as robust dimensionality reduction and deep neural networks with orthogonal weights (Stiefel manifold). Although many optimization algorithms for minimax problems have been developed in the Euclidean setting, it is difficult to convert them into Riemannian cases, and algorithms for nonconvex minimax problems with nonconvex constraints are even rare. On the other hand, to address the big data challenges, decentralized (serverless) training techniques have recently been emerging since they can reduce communications overhead and avoid the bottleneck problem on the server node. Nonetheless, the algorithm for decentralized Riemannian minimax problems has not been studied. In this paper, we study the distributed nonconvex-strongly-concave minimax optimization problem over the Stiefel manifold and propose both deterministic and stochastic minimax methods. The Steifel manifold is a non-convex set. The global function is represented as the finite sum of local functions. For the deterministic setting, we propose DRGDA and prove that our deterministic method achieves a gradient complexity of O( epsilon(-2)) under mild conditions. For the stochastic setting, we propose DRSGDA and prove that our stochastic method achieves a gradient complexity of O( epsilon(-4)). The DRGDA and DRSGDA are the first algorithms for distributed minimax optimization with nonconvex constraints with exact convergence. Extensive experimental results on the Deep Neural Networks (DNNs) training over the Stiefel manifold demonstrate the efficiency of our algorithms.