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We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the “sub-optimality” of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines—stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)—which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. [ABSTRACT FROM AUTHOR]

This paper develops a two-stage stochastic model to investigate the evolution of random fields on the unit sphere S 2 in R 3. The model is defined by a time-fractional stochastic diffusion equation on S 2 governed by a diffusion operator with a time-fractional derivative defined in the Riemann–Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on S 2 as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on S 2. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree L ≥ 1. The rate of convergence of the truncation errors as a function of L and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Coupled multi-physics simulations play a crucial role in the design and operation of nuclear reactors, particularly in assessing the behavior of used nuclear fuel. This study focuses on exploring the efficacy of coupled calculations for used nuclear fuel through the integration of neutron transport and thermal-hydraulics codes. Neutronics calculations were conducted using the Monte Carlo code Serpent, while thermal-hydraulic calculations utilized the Computational Fluid Dynamics (CFD) software OpenFOAM. The investigation was focused on a VVER-440 fuel pin situated in a hexagonal coolant flow area. Three computational grids were generated, containing 0.15 million, 0.39 million, and 1.1 million computational cells, along with three variants of axial material refinement featuring 42, 21, and 10 material layers. The purpose was to analyze the impact of spatial refinement on key parameters such as multiplication factor, power flux, and temperature fields. Several relaxation factors in Picard iterations were systematically compared to enhance the convergence speed of the coupling procedure. Notably, simulations without relaxation (α = 1) resulted in oscillations in predicted results, while a low value of α led to slow convergence. The investigation revealed that employing a stochastic approximation with a varying relaxation factor coupled with a varying number of simulated particles demonstrated superior performance compared to cases with a constant relaxation factor α or a stochastic approximation with a constant number of simulated particles. Furthermore, it was observed that the resolution of axial fuel segmentation significantly influenced predicted multiplication factor k i n f and temperature profiles. Interestingly, the spatial resolution of the computational grid exhibited minimal impact on the predicted results. • CFD code OpenFOAM and Monte Carlo neutron transport code Serpent were coupled. • Relaxation of power flux improved the simulation convergence. • The stochastic approximation converged faster than the fixed number of neutrons case. • Axial segmentation of material input had a significant effect on predicted results. [ABSTRACT FROM AUTHOR]

This article investigates an event-triggered strategy to deal with the bipartite consensus issue for stochastic multiagent systems (SMASs) in a communication noisy environment. The communication network can be modeled by a random signed graph and the communication noise is compound noises (additive noise and multiplicative noise). The stochastic event-triggered bipartite consensus control protocol for SMASs with compound noises is presented by the stochastic approximation (SA) method. Due to the control gain is time-varying and agent-dependent, the implementation of event-triggered control protocol may cause out-sync of the control gain. Meanwhile, the coexistence of stochastic antagonistic information and compound noises causes it hard to turn the noises term into an error equation, which leads to the fact that the traditional error transition method is invalid for our underlying system. To deal with these challenges, the state's boundedness for each agent is first constructed by the Lyapunov method, and then the event-triggered bipartite consensus can be demonstrated via a new semi-decomposition technique. Finally, the effectiveness of the proposed SA controller is verified by two examples. [ABSTRACT FROM AUTHOR]

Consensus of networked nonlinear nonparametric time-varying systems is studied, and the nonlinearity at input may possess arbitrary growth rate. In addition, the communication noise is also under consideration. The purely distributed control algorithm is designed on the basis of distributed stochastic approximation algorithm with expanding truncations (DSAAWET). The truncation mechanism neutralizes the divergent tendency caused by unknown nonlinearities and noises, which makes the algorithm well tackle more general nonlinearities with high growth rate and more complicated structure noises. We first prove the average consensus over fixed digraph, and then extend the results to averagely connected time-varying digraphs. Finally, the validity of the algorithm is justified by numerical simulations. [ABSTRACT FROM AUTHOR]

We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism. [ABSTRACT FROM AUTHOR]

In this paper we model the spreading of the SARS-CoV-2 in Mexico by introducing a new stochastic approximation constructed from first principles, where the number of new infected individuals caused by a single infectious individual per unit time (a day), is a random variable of a time-dependent Poisson distribution. The model, structured on the basis of a Latent-Infectious-(Recovered or Deceased) (LI(RD)) compartmental approximation together with a modulation of the mean number of new infections (the Poisson parameters), provides a good tool to study theoretical and real scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a novel expectation–maximization-based simultaneous localization and mapping (SLAM) algorithm for millimeter-wave (mmW) communication systems. By fully exploiting the geometric relationship among the access point (AP) positions, the angle difference of arrival (ADOA) from the APs and the mobile terminal (MT) position, and regarding the MT positions as the latent variable of the AP positions, the proposed algorithm first reformulates the SLAM problem as the maximum likelihood joint estimation over both the AP positions and the MT positions in a latent variable model. Then, it employs a feasible stochastic approximation expectation–maximization (EM) method to estimate the AP positions. Specifically, the stochastic Monte Carlo approximation is employed to obtain the intractable expectation of the MT positions' posterior probability in the E-step, and the gradient descent-based optimization is used as a viable substitute for estimating the high-dimensional AP positions in the M-step. Further, it estimates the MT positions and constructs the indoor map based on the estimated AP topology. Due to the efficient processing capability of the stochastic approximation EM method and taking full advantage of the abundant spatial information in the crowd-sourcing ADOA data, the proposed method can achieve a better positioning and mapping performance than the existing geometry-based mmW SLAM method, which usually has to compromise between the computation complexity and the estimation performance. The simulation results confirm the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Inspired by the fact that certain randomization schemes incorporated into the stochastic (proximal) gradient methods allow for a large reduction in computational time, we incorporate such a scheme into stochastic alternating direction method of multipliers (ADMM), yielding a faster stochastic alternating direction method (FSADM) for solving a class of large scale convex composite problems. In the numerical experiments, we observe a reduction of this method in computational time compared to previous methods. More importantly, we unify the stochastic ADMM for solving general convex and strongly convex composite problems (i.e., the iterative scheme does not change when the the problem goes from strongly convex to general convex). In addition, we establish the convergence rates of FSADM for these two cases. [ABSTRACT FROM AUTHOR]

It is well known that ergodicity and the strong law of large numbers (SLLN) play important roles in stochastic control and stochastic approximations. For a class of regime-switching functional diffusion processes with infinite delay, this paper establishes exponential ergodicity in a Wasserstein distance under certain "averaging conditions." It follows from such an ergodicity property that an SLLN for additive functionals of the regime-switching diffusions is obtained based on the property of uniform mixing. Finally, an example is presented to illustrate the application of ergodicity and the SLLN to a numerical algorithm for stochastic optimization. [ABSTRACT FROM AUTHOR]

Based on the stochastic maximum principle for the partially coupled forwardbackward stochastic control system (FBSCS), a modified method of successive approximations (MSA) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales, we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs. [ABSTRACT FROM AUTHOR]

The stochastic Landau–Lifshitz–Bloch equation describes the evolution of spins in ferromagnetic materials for temperatures both below and above the Curie temperature. We consider both exchange and anisotropy energy in dimensions 1 and 2. The equation is transformed into a non-linear partial differential equation with random coefficients, which has a pathwise unique solution. This solution depends continuously on the considered Wiener process. We transform back to the original equation to conclude that the solution to the original equation depends continuously on the Wiener process considered. [ABSTRACT FROM AUTHOR]

We consider the situation where multiple transportation service providers cooperate to offer an integrated multi-modal platform to enhance the convenience to the passengers through ease in multi-modal journey planning, payment, and first and last mile connectivity. This market structure allows the multi-modal platform to coordinate profits across modes and also provide incentives to the passengers. Accordingly, in this paper, we use cooperative game theory coupled with the hyperpath-based stochastic user equilibrium framework to study such a market. We assume that the platform sets incentives (price discount or excess charge on passengers) along every edge in the transportation network. We derive the continuity and monotonicity properties of the equilibrium flow with respect to the incentives along every edge. The optimal incentives that maximize the profit of the platform are obtained through a two time-scale stochastic approximation algorithm. We use the asymmetric Nash bargaining solution to design a fair profit sharing scheme among the service providers. We show that the profit for each service provider increases after cooperation on such a platform. Finally, we complement the theoretical results through two numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler–Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to optimize autonomous performance (i.e. both longitudinal and lateral) and endurance of the quadrotor type aerial vehicle simultaneously depending on the autopilot gain coefficients and battery weight. Design/methodology/approach: Quadrotor design processes are critical to performance. Unmanned aerial vehicle durability is an important performance parameter. One of the factors affecting durability is the battery. Battery weight, energy capacity and discharge rate are important design parameters of the battery. In this study, proper autopilot gain coefficients and battery weight are obtained by using a stochastic optimization method named as simultaneous perturbation stochastic approximation (SPSA). Because there is no direct correlation between battery weight and battery energy density, artificial neural network (ANN) is benefited to obtain battery energy density corresponding to resulted battery weight found from SPSA algorithm. By using the SPSA algorithm optimum performance index is obtained, then obtained data is used for longitudinal and lateral autonomous flight simulations. Findings: With SPSA, the best proportional integrator and derivative (PID) coefficients and battery weight, energy efficiency and endurance were obtained in case of morphing. Research limitations/implications: It takes a long time to find the most suitable battery values depending on quadrotor endurance. However, this situation can be overcome with the proposed SPSA. Practical implications: It is very useful to determine quadrotor endurance, PID coefficients and morphing rate using the optimization method. Social implications: Determining quadrotor endurance, PID coefficients and morphing rate using the optimization method provides advantages in terms of time, cost and practicality. Originality/value: The proposed method improves quadrotor endurance. In addition, with the SPSA optimization method and ANN, the parameters required for endurance will be obtained faster and more securely. In addition, the energy density according to the battery weight also contributes to the clean environment and energy efficiency. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimates for a fully discrete time-implicit finite element approximation of the stochastic total variaton flow (STVF) with additive space time noise. The estimates are first derived for an implementable fully discrete approximation of a regularized STVF. We then show that the derived a posteriori estimates remain valid for the unregularized flow up to a perturbation term that can be controlled by the regularization parameter. Based on the derived a posteriori estimates we propose a pathwise algorithm for the adaptive space-time refinement and perform numerical simulation for the regularized STVF to demonstrate the behavior of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Active Disturbance Rejection Control (ADRC) emerges as a promising control method that can effectively handle uncertainties and disturbances. However, many model-based ADRC tuning methods turn laborious to achieve satisfactory control performance, when the critical process parameters are difficult to accurately obtain, especially the time delay information. To this end, this paper aims to propose a data-driven iterative tuning method for time-delayed ADRC (TD-ADRC). Based on parameter scaling technique, the quantitative correlation among control performance, robustness and normalized controller parameters are investigated. It is then used to design robust nominal controller. Then, based on the TD-ADRC inner-loop equivalent structure, an iterative feedback tuning (IFT) method is proposed to optimally obtain the nominal first order plus time delay (FOPTD) process model. Its unbiasedness and convergence are also described. With the empirical relations and the IFT stochastic approximation algorithm, a data-driven iterative tuning method for TA-ADRC is proposed, which allows a reasonable trade-off between system performance and robustness. Simulation results validate the efficacy of the proposed method, and a water-tank control experiment depicts a promising prospect in control practice. • A data-driven iterative tuning method for time-delayed ADRC is proposed considering both control performance and robustness. • An unbiased gradient estimate with respect to the model parameters under noise conditions is derived. • Control parameters are tuned iteratively through a data-driven method: iterative feedback tuning method. [ABSTRACT FROM AUTHOR]

In this paper we study a class of nonconvex optimization which involves uncertainty in the objective and a large number of nonconvex functional constraints. Challenges often arise when solving this type of problems due to the nonconvexity of the feasible set and the high cost of calculating function value and gradient of all constraints simultaneously. To handle these issues, we propose a stochastic primal-dual method in this paper. At each iteration, a proximal subproblem based on a stochastic approximation to an augmented Lagrangian function is solved to update the primal variable, which is then used to update dual variables. We explore theoretical properties of the proposed algorithm and establish its iteration and sample complexities to find an ϵ -stationary point of the original problem. Numerical tests on a weighted maximin dispersion problem and a nonconvex quadratically constrained optimization problem demonstrate the promising performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well. [ABSTRACT FROM AUTHOR]

In this research paper, we set forward a non-parametric multivariate recursive kernel regression estimator under missing data using the propensity score approach in order to describe writing word production. Our main objective is to explore cognitive processes and mental representations mobilized when a human being prepares to write a word according to the idea developed in [21]. We investigate the asymptotic properties of the proposed recursive estimator and compare them to the well known Nadaraya-Watson's regression estimator. We calculate the bias and the variance of the proposed estimator which depend on the choice of some parameters such as the stepsize and the bandwidth. We examine some data-driven procedures to select these parameters. Thus, we demonstrate that, under some optimal choices of these parameters, the MSE (Mean Squared Error) of the proposed estimator can be smaller than the one obtained by using Nadaraya Watson's regression estimator. The elaborated estimator is then applied to the behavioral data to classify some participants in groups. This classification may stand for a departure point to tackle written behavior variations. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of the constant step Stochastic Gradient Descent for the minimization of an unknown function, defined as the expectation of a non convex, non smooth, locally Lipschitz random function. As the gradient may not exist, it is replaced by a certain operator: a reasonable choice is to use an element of the Clarke subdifferential of the random function; another choice is the output of the celebrated backpropagation algorithm, which is popular amongst practioners, and whose properties have recently been studied by Bolte and Pauwels. Since the expectation of the chosen operator is not in general an element of the Clarke subdifferential of the mean function, it has been assumed in the literature that an oracle of the Clarke subdifferential of the mean function is available. As a first result, it is shown in this paper that such an oracle is not needed for almost all initialization points of the algorithm. Next, in the small step size regime, it is shown that the interpolated trajectory of the algorithm converges in probability (in the compact convergence sense) towards the set of solutions of a particular differential inclusion: the subgradient flow. Finally, viewing the iterates as a Markov chain whose transition kernel is indexed by the step size, it is shown that the invariant distribution of the kernel converge weakly to the set of invariant distribution of this differential inclusion as the step size tends to zero. These results show that when the step size is small, with large probability, the iterates eventually lie in a neighborhood of the critical points of the mean function. [ABSTRACT FROM AUTHOR]

We study an online linear programming (OLP) problem under a random input model in which the columns of the constraint matrix along with the corresponding coefficients in the objective function are independently and identically drawn from an unknown distribution and revealed sequentially over time. Virtually all existing online algorithms were based on learning the dual optimal solutions/prices of the linear programs (LPs), and their analyses were focused on the aggregate objective value and solving the packing LP, where all coefficients in the constraint matrix and objective are nonnegative. However, two major open questions were as follows. (i) Does the set of LP optimal dual prices learned in the existing algorithms converge to those of the "offline" LP? (ii) Could the results be extended to general LP problems where the coefficients can be either positive or negative? We resolve these two questions by establishing convergence results for the dual prices under moderate regularity conditions for general LP problems. Specifically, we identify an equivalent form of the dual problem that relates the dual LP with a sample average approximation to a stochastic program. Furthermore, we propose a new type of OLP algorithm, action-history-dependent learning algorithm, which improves the previous algorithm performances by taking into account the past input data and the past decisions/actions. We derive an O (log n log log n) regret bound (under a locally strong convexity and smoothness condition) for the proposed algorithm, against the O (n) bound for typical dual-price learning algorithms, where n is the number of decision variables. Numerical experiments demonstrate the effectiveness of the proposed algorithm and the action-history-dependent design. [ABSTRACT FROM AUTHOR]

In this study, we propose the use of a first-order gradient framework, the adaptive moment estimation (Adam), in conjunction with a stochastic gradient approximation, to well location and trajectory optimization problems. The Adam framework allows the incorporation of additional information from previous gradients to calculate variable-specific progression steps. As a result, this assists the search progression to be adjusted further for each variable and allows a convergence speed-up in problems where the gradients need to be approximated. We argue that under computational budget constraints, local optimization algorithms provide suitable solutions from a heuristic initial guess. Nonlinear constraints are taken into account to ensure the proposed solutions are not in violation of practical field considerations. The performance of the proposed algorithm is compared against steepest descent and generalized pattern search, using two case studies — the placement of four vertical wells and placement of 20 nonconventional (deviated, horizontal and/or slanted) wells. The results indicate that the proposed algorithm consistently outperforms the tested methods in terms computational efficiency and final optimum value. Additional discussions regarding nonconventional parameterization provide insights into simultaneous perturbation gradient approximations. [ABSTRACT FROM AUTHOR]

Field development and control optimization aim to maximize the economic profit of oil and gas production while considering several sources of uncertainty. This results in a high-dimensional optimization problem with a computationally demanding and uncertain objective function based on the simulated reservoir model. The limitations of many current robust optimization methods are: 1) it is single-level optimization (e.g. optimization of well locations/placement only; or of well production/injection control variables only) that ignores interference between the control variables from different levels; and 2) they provide a single optimal solution, whereas operational problems often add unexpected constraints likely to reduce that optimal, inflexible solution to a sub-optimal scenario. This paper presents a robust, multi-solution framework based on sequential iterative optimization of control variables at multiple levels using the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm. A systematic realization selection process, tailored to the objective of the subsequent optimization stage, is used to select a small representative ensemble of reservoir model realizations to be used for calculating the expected objective value. The estimated gradients are calculated using a 1:1 ratio mapping ensemble of control variables perturbations at each iteration onto the ensemble of selected reservoir model realizations to reduce the computational cost. An ensemble of close-to-optimum solutions is then chosen from each level (e.g. from the well placement optimization level) and transferred to the next level of optimization (e.g. where the control settings are optimized), and this loop continues until no significant improvement is observed in the expected objective value. Fit-for-purpose clustering techniques are developed to systematically select an ensemble of solutions, with maximum differences in control variables but close-to-optimum objective values, at each optimization level. The proposed framework has been tested on a benchmark case study (Brugge field). Multiple solutions are obtained with different well locations and control settings but close-to-optimum objective values. We show that suboptimal solutions from an early optimization level can approach and even outdo the optimal one at the next level(s). Results demonstrate the advantage of the developed framework in more efficient exploration of the search space and providing the much-needed operational flexibility to field operators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a stochastic self-adaptive subgradient extragradient approximation algorithm for solving the stochastic pseudomonotone variational inequality problem. The new method uses a variable stepsize generated by the simple computation at each iteration. Contrary to many known algorithms, the resulting algorithm can be easily implemented without prior knowledge of the Lipschitz constant of the mapping, and also without any line search procedure. The convergence and convergence rate of the algorithm are shown. Some numerical examples are given to illustrate the effectiveness of the proposed algorithm. Computation results show that our algorithm has the competitiveness over other related algorithms in the literature. Finally, we apply this algorithm to solve a traffic equilibrium problem. [ABSTRACT FROM AUTHOR]

We propose an adaptively weighted stochastic gradient Langevin dynamics (AWSGLD) algorithm for Bayesian learning of big data problems. The proposed algorithm is scalable and possesses a self-adjusting mechanism: It adaptively flattens the high-energy region and protrudes the low-energy region during simulations such that both Monte Carlo simulation and global optimization tasks can be greatly facilitated in a single run. The self-adjusting mechanism enables the proposed algorithm to be essentially immune to local traps. Theoretically, by showing the stability of the mean-field system and verifying the existence and regularity properties of the solution of Poisson equation, we establish the convergence of the AWSGLD algorithm, including both the convergence of the self-adapting parameters and the convergence of the weighted averaging estimators. Empirically, the AWSGLD algorithm is tested on multiple benchmark datasets including CIFAR100 and SVHN for both optimization and uncertainty estimation tasks. The numerical results indicate its great potential in Monte Carlo simulation and global optimization for modern machine learning tasks. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic iterative counterpart of the value iteration scheme wherein only noisy and possibly biased approximations of the Bellman operator are available. We call this counterpart the approximate value iteration (AVI) scheme. Neural networks are often used as function approximators, in order to counter Bellman's curse of dimensionality. In this paper, they are used to approximate the Bellman operator. Because neural networks are typically trained using sample data, errors and biases may be introduced. The design of AVI accounts for implementations with biased approximations of the Bellman operator and sampling errors. We present verifiable sufficient conditions under which AVI is stable (almost surely bounded) and converges to a fixed point of the approximate Bellman operator. To ensure the stability of AVI, we present three different yet related sets of sufficient conditions that are based on the existence of an appropriate Lyapunov function. These Lyapunov function–based conditions are easily verifiable and new to the literature. The verifiability is enhanced by the fact that a recipe for the construction of the necessary Lyapunov function is also provided. We also show that the stability analysis of AVI can be readily extended to the general case of set-valued stochastic approximations. Finally, we show that AVI can also be used in more general circumstances, that is, for finding fixed points of contractive set-valued maps. [ABSTRACT FROM AUTHOR]

The recent surge of breakthroughs in machine learning and artificial intelligence has sparked renewed interest in large-scale stochastic optimization problems that are universally considered hard. One of the most widely used methods for solving such problems is distributed asynchronous stochastic gradient descent (DASGD), a family of algorithms that result from parallelizing stochastic gradient descent on distributed computing architectures (possibly) asychronously. However, a key obstacle in the efficient implementation of DASGD is the issue of delays: when a computing node contributes a gradient update, the global model parameter may have already been updated by other nodes several times over, thereby rendering this gradient information stale. These delays can quickly add up if the computational throughput of a node is saturated, so the convergence of DASGD may be compromised in the presence of large delays. Our first contribution is that, by carefully tuning the algorithm's step size, convergence to the critical set is still achieved in mean square, even if the delays grow unbounded at a polynomial rate. We also establish finer results in a broad class of structured optimization problems (called variationally coherent), where we show that DASGD converges to a global optimum with a probability of one under the same delay assumptions. Together, these results contribute to the broad landscape of large-scale nonconvex stochastic optimization by offering state-of-the-art theoretical guarantees and providing insights for algorithm design. [ABSTRACT FROM AUTHOR]

This work addresses the problem of traffic splitting for improving the overall delay jitter performance in the uplink multi-access system. We propose a packet-scheduling paradigm based on stochastic approximation algorithm to distribute the source traffic across the multiple network paths/interfaces. We first provide an analytical model and the delay jitter analysis for an individual interface. Later we formulate the traffic splitting problem as an optimization problem to learn the optimal split across the interfaces. We share the experimental results for the video and constant bit rate traffic on real networks (Wi-Fi or cellular networks) and convergence of our system using the proposed scheme in the dynamic network environment. The paradigm proposed in the paper is general and can be adapted to different objective functions. [ABSTRACT FROM AUTHOR]

The stochastic auxiliary problem principle (APP) algorithm is a general stochastic approximation (SA) scheme that turns the resolution of an original convex optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objective. In order to derive convergence results or efficiency estimates for an SA scheme, the iterates must be random variables. This is why we prove the measurability of the iterates of the stochastic APP algorithm. Then, we extend convergence results from the Hilbert space case to the reflexive separable Banach space case. Finally, we derive efficiency estimates for the function values taken at the averaged sequence of iterates or at the last iterate, the latter being obtained by adapting the concept of modified Fejér monotonicity to our framework. [ABSTRACT FROM AUTHOR]

The two-stage stochastic linear complementarity problem (TSLCP), which can be regarded as a special and important reformulation of two-stage stochastic linear programming, has arisen in various fields, such as stochastic programming, game theory, traffic equilibrium, and theoretical economics. Considerable effort has been devoted to designing numerical methods for solving TSLCPs. A popular approach is to integrate the progressive hedging algorithm (PHA) as a sub-algorithm into a discretization framework. In this paper, aiming to solve large-scale TSLCPs, we propose two kinds of stochastic methods: the stochastic approximation method based on projection (SAP) and the dynamic sampling SAP (DS-SAP), both of which offering more direct and improved control of the computational costs of the involved subproblems, especially compared with the PHA. In particular, the linear complementarity subproblems are solved inexactly during each iteration, and the convergence analysis of both SAP and DS-SAP with an inexactness criterion is presented. Moreover, numerical implementations and practical applications demonstrate the efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

Deep neural networks (DNNs) have shown their success as high-dimensional function approximators in many applications; however, training DNNs can be challenging in general. DNN training is commonly phrased as a stochastic optimization problem whose challenges include nonconvexity, nonsmoothness, insufficient regularization, and complicated data distributions. Hence, the performance of DNNs on a given task depends crucially on tuning hyperparameters, especially learning rates and regularization parameters. In the absence of theoretical guidelines or prior experience on similar tasks, this requires solving a series of repeated training problems which can be time-consuming and demanding on computational resources. This can limit the applicability of DNNs to problems with nonstandard, complex, and scarce datasets, e.g., those arising in many scientific applications. To remedy the challenges of DNN training, we propose slimTrain, a stochastic optimization method for training DNNs with reduced sensitivity to the choice of hyperparameters and fast initial convergence. The central idea of slimTrain is to exploit the separability inherent in many DNN architectures; that is, we separate the DNN into a nonlinear feature extractor followed by a linear model. This separability allows us to leverage recent advances made for solving large-scale, linear, ill-posed inverse problems. Crucially, for the linear weights, slimTrain does not require a learning rate and automatically adapts the regularization parameter. In our numerical experiments using function approximation tasks arising in surrogate modeling and dimensionality reduction, slimTrain outperforms existing DNN training methods with the recommended hyperparameter settings and reduces the sensitivity of DNN training to the remaining hyperparameters. Since our method operates on mini-batches, its computational overhead per iteration is modest and savings can be realized by reducing the number of iterations (due to quicker initial convergence) or the number of training problems that need to be solved to identify effective hyperparameters. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of the regression estimation near the edges under censoring. For this purpose, we consider a new recursive estimator based on the stochastic approximation algorithm and Bernstein polynomials of the regression function when the response random variable is subject to random right censoring. We give the central limit theorem and the strong pointwise convergence rate for our proposed nonparametric recursive estimators under some mild conditions. Finally, we provide pointwise moderate deviation principles (MDP) for the proposed estimators. We corroborate these theoretical results through simulations as well as the analysis of a real data set. [ABSTRACT FROM AUTHOR]

In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation with additional stochastic proximal gradient steps to guarantee convergence. Based on suitable bounds on the step sizes, we establish global convergence to stationary points in expectation and an extension of the approach using variance reduction techniques is discussed. Motivated by large-scale and big data applications, we investigate a stochastic coordinate-type quasi-Newton scheme that allows to generate cheap and tractable stochastic higher order directions. Finally, numerical results on large-scale logistic regression and deep learning problems show that our proposed algorithm compares favorably with other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals. [ABSTRACT FROM AUTHOR]

We analyze the performance of dynamic mode decomposition (DMD)-based approximations of the stochastic Koopman operator for random dynamical systems where either the dynamics or observables are affected by noise. For many DMD algorithms, the presence of noise can introduce a bias in the DMD operator, leading to poor approximations of the dynamics. In particular, methods using time delayed observables, such as Hankel DMD, are biased when the dynamics are random. We introduce a new, robust DMD algorithm that can approximate the stochastic Koopman operator despite the presence of noise. We then demonstrate how this algorithm can be applied to time delayed observables, which allows us to generate a Krylov subspace from a single observable. This allows us to compute a realization of the stochastic Koopman operator using a single observable measured over a single trajectory. We test the performance of the algorithms over several examples. [ABSTRACT FROM AUTHOR]

We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces. [ABSTRACT FROM AUTHOR]

A Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order |$ \alpha \in (0, 1)$|⁠ , driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function in time, in order to obtain a regularized stochastic fractional subdiffusion problem. The regularized problem is then approximated by using the finite element method in spatial direction. The mean squared errors are proved based on the sharp estimates of the various Mittag–Leffler functions involved in the integrals. Numerical experiments are conducted to show that the numerical results are consistent with the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic age-structured HIV/AIDS model with nonlinear incidence rates is proposed. It is of great importance to develop efficient numerical approximation methods to solve this HIV/AIDS model since most stochastic partial differential equations (SPDEs) cannot be solved analytically. From the perspective of biological significance, the exact solution of the HIV/AIDS model must be nonnegative and bounded. Then a modified explicit Euler–Maruyama (EM) scheme is constructed based on a projection operator. The EM scheme could preserves the nonnegativity of the numerical solutions and also make the numerical solutions not outside the domain of the exact solutions. The convergence results between the numerical solutions and the exact solutions are analyzed, and some numerical examples are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Customers are sensitive to price when they shop from a group of substitutable products. Specifically, they are often attracted to the cheapest product in the group, even when the price difference to the next cheapest product is small. Such a phenomenon has been long noticed by the airline industry when customers purchase airline tickets. In "Network Revenue Management under a Spiked Multinomial Logit Choice Model," Cao, Kleywegt, and Wang consider a new choice model to incorporate customers' preference toward the cheapest product, which extends the popular multinomial logit choice model. They find that using this new choice model may improve airlines' revenue. Moreover, (near) optimal revenue management policies under this choice model can be computed in an efficient manner. Airline booking data have shown that the fraction of customers who choose the cheapest available fare class often is much greater than that predicted by the multinomial logit choice model calibrated with the data. For example, the fraction of customers who choose the cheapest available fare class is much greater than the fraction of customers who choose the next cheapest available one, even if the price difference is small. To model this spike in demand for the cheapest available fare class, a choice model called the spiked multinomial logit (spiked-MNL) model was proposed. We study a network revenue management problem under the spiked-MNL choice model. We show that efficient sets, that is, assortments that offer a Pareto-optimal tradeoff between revenue and resource use, are nested-by-revenue when the spike effect is nonnegative. We use this result to show how a deterministic approximation of the stochastic dynamic program can be solved efficiently by solving a small linear program. The solution of the small linear program is used to construct a booking limit policy, and we prove that the policy is asymptotically optimal. This is the first such result for a booking limit policy under a choice model, and our proof uses an approach that is different from those used for previous asymptotic optimality results. Finally, we evaluate different revenue management policies in numerical experiments using both synthetic and airline data. [ABSTRACT FROM AUTHOR]

Model space quantum Monte Carlo (MSQMC) is an extension of full configuration interaction QMC that allows us to calculate quasi-degenerate and excited electronic states by sampling the effective Hamiltonian in the model space. We introduce a novel algorithm based on the state-selective partitioning for the effective Hamiltonian using left eigenvectors to calculate several electronic states simultaneously at much less computational cost than the original MSQMC with the energy-dependent partitioning. The sampling of walkers in MSQMC is analyzed in the single reference limit using a stochastic algorithm for higher-order perturbation energies by the analogy of the deterministic case utilizing a full configuration interaction program. We further develop size-consistency corrections of the initiator adaptation (i-MSQMC) in three different ways, i.e., the coupled electron pair approximation, a posteriori, and second-order perturbative corrections. It is clearly demonstrated that most of the initiator error is originating from the deficiency of proper scaling of correlation energy due to its truncated CI nature of the initiator approximation and that the greater part of the error can be recovered by the size-consistency corrections developed in this work. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with low regular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in some fractional Sobolev space. [ABSTRACT FROM AUTHOR]

To better respond to the impact of power system-uncertain parameters on transient stability, a novel model named the parametric transient stability constrained optimal power flow (parametric TSCOPF) is proposed. It seeks the optimal control scheme of transient stability constrained optimal power flow (TSCOPF) expressed by the function of uncertain parameters in power systems. The key difficulty to solve this model lies in that the relationship between the parametric TSCOPF solution and uncertain parameters is implicit, which is hard to derive generally. To this end, this paper approximates the optimal solution of parametric TSCOPF by polynomial expressions of uncertain parameters based on the stochastic collocation method. First, the parametric TSCOPF model includes both uncertain parameters and transient stability constraints, in which the transient stability constraint is constructed as a set of polynomial expressions using the SCM. Then, to derive the relationship between the parametric TSCOPF solution and uncertain parameters, the SCM is applied to the parametric Karush–Kuhn–Tucker (KKT) conditions of the parametric TSCOPF model, so that the optimal solution of the parametric TSCOPF is approximated by using polynomial expressions with respect to uncertain parameters. The proposed parametric TSCOPF model has been tested on a 3-machine, 9-bus system, and the IEEE 145-bus system, which verifies the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]