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We consider minimizing a class of nonconvex composite stochastic optimization problems, and deterministic optimization problems whose objective function consists of an expectation function (or an average of finitely many smooth functions) and a weakly convex but potentially nonsmooth function. And in this paper, we focus on the theoretical properties of two types of stochastic splitting methods for solving these nonconvex optimization problems: stochastic alternating direction method of multipliers and stochastic proximal gradient descent. In particular, several inexact versions of these two types of splitting methods are studied. At each iteration, the proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules, which allows our approaches to handle some complex regularized problems in statistics and machine learning. Under mild conditions, we obtain the convergence of the schemes and their computational complexity related to the evaluations on the component gradient of the smooth function, and find that some conclusions of their exact counterparts can be recovered. [ABSTRACT FROM AUTHOR]

We study the effective approximation for a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behavior as the scaling parameter tends to 0. This is motivated by data assimilation with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Lévy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With the help of a two-scale convergence technique, we establish effective approximation for this nonlocal stochastic partial differential equation. More precisely, we show that a nonlocal stochastic Schrödinger equation has a nonlocal effective equation. We show that it approximates the original stochastic Schrödinger equation weakly in a Sobolev-type space and strongly in L 2 space. In particular, this effective approximation holds when the nonlocal operator is the fractional Laplacian. [ABSTRACT FROM AUTHOR]

Aim: nlmixr offers first‐order conditional estimation (FOCE), FOCE with interaction (FOCEi) and stochastic approximation estimation‐maximisation (SAEM) to fit nonlinear mixed‐effect models (NLMEM). We modelled metformin's pharmacokinetic data using nlmixr and investigated SAEM and FOCEi's performance with respect to bias and precision of parameter estimates, and robustness to initial estimates. Method: Compartmental models were fitted. The final model was determined based on the objective function value and inspection of goodness‐of‐fit plots. The bias and precision of parameter estimates were compared between SAEM and FOCEi using stochastic simulations and estimations. For robustness, parameters were re‐estimated as the initial estimates were perturbed 100 times and resultant changes evaluated. Results: The absorption kinetics of metformin depend significantly on food status. Under the fasted state, the first‐order absorption into the central compartment was preceded by zero‐order infusion into the depot compartment, whereas for the fed state, the absorption into the depot was instantaneous followed by first‐order absorption from depot into the central compartment. The means of relative mean estimation error (rMEE) (MEESAEMMEEFOCEi) and rRMSE (RMSESAEMRMSEFOCEi) were 0.48 and 0.35, respectively. All parameter estimates given by SAEM appeared to be narrowly distributed and were close to the true value used for simulation. In contrast, the distribution of estimates from FOCEi were skewed and more biased. When initial estimates were perturbed, FOCEi estimates were more biased and imprecise. Discussion: nlmixr is reliable for NLMEM. SAEM was superior to FOCEi in terms of bias and precision, and more robust against initial estimate perturbations. [ABSTRACT FROM AUTHOR]

We consider a symmetric system of conservation laws and its stochastic approximation with stochastic multiplicative noise. Using the vanishing viscosity with the zero-noise limit we obtain a deterministic weak solution for some time interval. [ABSTRACT FROM AUTHOR]

Simultaneous perturbation stochastic approximation (SPSA) algorithm, an algorithm without secondary path modeling, has been applied to active noise control by some researchers. Some extended versions of this algorithm have been also developed to improve its performance. However, these existing algorithms are mostly dedicated to controlling the periodic noise instead of the broadband noise. In particular, background noise is not taken into account when SPSA algorithms are applied to control broadband noise. In this paper, an algorithm combining the cost function with the SPSA algorithm to control broadband noise has been proposed. The suggested cost function is an inner product of the estimated cross-correlation function between a reference vector and the error signal. The elements of the reference vector are composed of the reference signals at different times. Moreover, the algorithm analysis is performed and the numerical simulations are carried out to demonstrate the validity of the proposed algorithm. The results illustrate that the proposed algorithm can effectively reduce broadband noise when interference noise exists in the control system. Furthermore, the proposed algorithm has better convergence performance than other SPSA algorithms. [ABSTRACT FROM AUTHOR]

This paper analyzes a two-timescale stochastic algorithm framework for bilevel optimization. Bilevel optimization is a class of problems which exhibits a two-level structure, and its goal is to minimize an outer objective function with variables which are constrained to be the optimal solution to an (inner) optimization problem. We consider the case when the inner problem is unconstrained and strongly convex, while the outer problem is constrained and has a smooth objective function. We propose a two-timescale stochastic approximation (TTSA) algorithm for tackling such a bilevel problem. In the algorithm, a stochastic gradient update with a larger step size is used for the inner problem, while a projected stochastic gradient update with a smaller step size is used for the outer problem. We analyze the convergence rates for the TTSA algorithm under various settings: when the outer problem is strongly convex (resp. weakly convex), the TTSA algorithm finds an O (Kmax-2/3)-optimal (resp. O (Kmax-2/5)-stationary) solution, where Kmax is the total iteration number. As an application, we show that a two-timescale natural actor-critic proximal policy optimization algorithm can be viewed as a special case of our TTSA framework. Importantly, the natural actor-critic algorithm is shown to converge at a rate of O (Kmax-1/4) in terms of the gap in expected discounted reward compared to a global optimal policy. [ABSTRACT FROM AUTHOR]

In this paper, we consider discrete time approximations for stochastic differential equations with the form: X t = X 0 + ∫ 0 t f (X s) d h s + ∫ 0 t g (X s) d Y s H , t > 0 , where h : R + → R is a continuous function with locally bounded variation, f , g : R → R are measurable functions, and the integral with respect to Y t H = ∫ 0 t σ s d S s H is the pathwise Riemann-Stieltjes integral, SH is a subfractional Brownian motion with H ∈ (1 2 , 1) , σ is a deterministic (possibly discontinuous) function. [ABSTRACT FROM AUTHOR]

We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an N-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model. [ABSTRACT FROM AUTHOR]

We introduce a streaming framework for analyzing stochastic approximation/optimization problems. This streaming framework is analogous to solving optimization problems using time-varying mini-batches that arrive sequentially. We provide non-asymptotic convergence rates of various gradientbased algorithms; this includes the famous Stochastic Gradient (SG) descent (a.k.a. Robbins-Monro algorithm), mini-batch SG and time-varying mini-batch SG algorithms, as well as their iterated averages (a.k.a. Polyak-Ruppert averaging). We show (i) how to accelerate convergence by choosing the learning rate according to the time-varying mini-batches, (ii) that Polyak-Ruppert averaging achieves optimal convergence in terms of attaining the Cramer-Rao lower bound, and (iii) how time-varying mini-batches together with Polyak-Ruppert averaging can provide variance reduction and accelerate convergence simultaneously, which is advantageous for many learning problems, such as online, sequential, and large-scale learning. We further demonstrate these favorable effects for various time-varying minibatches. [ABSTRACT FROM AUTHOR]

This paper establishes the strong convergence of the tamed Euler–Maruyama (EM) approximation for McKean–Vlasov stochastic differential equations (SDEs) with super-linear drift and Hölder diffusion coefficients. By the Yamada-Watanable technique, this paper deals with the Hölder diffusion coefficient. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean-Vlasov SDEs. When one-sided local Lipschitz condition is replaced by the global condition, the convergence rate can be obtained. Finally, two examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to offer a simultaneous design approach for helicopter having swept anhedral blade tip shape and helicopter flight control system (HFCS) to minimize controller cost. Design/methodology/approach: By considering previously stated offer, control-oriented models and a stochastic optimization method are applied to minimize controller cost of the HFCS. Findings: Using simultaneous design approach for helicopters having blade tip swept and blade tip anhedral causes considerably less control effort than the helicopters not benefiting this related design approach. Practical implications: Simultaneous design approach for helicopters having blade tip swept and blade tip anhedral is applicable to consider fuel economy. Originality/value: One important novelty of this paper is using simultaneous approach for determining optimum shape of blade tip swept and anhedral. Another considerable novelty of this paper is also using a stochastic optimization method called simultaneous perturbation stochastic approximation for previously mentioned purpose. In this paper, it is also reached that using simultaneous design approach for swept anhedral helicopter blade tip shape and HFCS causes less control effort than the helicopters not using this approach. This leads to less fuel consumption and green environment. [ABSTRACT FROM AUTHOR]

Stochastic gradient descent (SGD) is an immensely popular approach for online learning in settings where data arrives in a stream or data sizes are very large. However, despite an ever-increasing volume of work on SGD, much less is known about the statistical inferential properties of SGD-based predictions. Taking a fully inferential viewpoint, this paper introduces a novel procedure termed HiGrad to conduct statistical inference for online learning, without incurring additional computational cost compared with SGD. The HiGrad procedure begins by performing SGD updates for a while and then splits the single thread into several threads, and this procedure hierarchically operates in this fashion along each thread. With predictions provided by multiple threads in place, a t-based confidence interval is constructed by decorrelating predictions using covariance structures given by a Donsker-style extension of the Ruppert-Polyak averaging scheme, which is a technical contribution of independent interest. Under certain regularity conditions, the HiGrad confidence interval is shown to attain asymptotically exact coverage probability. Finally, the performance of HiGrad is evaluated through extensive simulation studies and a real data example. An R package higrad has been developed to implement the method. [ABSTRACT FROM AUTHOR]

Using counterdiabatic (CD) driving—aiming at suppression of diabatic transition—in digitized adiabatic evolution has garnered immense interest in quantum protocols and algorithms. However, improving the approximate CD terms with a nested commutator ansatz is a challenging task. In this work, we propose a technique of finding optimal coefficients of the CD terms using a variational quantum circuit. By classical optimization routines, the parameters of this circuit are optimized to provide the coefficients corresponding to the CD terms. Then their improved performance is exemplified in Greenberger–Horne–Zeilinger state preparation on the nearest-neighbour Ising model. Finally, we also show the advantage over the usual quantum approximation optimization algorithm, in terms of fidelity with bounded time. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extension of the generalized class of recursive regression estimators to the case of strong mixing data. Then, we study the properties of these estimators and compare them with the well known Nadaraya-Watson estimator. The Bias, variance and Mean Integrated Square Error are computed explicitly. Using a selected bandwidth and a special stepsize, we showed that the proposed recursive estimators allowed us to obtain quite better results compared to the non-recursive regression estimator under α-mixing condition in terms of estimation error and much better in terms of computational costs. [ABSTRACT FROM AUTHOR]

Bayesian response adaptive clinical trials are currently evaluating experimental therapies for several diseases. Adaptive decisions, such as pre-planned variations of the randomization probabilities, attempt to accelerate the development of new treatments. The design of response adaptive trials, in most cases, requires time consuming simulation studies to describe operating characteristics, such as type I/II error rates, across plausible scenarios. We investigate large sample approximations of pivotal operating characteristics in Bayesian Uncertainty directed trial Designs (BUDs). A BUD trial utilizes an explicit metric u to quantify the information accrued during the study on parameters of interest, for example the treatment effects. The randomization probabilities vary during time to minimize the uncertainty summary u at completion of the study. We provide an asymptotic analysis (i) of the allocation of patients to treatment arms and (ii) of the randomization probabilities. For BUDs with outcome distributions belonging to the natural exponential family with quadratic variance function, we illustrate the asymptotic normality of the number of patients assigned to each arm and of the randomization probabilities. We use these results to approximate relevant operating characteristics such as the power of the BUD. We evaluate the accuracy of the approximations through simulations under several scenarios for binary, time-to-event and continuous outcome models. [ABSTRACT FROM AUTHOR]

In the process of gas lift design and condition diagnosis, the accuracy and timeliness of wellbore multiphase flow model prediction results are the basis for all subsequent work. However, for the commonly used wellbore multiphase flow pressure drop prediction models, there is a big deviation between the predicted value and the measured one, and the optimization based on the measured data is time-consuming, and it is difficult to obtain the optimal parameters of the model. Therefore, based on the well bore pressure distribution data measured quickly in area R of an oil field in Kazakhstan, a better prediction model of multiphase flow in the well bore was selected at first. Then, the simultaneous perturbation stochastic approximation (SPSA) algorithm was incorporated in the wellbore multiphase flow model to optimize the liquid holdup, which is the leading factor in the model. After repeated single well optimization and greedy selection, the optimal parameters suitable for the whole block were obtained. The example shows that the optimization speed is 10 times faster than that of Particle Swarm Optimization (PSO). After that, the optimized model was used to predict the wellbore pressure distribution, and it was found that the relative error between the measured value and the predicted one was less than 15%. [ABSTRACT FROM AUTHOR]

Stochastic epidemic models (SEMs) fit to incidence data are critical to elucidating outbreak dynamics, shaping response strategies, and preparing for future epidemics. SEMs typically represent counts of individuals in discrete infection states using Markov jump processes (MJPs), but are computationally challenging as imperfect surveillance, lack of subject‐level information, and temporal coarseness of the data obscure the true epidemic. Analytic integration over the latent epidemic process is impossible, and integration via Markov chain Monte Carlo (MCMC) is cumbersome due to the dimensionality and discreteness of the latent state space. Simulation‐based computational approaches can address the intractability of the MJP likelihood, but are numerically fragile and prohibitively expensive for complex models. A linear noise approximation (LNA) that approximates the MJP transition density with a Gaussian density has been explored for analyzing prevalence data in large‐population settings, but requires modification for analyzing incidence counts without assuming that the data are normally distributed. We demonstrate how to reparameterize SEMs to appropriately analyze incidence data, and fold the LNA into a data augmentation MCMC framework that outperforms deterministic methods, statistically, and simulation‐based methods, computationally. Our framework is computationally robust when the model dynamics are complex and applies to a broad class of SEMs. We evaluate our method in simulations that reflect Ebola, influenza, and SARS‐CoV‐2 dynamics, and apply our method to national surveillance counts from the 2013–2015 West Africa Ebola outbreak. [ABSTRACT FROM AUTHOR]

In this paper, we intend to identify two significant parameters – expected return and absolute risk aversion – in the Merton portfolio optimization problem under an exponential utility function where volatility is driven by a slow mean-reverting diffusion process. First, we find the approximate solution of the fully nonlinear Hamilton–Jacobi–Bellman equation for the Merton model by the stochastic asymptotic approximation method. Second, we estimate parameters – expected return and absolute risk aversion – through the approximate solution and prove the uniqueness and stability of the parameter identification problem. Finally, we provide an illustrative example to demonstrate the capacity and efficiency of our method. [ABSTRACT FROM AUTHOR]

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue 'Non-smooth variational problems and applications'. [ABSTRACT FROM AUTHOR]

In this article, we establish a cluster‐based gradient method (CGM) by combining K‐means clustering algorithm and stochastic gradient descent (SGD). By clustering the sampling solutions, we use the cluster centroids to represent sampling data and give an estimate to the full gradient. It is well known that the full gradient descent (FGD) can provide the steepest descent direction for finding a local minimum of the desired stochastic control problems. However, the huge computational requirements, which is proportional to the product of sample size and the numerical cost for each sample, often makes FGD cost prohibitive for large scale optimization problems. To reduce the formidable cost and the risks of getting stuck in a local minimum, SGD is proposed and can be regarded as a stochastic approximation of FGD. This, however, would result in a slow convergence due to the incorrect approximation of the iteratively update parameters. Our study shows that CGM could provide a good stochastic approximation to the full gradient with small sample size while has a more stable and faster convergence than SGD. To verify our algorithm, a stochastic elliptic control problem is selected and tested. The numerical results validate our method as a reliable gradient descent method with great potential applications in optimization problems. [ABSTRACT FROM AUTHOR]

We investigate sample average approximation (SAA) for two-stage stochastic programs without relatively complete recourse, i.e., for problems in which there are first-stage feasible solutions that are not guaranteed to have a feasible recourse action. As a feasibility measure of the SAA solution, we consider the "recourse likelihood", which is the probability that the solution has a feasible recourse action. For ϵ ∈ (0 , 1) , we demonstrate that the probability that a SAA solution has recourse likelihood below 1 - ϵ converges to zero exponentially fast with the sample size. Next, we analyze the rate of convergence of optimal solutions of the SAA to optimal solutions of the true problem for problems with a finite feasible region, such as bounded integer programming problems. For problems with non-finite feasible region, we propose modified "padded" SAA problems and demonstrate in two cases that such problems can yield, with high confidence, solutions that are certain to have a feasible recourse decision. Finally, we conduct a numerical study on a two-stage resource planning problem that illustrates the results, and also suggests there may be room for improvement in some of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the limit theory of stochastic McKean–Vlasov equations. First, we prove the optimal L p ( p ⩾ 2 ) strong convergence rate of the Wong–Zakai approximation for stochastic McKean–Vlasov equations. Then we show the support theorem for stochastic McKean–Vlasov equations. [ABSTRACT FROM AUTHOR]

This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We propose a stochastic augmented Lagrangian-type algorithm—namely, the stochastic linearized proximal method of multipliers—to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits O (K − 1 / 2) expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where K denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has a O (log (K) K − 1 / 2) constraint violation bound and O (log 3 / 2 (K) K − 1 / 2) objective bound. Numerical results demonstrate that the proposed algorithm is efficient. History: Accepted byAntonio Frangioni, Area Editor for Design & Analysis ofAlgorithms—Continuous. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11971089, 11731013, 12071055, and 11871135]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1228] or is available from the IJOC GitHub software repository (https://github.com/INFORMSJoC) at http://dx.doi.org/10.5281/zenodo.6818229. [ABSTRACT FROM AUTHOR]

We present a gradient-based algorithm for solving a class of simulation optimization problems in which the objective function is the quantile of a simulation output random variable. In contrast with existing quantile (quantile derivative) estimation techniques, which aim to eliminate the estimator bias by gradually increasing the simulation sample size, our algorithm incorporates a novel recursive procedure that only requires a single simulation sample at each step to simultaneously obtain quantile and quantile derivative estimators that are asymptotically unbiased. We show that these estimators, when coupled with the standard gradient descent method, lead to a multitime-scale stochastic approximation type of algorithm that converges to an optimal quantile value with probability one. In our numerical experiments, the proposed algorithm is applied to optimal investment portfolio problems, resulting in new solutions that complement those obtained under the classical Markowitz mean-variance framework. History: Accepted by Alice E. Smith, Editor-in-Chief; Bruno Tuffin, Area Editor for Simulation. Funding: The work of Y. Peng was supported in part by the National Natural Science Foundation of China (NSFC) [Grants 72022001, 92146003, and 71901003], and by the Key Research and Development Programof Beijing Municipal Science and Technology Commission. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1214. [ABSTRACT FROM AUTHOR]

Joint models of longitudinal process and time‐to‐event data have recently gained attention, notably to provide individualized dynamic predictions. In the presence of competing risks, models published mostly involve cause‐specific hazard functions jointly estimated with a linear or generalized linear model. Here we propose to extend the modeling to full parametric joint estimation of a nonlinear mixed‐effects model and a subdistribution hazard model. We apply this approach on 6046 patients admitted in intensive care unit (ICU) for sepsis with daily Sequential Organ Failure Assessment (SOFA) score measurements. The joint model is built on a randomly selected training set of two thirds of patients and links the current predicted SOFA measurement to the instantaneous risks of ICU death and discharge from ICU, both adjusted on the patient age. Stochastic Approximation Expectation Maximization algorithm in Monolix is used for estimation. SOFA evolution is significantly associated with both risks: 0.37, 95% confidence interval (CI) = [0.35, 0.39] for the risk of death and −0.38, 95% CI = [−0.39, −0.36] for the risk of discharge. A simulation study, inspired from the real data, shows the good estimation properties of the parameters. We assess on the validation set the added value of modeling the longitudinal SOFA follow‐up for the prediction of death compared with a model that includes only SOFA at baseline. Time‐dependent receiver operating characteristic area under the curve and Brier scores show that when enough longitudinal individual information is available, joint modeling provides better predictions. The methodology can easily be applied to other clinical applications because of the general form of the model. [ABSTRACT FROM AUTHOR]

Antenna arrays have gained significant interest in millimetre‐wave communication systems as an enabling technology to achieve higher capacity and mitigate the high propagation loss. Such arrays with a large bandwidth need to be efficiently calibrated to maximise their performance. An antenna array calibration method based on a stochastic approximation algorithm and simultaneous perturbation has been developed and the procedures to implement it in both frequency and time domains have been presented. The approaches to define objective functions and establish gradient approximations to fulfill a successful convergence for acquiring calibration coefficients in both domains have been explored. In the time domain implementation, only a fraction of the measurement time was required to calibrate an antenna array of ultrawide bandwidth compared with other methods using a perturbation technique. The effectiveness of the proposed method has been validated via numerical experiments in both domains. [ABSTRACT FROM AUTHOR]

Identifying underlying governing equations and relevant information from high-dimensional observable data has always been a challenge in physical sciences. With the recent advances in sensing technology and available datasets, various machine learning techniques have made it possible to distill underlying mathematical models from sufficiently clean and usable datasets. However, most of these techniques rely on prior knowledge of the system and noise free data obtained by simulation of physical system or by direct measurements of the signals. Hence, the inference obtained by using these techniques is often unreliable to be used in the real world where observed data are noisy and require feature engineering to extract relevant features. In this work, we provide a deep-learning framework that extracts relevant information from real-world videos of highly stochastic systems, with no prior knowledge and distills the underlying governing equation representing the system. We demonstrate this approach on videos of confined multi-agent/particle systems of ants, termites, fishes as well as a simulated confined multi-particle system with elastic collision interactions. Furthermore, we explore how these seemingly diverse systems have predictable underlying behavior. In this study, we have used motion tracking to extract spatial trajectories of individual agents/particles in a system, and by using LSTM VAE we projected these features on to a low-dimensional latent space from which the underlying differential equation representing the data was extracted using SINDy framework. [ABSTRACT FROM AUTHOR]

This article investigates resilient distributed online estimation (DOE) in unreliable directed networks with differential privacy requirements. In the network considered, some agents are subject to Byzantine attacks and thus could send arbitrary incorrect messages to their neighbors. The remaining agents aim to collaboratively estimate the value of an unknown vector parameter while protecting their private data. In this article, by adding private noises to mask the estimate, a stochastic approximation‐type resilient differentially private DOE algorithm is proposed to protect the privacy of sensitive information. A time‐varying step size is introduced to attenuate the divergence caused by the private noise, and furthermore, guarantees the convergence of the algorithm. When the directed graph is (2F$$ \Big(2F $$+1)‐robust, the algorithm is shown to be both mean square and almost sure convergence in the sense of ϵ$$ \epsilon $$‐differential privacy. A simulation example is given to verify the effectiveness and superiority of the algorithm. [ABSTRACT FROM AUTHOR]

Background: Current dose-finding designs for phase I clinical trials can correctly select the MTD in a range of 30-80% depending on various conditions based on a sample of 30 subjects. However, there is still an unmet need for efficiency and cost saving.Methods: We propose a novel dose-finding design based on Bayesian stochastic approximation. The design features utilization of dose level information through local adaptive modelling and free assumption of toxicity probabilities and hyper-parameters. It allows a flexible target toxicity rate and varying cohort size. And we extend it to accommodate historical information via prior effective sample size. We compare the proposed design to some commonly used methods in terms of accuracy and safety by simulation.Results: On average, our design can improve the percentage of correct selection to about 60% when the MTD resides at a early or middle position in the search domain and perform comparably to other competitive methods otherwise. A free online software package is provided to facilitate the application, where a simple decision tree for the design can be pre-printed beforehand.Conclusion: The paper proposes a novel dose-finding design for phase I clinical trials. Applying the design to future cancer trials can greatly improve the efficiency, consequently save cost and shorten the development period. [ABSTRACT FROM AUTHOR]

In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of the oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on a specific problem, however, starting from the work [A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM. J. Opt. 19 (2009), pp. 1574–1609] it was generally accepted that the SA is better than the SAA. We show that for the Wasserstein barycenter problem, this superiority can be inverted. We provide a detailed comparison by stating the complexity bounds for the SA and SAA implementations calculating barycenters defined with respect to optimal transport distances and entropy-regularized optimal transport distances. As a byproduct, we also construct confidence intervals for the barycenter defined with respect to entropy-regularized optimal transport distances in the ℓ 2 -norm. The preliminary results are derived for a general convex optimization problem given by the expectation to have other applications besides the Wasserstein barycenter problem. [ABSTRACT FROM AUTHOR]

This paper studies a Gaussian process approximation for a class of stochastic traffic flow models. It can be used to efficiently and accurately evaluate the joint (in the spatial and temporal sense) distribution of vehicle-density distributions in road traffic networks of arbitrary topology. The Gaussian approximation follows, via a scaling-limit argument, from a Markovian model that is consistent with discrete-space kinematic wave models. We describe in detail how this formal result can be converted into a computational procedure. The performance of our approach is demonstrated through a series of experiments that feature various realistic scenarios. Moreover, we discuss the computational complexity of our approach by assessing how computation times depend on the network size. We also argue that the (debatable) assumption that the vehicles' headways are exponentially distributed does not negatively impact the accuracy of our approximation. • A Gaussian method to evaluate the joint distribution of vehicles in traffic networks. • Detailed implementation guidelines. • Numerical examples demonstrating the use of the methodology in realistic scenarios. • A demonstration of the methodology's low computational complexity. • Relaxation of assumptions underlying the theory to include more realistic dynamics. [ABSTRACT FROM AUTHOR]

The steam generator (SG) is a critical component of the steam supply system in the nuclear power plant (NPP). Hence, it is necessary to control the SG level well to ensure the stable operation of the NPPs. However, its dynamic level response process has significant nonlinearity (such as the 'swell and shrinks' effect) and time-varying properties. As most of the SG level control systems (SGLCS) are constructed based on the Proportional-Integral-Derivative (PID) controllers with fixed parameters, the controller parameters should be optimized to improve the performance of the SGLCS. However, traditional parameters tuning methods are generally experience-based, cumbersome, and time-consuming, and it is difficult to obtain the optimal parameters. To address the challenge, this study adopts a knowledge-informed simultaneous perturbation stochastic approximation (IK-SPSA) based on adjacent iteration points information to improve the performance of the SGLCS. Rather than the traditional controller parameter tuning method, the IK-SPSA method optimizes the control system directly by using measurements of control performance. The method's efficiency lies in the following aspects. Firstly, with the help of historical information during the optimization process, the IK-SPSA can dynamically sense the current status of the optimization process. Secondly, it can accomplish the iteration step size tuning adaptively according to the optimization process's current status, reducing the optimization cost. Thirdly, it has the stochastic characteristic of simultaneous perturbation, which gives it high optimization efficiency to optimize high dimensional controller parameters. Fourthly, it incorporates an intelligent termination control mechanism to accomplish optimization progress control. This mechanism could terminate the optimization process intelligently through historical iterative process information, avoiding unnecessary iterations. The optimization method can improve the stability, safety, and economy of SGLCS. The simulation results demonstrated the effectiveness and efficiency of the method. [ABSTRACT FROM AUTHOR]

We introduce a class of Markov chains that includes models of stochastic approximation by averaging and non-averaging. Using a martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences. Finally, we apply these inequalities to stochastic approximation by averaging and empirical risk minimization. [ABSTRACT FROM AUTHOR]

We observe that embeddings into random metrics can be fruitfully used to study the L1$L_1$‐embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L1$L_1$‐distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree‐metrics. For instance, every lamplighter metric on an n$n$‐point metric space embeds bi‐Lipschitzly into L1$L_1$ with distortion O(logn)$O(\log n)$. In particular, for every finite group G$G$ the lamplighter group H=Z2≀G$H=\mathbb {Z}_2\wr G$ bi‐Lipschitzly embeds into L1$L_1$ with distortion O(loglog|H|)$O(\log \log |H|)$. In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L1$L_1$ of the lamplighter group over the d$d$‐dimensional infinite lattice Zd$\mathbb {Z}^d$ can be constructed from bi‐Lipschitz embeddings of the lamplighter graphs over finite d$d$‐dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces. [ABSTRACT FROM AUTHOR]

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in O (n log ⁡ n) operations and O(n) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. for this article are available online. [ABSTRACT FROM AUTHOR]

Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility, or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behavior of the algorithm is investigated on the uniform electron gas and the water dimer at coupled-cluster levels including up to quadruple excitations. We also describe two improvements to the original sampling algorithm, full non-composite, and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion. [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]

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]

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]

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]

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]

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]

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]

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 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]