Asymptotic Optimality Research Articles

UAV trajectory planning based on bi-directional APF-RRT* algorithm with goal-biased

In recent decades, RRT* algorithm has attracted much attention because of its asymptotic optimization. However, the RRT* algorithm still suffers from slow convergence rate and large randomness of search range. To overcome the shortcomings of this algorithm, this paper proposes UAV trajectory planning based on bi-directional APF-RRT* algorithm with goal-biased. Firstly, goal-biased strategy is used to guide the generation of random sampling points, and two mutually alternating random search trees are established by the bi-directional RRT* algorithm to perform the search, thus increasing the convergence rate of the algorithm. Secondly, the number of iterations is greatly reduced by incorporating an modified artificial potential field method into the bi-directional growth tree. In the process of smoothing the paths, a cubic spline interpolation algorithm is applied to optimize the paths to obtain the best trajectory. The combination of the two algorithms improves the direction of new node generation and reduces the path cost. Finally, the algorithm of this paper is compared with Informed-RRT*, Bi-RRT* and improved P-RRT* algorithms, and it enhances the search performance of the growing tree.

Mixed-integer simulation optimization for multi-echelon inventory problems with lost sales

We propose a mixed-integer simulation optimization framework for solving multi-echelon inventory problems with lost sales. We want to seek optimal settings of the order-up-to levels and the review intervals for warehouse and retailers. The aim is to minimize the total expected costs including the inventory holding cost, the ordering cost and the penalty cost. The proposed optimization method represents a complementary combination of ranking-and-selection procedures and stochastic-approximation algorithms for both integer-valued and real-valued variables. We provide a proof for the finite-time statistical validity of the developed algorithm. We also discuss the convergence conditions for the asymptotic optimality of our algorithm. The algorithmic performance is examined with experiments under different parameter settings and stopping conditions. During the experiments, our algorithm performs favorably in comparison to the popular Arena optimization tool, OptQuest.

Online System Identification and Optimal Control for Mission-Critical IoT Systems Over MIMO Fading Channels

With the rapid development of mobile computing, mission-critical Internet of Things (IoT) systems have become popular. Typical mission-critical IoT systems may contain complicated unknown and unstable elements and it is of particular importance to identify and stabilize them as unstable systems may experience catastrophic consequences. We consider the identification and optimal control for a mission-critical IoT system over multiple-input–multiple-output (MIMO) fading channels. First, we focus on the optimal control of the mission-critical IoT system, assuming that the system dynamics are known, and propose a novel stochastic-approximation-based algorithm to learn the optimal control solution for the IoT controller in an online manner. Second, we extend the optimal control framework to deal with the unknown mission-critical IoT system and propose a novel normalized-stochastic-gradient-descent-based algorithm to simultaneously identify and control the system in an online manner. Using the Lyapunov stability analysis, we theoretically show the asymptotic optimality of the proposed learning algorithms. Numerical results are analyzed for our proposed scheme and for several state-of-the-art learning schemes in terms of the computational complexity, convergence, and stability performance. Specifically, the proposed scheme can be implemented more than 50% faster than the state-of-the-art learning schemes. Moreover, the system identification performance of the proposed scheme can achieve a normalized system identification mean square error (MSE) of around 0.01 in 100 iterations. This is a substantial improvement compared to the baseline algorithms, where the normalized system identification MSE diverges.

Online Multivariate Anomaly Detection and Localization for High-Dimensional Settings.

This paper considers the real-time detection of abrupt and persistent anomalies in high-dimensional data streams. The goal is to detect anomalies quickly and accurately so that the appropriate countermeasures could be taken in time before the system possibly gets harmed. We propose a sequential and multivariate anomaly detection method that scales well to high-dimensional datasets. The proposed method follows a nonparametric, i.e., data-driven, and semi-supervised approach, i.e., trains only on nominal data. Thus, it is applicable to a wide range of applications and data types. Thanks to its multivariate nature, it can quickly and accurately detect challenging anomalies, such as changes in the correlation structure. Its asymptotic optimality and computational complexity are comprehensively analyzed. In conjunction with the detection method, an effective technique for localizing the anomalous data dimensions is also proposed. The practical use of proposed algorithms are demonstrated using synthetic and real data, and in variety of applications including seizure detection, DDoS attack detection, and video surveillance.

Asymptotically Optimal Inventory Control for Assemble-to-Order Systems

We consider assemble-to-order (ATO) inventory systems with a general bill of materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multistage stochastic program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate nonidentical lead times. Our component allocation policy differentiates demands based on backlog costs, bill of materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad stochastic tracking model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries. Funding: This study is based on work supported by the National Science Foundation [Grant CMMI-1363314].

The LAN property for McKean–Vlasov models in a mean-field regime

We establish the local asymptotic normality (LAN) property for estimating a multidimensional parameter in the drift of a system of N interacting particles observed over a fixed time horizon in a mean-field regime N→∞. By implementing the classical theory of Ibragimov and Hasminski, we obtain in particular sharp results for the maximum likelihood estimator that go beyond its simple asymptotic normality thanks to Hájek’s convolution theorem and strong controls of the likelihood process that yield asymptotic minimax optimality (up to constants). Our structural results shed some light to the accompanying nonlinear McKean–Vlasov experiment, and enable us to derive simple and explicit criteria to obtain identifiability and non-degeneracy of the Fisher information matrix. These conditions are also of interest for other recent studies on the topic of parametric inference for interacting diffusions.

Optimal Model Averaging of Mixed-Data Kernel-Weighted Spline Regressions

Model averaging has a rich history dating from its use for combining forecasts from time-series models (Bates and Granger) and presents a compelling alternative to model selection methods. We propose a frequentist model averaging procedure defined over categorical regression splines (Ma, Racine, and Yang) that allows for mixed-data predictors, as well as nonnested and heteroscedastic candidate models. We demonstrate the asymptotic optimality of the proposed model averaging estimator, and develop a post-averaging inference theory for it. Theoretical underpinnings are provided, finite-sample performance is evaluated, and an empirical illustration reveals that the method is capable of outperforming a range of popular model selection criteria in applied settings. An R package is available for practitioners (Racine).

Online Mapping and Motion Planning Under Uncertainty for Safe Navigation in Unknown Environments

Safe autonomous navigation is an essential and challenging problem for robots operating in highly unstructured or completely unknown environments. Under these conditions, not only robotic systems must deal with limited localization information but also their maneuverability is constrained by their dynamics and often suffers from uncertainty. In order to cope with these constraints, this article proposes an uncertainty-based framework for mapping and planning feasible motions online with probabilistic safety guarantees. The proposed approach deals with the motion, probabilistic safety, and online computation constraints by: 1) incrementally mapping the surroundings to build an uncertainty-aware representation of the environment and 2) iteratively (re)planning trajectories to goal that is kinodynamically feasible and probabilistically safe through a multilayered sampling-based planner in the belief space. In-depth empirical analyses illustrate some important properties of this approach, namely: 1) the multilayered planning strategy enables rapid exploration of the high-dimensional belief space while preserving asymptotic optimality and completeness guarantees and 2) the proposed routine for probabilistic collision checking results in tighter probability bounds in comparison to other uncertainty-aware planners in the literature. Furthermore, real-world in-water experimental evaluation on a nonholonomic torpedo-shaped autonomous underwater vehicle and simulated trials in an urban environment on an unmanned aerial vehicle demonstrate the efficacy of the method and its suitability for systems with limited onboard computational power. Note to Practitioners—Emergent robotic applications require operating in previously unmapped scenarios. This article presents a unified mapping–planning strategy that enables robots to navigate autonomously and safely in harsh environments.

Online Resource Management of Heterogeneous Cellular Networks Powered by Grid-Connected Smart Micro Grids

This paper investigates a long-term average total energy cost minimization problem via resource management, including admission control, power allocation, and Energy Sharing (ES) of renewable energy in Heterogeneous Cellular Networks powered by Grid-connected Smart Micro Grids (GSMG-HCNs). In GSMG-HCNs, both renewable and grid energy power the base stations. Unlike existing works, we consider the cost of both renewable and grid energy and formulate the power line loss process caused by ES into our model. To solve the proposed problem, we transform it into a real-time issue by the Lyapunov technique. The proposed Cost-Aware Online Resource Management (CAORM) algorithm decouples the real-time issue into two sub-problems, one of which is linear and the other is addressed based on the successive convex approximation approach. We theoretically prove the asymptotic optimality of the CAORM algorithm and a tradeoff between the average total energy cost and the average queue length. Simulation results reveal that the CAORM algorithm outperforms benchmarks in reducing total energy cost and can make appropriate decisions according to different unit costs of renewable energy. Besides, the designed distance-related ES loss rate can help obtain better solutions with lower ES losses.

Asymptotic optimality of speed-aware JSQ for heterogeneous service systems

The Join-the-Shortest-Queue (JSQ) load-balancing scheme is known to minimise the average delay of jobs in homogeneous systems consisting of identical servers. However, it performs poorly in heterogeneous systems where servers have different processing rates. Finding a delay optimal scheme remains an open problem for heterogeneous systems. In this paper, we consider a speed-aware version of the JSQ scheme for heterogeneous systems and show that it achieves delay optimality in the fluid limit. One of the key issues in establishing this optimality result for heterogeneous systems is to show that the sequence of steady-state distributions indexed by the system size is tight in an appropriately defined space. The usual technique for showing tightness by coupling with a suitably defined dominant system does not work for heterogeneous systems. To prove tightness, we devise a new technique that uses the drift of exponential Lyapunov functions. Using the non-negativity of the drift, we show that the stationary queue length distribution has an exponentially decaying tail — a fact we use to prove tightness. Another technical difficulty arises due to the complexity of the underlying state-space and the separation of two time-scales in the fluid limit. Due to these factors, the fluid-limit turns out to be a function of the invariant distribution of a multi-dimensional Markov chain which is hard to characterise. By using some properties of this invariant distribution and using the monotonicity of the system, we show that the fluid limit is has a unique and globally attractive fixed point.

An asymptotically optimal public parking lot location algorithm based on intuitive reasoning

In order to solve the problems of road traffic congestion and the increasing parking time caused by the imbalance of parking lot supply and demand, this paper proposes an asymptotically optimal public parking lot location algorithm based on intuitive reasoning to optimize the parking lot location problem. Guided by the idea of intuitive reasoning, we use walking distance as indicator to measure the variability among location data and build a combinatorial optimization model aimed at guiding search decisions in the solution space of complex problems to find optimal solutions. First, Selective Attention Mechanism (SAM) is introduced to reduce the search space by adaptively focusing on the important information in the features. Then, Quantum Annealing (QA) algorithm with quantum tunneling effect is used to jump out of the local extremum in the search space with high probability and further approach the global optimal solution. Experiments on the parking lot location dataset in Luohu District, Shenzhen, show that the proposed method has improved the accuracy and running speed of the solution, and the asymptotic optimality of the algorithm and its effectiveness in solving the public parking lot location problem are verified.

Calibrating the Scan Statistic: Finite Sample Performance Versus Asymptotics

Abstract We consider the problem of detecting an elevated mean on an interval with unknown location and length in the univariate Gaussian sequence model. Recent results have shown that using scale-dependent critical values for the scan statistic allows to attain asymptotically optimal detection simultaneously for all signal lengths, thereby improving on the traditional scan, but this procedure has been criticised for losing too much power for short signals. We explain this discrepancy by showing that these asymptotic optimality results will necessarily be too imprecise to discern the performance of scan statistics in a practically relevant way, even in a large sample context. Instead, we propose to assess the performance with a new finite sample criterion. We then present three calibrations for scan statistics that perform well across a range of relevant signal lengths: The first calibration uses a particular adjustment to the critical values and is therefore tailored to the Gaussian case. The second calibration uses a scale-dependent adjustment to the significance levels rather than to the critical values and this adjustment is therefore applicable to arbitrary known null distributions. The third calibration restricts the scan to a particular sparse subset of the scan windows and then applies a weighted Bonferroni adjustment to the corresponding test statistics. This Bonferroni scan is also applicable to arbitrary null distributions and in addition is very simple to implement. We show how to apply these calibrations for scanning in a number of distributional settings: for normal observations with an unknown baseline and a known or unknown constant variance, for observations from a natural exponential family, for potentially heteroscadastic observations from a symmetric density by employing self-normalisation in a novel way, and for exchangeable observations using tests based on permutations, ranks or signs.

Cross-Validated Loss-Based Covariance Matrix Estimator Selection in High Dimensions

The covariance matrix plays a fundamental role in many modern exploratory and inferential statistical procedures, including dimensionality reduction, hypothesis testing, and regression. In low-dimensional regimes, where the number of observations far exceeds the number of variables, the optimality of the sample covariance matrix as an estimator of this parameter is well-established. High-dimensional regimes do not admit such a convenience. Thus, a variety of estimators have been derived to overcome the shortcomings of the canonical estimator in such settings. Yet, selecting an optimal estimator from among the plethora available remains an open challenge. Using the framework of cross-validated loss-based estimation, we develop the theoretical underpinnings of just such an estimator selection procedure. We propose a general class of loss functions for covariance matrix estimation and establish accompanying finite-sample risk bounds and conditions for the asymptotic optimality of the cross-validation selector. In numerical experiments, we demonstrate the optimality of our proposed selector in moderate sample sizes and across diverse data-generating processes. The practical benefits of our procedure are highlighted in a dimension reduction application to single-cell transcriptome sequencing data.

Scalable Load Balancing in Networked Systems: A Survey of Recent Advances

In this survey we provide an overview of recent advances on scalable load balancing schemes which provide favorable delay performance and yet require minimal implementation overhead. The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. The join-the-shortest-queue (JSQ) policy yields vanishing delays as $N$ grows large, as in a centralized queuing arrangement, but involves a prohibitive communication burden. In contrast, JSQ($d$) schemes that assign an incoming task to a server with the shortest queue among $d$ servers selected uniformly at random require little communication, but lead to constant delays. In order to examine this fundamental trade-off between delay performance and implementation overhead, we discuss a body of recent research on JSQ($d(N)$) schemes in which the diversity parameter $d(N)$ depends on $N$ and investigate the growth rate of $d(N)$ required to match the optimal JSQ performance on fluid and diffusion scales. Stochastic coupling techniques and scaling limits play an instrumental role in establishing this asymptotic optimality. We demonstrate how this methodology carries over to infinite-server settings, finite buffers, multiple dispatchers, servers arranged on graph topologies, and token-based load balancing schemes such as join-the-idle-queue (JIQ), thus providing a broad overview of the main trends in the field.

Model averaging for generalized linear models in fragmentary data prediction

ABSTRACT Fragmentary data is becoming more and more popular in many areas which brings big challenges to researchers and data analysts. Most existing methods dealing with fragmentary data consider a continuous response while in many applications the response variable is discrete. In this paper, we propose a model averaging method for generalized linear models in fragmentary data prediction. The candidate models are fitted based on different combinations of covariate availability and sample size. The optimal weight is selected by minimizing the Kullback–Leibler loss in the completed cases and its asymptotic optimality is established. Empirical evidences from a simulation study and a real data analysis about Alzheimer disease are presented.

UAV trajectory planning in cluttered environments based on PF-RRT* algorithm with goal-biased strategy

In recent decades, Rapidly-exploring Random Tree star(RRT*) with asymptotic optimality has attracted much attention in path planning algorithm, but it suffers from slow convergence. Hence to solve the drawback, this paper proposes a novel Unmanned Aerial Vehicle(UAV) trajectory planning in cluttered environments based on PF-RRT* algorithm with goal-biased strategy. It creates a novel parent node for the new node near the obstacle by dichotomy method, instead of updating the parent node in the existing random tree nodes, which considerably decreases the path cost. The improved artificial potential field(APF) is proposed to guide the growth of the random tree towards the target point by adding random point attraction, target point attraction and obstacle repulsion, which not only addresses the local minimum problem, but also boosts the search rate of the random tree. The algorithm proposed in this paper combines with goal-biased strategy to obtain higher quality sampling points during the sampling process. Finally, the simulation verifies that the proposed algorithm is greatly optimized in terms of the number of iterations, convergence rate and path cost.

Path Planning for Multiple Unmanned Vehicles (MUVs) Formation Shape Generation Based on Dual RRT Optimization

In this paper, dual RRT optimization is proposed to solve the formation shape generation problem for a large number of MUVs. Since large numbers of MUVs are prone to collision during formation shape generation, this paper considers the use of path planning algorithms to solve the collision avoidance problem. Additionally, RRT as a commonly used path planning algorithm has non-optimal solutions and strong randomness. Therefore, this paper proposes a dual RRT optimization to improve the drawbacks of RRT, which is applicable to the formation shape generation of MUVs. First, an initial global path can be obtained quickly by taking advantage of RRT-connect. After that, RRT* is used to optimize the initial global path locally. After finding the section that needs to be optimized, RRT* performs a new path search on the section and replaces the original path. Due to its asymptotic optimality, the path obtained by RRT* is shorter and smoother than the initial path. Finally, the algorithm can further optimize the path results by introducing a path evaluation function to determine the results of multiple runs. The experimental results show that the dual RRT operation optimization can greatly reduce the running time while avoiding obstacles and obtaining better path results than the RRT* algorithm. Moreover, multiple runs still ensure stable path results. The formation shape generation of MUVs can be completed in the shortest time using dual RRT optimization.

Model averaging for asymptotically optimal combined forecasts

We propose a model-averaging (MA) method for constructing an asymptotically optimal combination of a set of point forecast sequences generated from a class of predictive regressions. The asymptotic optimality is defined in terms of approximating an unknown conditional-mean sequence based on the local(-to-zero) asymptotics. Our method has the following essential features. First, it is more general than combining a set of single point forecasts. Second, the asymptotic optimality is generally dependent on the estimation scheme and the asymptotic ratio of the length of forecast sequence relative to the in-sample size. Third, the asymptotically optimal weights may be consistently estimated under suitable conditions, while it needs the time series to be sufficiently long. We also assess the forecasting performance of our method using simulation data and real data.

Sequential Multi-Hypothesis Testing in Multi-Armed Bandit Problems: An Approach for Asymptotic Optimality

We consider a multi-hypothesis testing problem involving a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -armed bandit. Each arm’s signal follows a distribution from a vector exponential family. The actual parameters of the arms are unknown to the decision maker. The decision maker incurs a delay cost for delay until a decision and a switching cost whenever he switches from one arm to another. His goal is to minimise the overall cost until a decision is reached on the true hypothesis. Of interest are policies that satisfy a given constraint on the probability of false detection. This is a sequential decision making problem where the decision maker gets only a limited view of the true state of nature at each stage, but can control his view by choosing the arm to observe at each stage. An information-theoretic lower bound on the total cost (expected time for a reliable decision plus total switching cost) is first identified, and a variation on a sequential policy based on the generalised likelihood ratio statistic is then studied. Due to the vector exponential family assumption, the signal processing at each stage is simple; the associated conjugate prior distribution on the unknown model parameters enables easy updates of the posterior distribution. The proposed policy, with a suitable threshold for stopping, is shown to satisfy the given constraint on the probability of false detection. Under a continuous selection assumption, the policy is also shown to be asymptotically optimal in terms of the total cost among all policies that satisfy the constraint on the probability of false detection.

A Unifying Theory of Thompson Sampling for Continuous Risk-Averse Bandits

This paper unifies the design and the analysis of risk-averse Thompson sampling algorithms for the multi-armed bandit problem for a class of risk functionals ρ that are continuous and dominant. We prove generalised concentration bounds for these continuous and dominant risk functionals and show that a wide class of popular risk functionals belong to this class. Using our newly developed analytical toolkits, we analyse the algorithm ρ-MTS (for multinomial distributions) and prove that they admit asymptotically optimal regret bounds of risk-averse algorithms under the CVaR, proportional hazard, and other ubiquitous risk measures. More generally, we prove the asymptotic optimality of ρ-MTS for Bernoulli distributions for a class of risk measures known as empirical distribution performance measures (EDPMs); this includes the well-known mean-variance. Numerical simulations show that the regret bounds incurred by our algorithms are reasonably tight vis-à-vis algorithm-independent lower bounds.