Asymptotic Optimality Research Articles

Asymptotically Optimal Sampling Policy for Selecting Top-m Alternatives

We consider selecting the top-m alternatives from a finite number of alternatives via Monte Carlo simulation. Under a Bayesian framework, we formulate the sampling decision as a stochastic dynamic programming problem and develop a sequential sampling policy that maximizes a value function approximation one-step look ahead. To show the asymptotic optimality of the proposed procedure, the asymptotically optimal sampling ratios that optimize the large deviations rate of the probability of false selection for selecting the top-m alternatives have been rigorously defined. The proposed sampling policy is not only proved to be consistent but also achieve the asymptotically optimal sampling ratios. Numerical experiments demonstrate superiority of the proposed allocation procedure over existing ones. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72250065, 72293582, 72022001, and 71901003], and the National Science Foundation [Grant DMS-2053489], the major project of the National Natural Science Foundation of China [Grant 72293582], and the China Scholarship Council [Grant CSC202206010152]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0333 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0333 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Multiple Linear-Combination Security Network Coding.

In this paper, we put forward the model of multiple linear-combination security multicast network coding, where the wiretapper desires to obtain some information about a predefined set of multiple linear combinations of the source symbols by eavesdropping any one (but not more than one) channel subset up to a certain size r, referred to as the security level. For this model, the security capacity is defined as the maximum average number of source symbols that can be securely multicast to all sink nodes for one use of the network under the linear-combination security constraint. For any security level and any linear-combination security constraint, we fully characterize the security capacity in terms of the ratio of the rank of the linear-combination security constraint to the number of source symbols. Also, we develop a general construction of linear security network codes. Finally, we investigate the asymptotic behavior of the security capacity for a sequence of linear-combination security models and discuss the asymptotic optimality of our code construction.

Spectral CUSUM for Online Network Structure Change Detection

Detecting abrupt changes in the community structure of a network from noisy observations is a fundamental problem in statistics and machine learning. This paper presents an online change detection algorithm called Spectral-CUSUM to detect unknown network structure changes through a generalized likelihood ratio statistic. We characterize the average run length (ARL) and the expected detection delay (EDD) of the Spectral-CUSUM procedure and prove its asymptotic optimality. Finally, we demonstrate the good performance of the Spectral-CUSUM procedure and compare it with several baseline methods using simulations and real data examples on seismic event detection using sensor network data.

Asymptotic Optimality of Adaptive Systems with Alternative Standards in Unclassified Learning Situations

Asymptotic Optimality of Adaptive Systems with Alternative Standards in Unclassified Learning Situations

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in “Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times,” Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal.

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in “Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times,” Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal.

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

A robust control approach to asymptotic optimality of the heavy ball method for optimization of quadratic functions

Among first order optimization methods, Polyak’s heavy ball method has long been known to guarantee the asymptotic rate of convergence matching Nesterov’s lower bound for functions defined in an infinite-dimensional space. In this paper, we use results on the robust gain margin of linear uncertain feedback control systems to show that the heavy ball method is provably worst-case asymptotically optimal when applied to quadratic functions in a finite dimensional space.

Gaussian Belief Space Path Planning for Minimum Sensing Navigation

In this article, we propose a path planning methodology for a mobile robot navigating through an obstacle-filled environment to generate a reference path that is traceable with moderate sensing efforts. The desired reference path is characterized as the shortest path in an obstacle-filled Gaussian belief manifold equipped with a novel information-geometric distance function. The distance function we introduce is shown to be an asymmetric quasi-pseudometric and can be interpreted as the minimum information gain required to steer the Gaussian belief. An RRT*-based numerical solution algorithm is presented to solve the formulated shortest-path problem. To gain insight into the asymptotic optimality of the proposed algorithm, we show that the considered path length function is continuous with respect to the topology of total variation. Simulation results demonstrate that the proposed method is effective in various robot navigation scenarios to reduce sensing costs, such as the required frequency of sensor measurements and the number of sensors that must be operated simultaneously.

A robust model averaging approach for partially linear models with responses missing at random

AbstractIn this paper, with an assumed parametric model for the selection probability function, a robust model averaging estimation method is proposed for partially linear models with responses missing at random. The method is based on a weighted Mallows‐type criterion. The method is robust in the sense that the asymptotic optimality holds true as long as the true model of the selection probability function is some measurable function of its assumed model. The optimal weight vector for model averaging is obtained by minimizing the weighted Mallows‐type criterion. It is shown that the robust model averaging method achieves the lowest possible squared error asymptotically. Some simulation studies were conducted to evaluate the proposed method. An application to two real examples are provided as illustration.

Exponential asymptotic optimality of Whittle index policy

Exponential asymptotic optimality of Whittle index policy

Only Classical Parameterised States have Optimal Measurements under Least Squares Loss

Measurements of quantum states form a key component in quantum-information processing. It is therefore an important task to compare measurements and furthermore decide if a measurement strategy is optimal. Entropic quantities, such as the quantum Fisher information, capture asymptotic optimality but not optimality with finite resources. We introduce a framework that allows one to conclusively establish if a measurement is optimal in the non-asymptotic regime. Our method relies on the fundamental property of expected errors of estimators, known as risk, and it does not involve optimisation over entropic quantities. The framework applies to finite sample sizes and lack of prior knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go theorem that shows that only classical states admit optimal measurements under the most common choice of error measurement: least squares. We further consider the less restrictive notion of an approximately optimal measurement and give sufficient conditions for such measurements to exist. Finally, we generalise the notion of when an estimator is inadmissible (i.e. strictly worse than an alternative), and provide two sufficient conditions for a measurement to be inadmissible.

Directional fault classification for correlated High-Dimensional data streams using hidden Markov models

Modern manufacturing systems are often installed with sensor networks which generate high-dimensional data at high velocity. These data streams offer valuable information about the industrial system’s real-time performance. If a shift occurs in the manufacturing process, fault diagnosis based on the data streams becomes a fundamental task as it identifies the affected data streams and provides insights into the root cause. Existing fault diagnostic methods either ignore the correlation between different streams or fail to determine the shift directions. In this paper, we propose a directional fault classification procedure that incorporates the between-stream correlations. We suggest a three-state hidden Markov model that captures the correlation structure and enables inference about the shift direction. We show that our procedure is optimal in the sense that it minimizes the expected number of false discoveries while controlling the proportion of missed signals at a desired level. We also propose a deconvolution-expectation-maximization (DEM) algorithm for estimating the model parameters and establish the asymptotic optimality for the data-driven version of our procedure. Numerical comparisons with an existing approach and an application to a semiconductor production study show that the proposed procedure works well in practice.

Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

We consider two Gaussian measures μ,μ˜ on a separable Hilbert space, with fractional-order covariance operators A−2β and A˜−2β˜, respectively, and derive necessary and sufficient conditions on A,A˜ and β,β˜>0 for I. equivalence of the measures μ and μ˜, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure μ˜. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and A˜ are elliptic second-order differential operators, formulated on a bounded Euclidean domain D⊂Rd and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

Performance Analysis of Modified SRPT in Multiple-Processor Multitask Scheduling

In this paper we study the multiple-processor multitask scheduling problem in both deterministic and stochastic models, where each job have several tasks and is complete only when all its tasks are finished. We consider and analyze Modified Shortest Remaining Processing Time (M-SRPT) scheduling algorithm, a simple modification of SRPT, which always schedules jobs according to SRPT whenever possible, while satisfying the non-preemptive constraint. The M-SRPT algorithm is proved to achieve a competitive ratio of θ(log α + β) for minimizing response time, where α denotes the ratio between maximum job workload and minimum job workload, β represents the ratio between maximum non-preemptive task workload and minimum job workload. In addition, the competitive ratio achieved is shown to be optimal (up to a constant factor), when there are constant number of machines. We further consider the problem under Poisson arrival and general workload distribution (i. e., M/GI/N system), and show that M-SRPT achieves asymptotic optimal mean response time when the traffic intensity ρ approaches 1, if job size distribution has finite support. Beyond finite job workload, the asymptotic optimality of M-SRPT also holds for infinite job size distributions with certain probabilistic assumptions, for example, M/M/N system with finite task workload.

Improving path planning efficiency for underwater gravity-aided navigation based on a new depth sorting fast search algorithm

This study focuses on the improvement of path planning efficiency for underwater gravity-aided navigation. Firstly, a Depth Sorting Fast Search (DSFS) algorithm was proposed to improve the planning speed of the Quick Rapidly-exploring Random Trees∗ (Q-RRT∗) algorithm. A cost inequality relationship between an ancestor and its descendants was derived, and the ancestors were filtered accordingly. Secondly, the underwater gravity-aided navigation path planning system was designed based on the DSFS algorithm, taking into account the fitness, safety, and asymptotic optimality of the routes, according to the gravity suitability distribution of the navigation space. Finally, experimental comparisons of the computing performance of the ChooseParent procedure, the Rewire procedure, and the combination of the two procedures for Q-RRT∗ and DSFS were conducted under the same planning environment and parameter conditions, respectively. The results showed that the computational efficiency of the DSFS algorithm was improved by about 1.2 times compared with the Q-RRT∗ algorithm while ensuring correct computational results.

Self-Supervised Learning for Online Anomaly Detection in High-Dimensional Data Streams

In this paper, we address the problem of detecting and learning anomalies in high-dimensional data-streams in real-time. Following a data-driven approach, we propose an online and multivariate anomaly detection method that is suitable for the timely and accurate detection of anomalies. We propose our method for both semi-supervised and supervised settings. By combining the semi-supervised and supervised algorithms, we present a self-supervised online learning algorithm in which the semi-supervised algorithm trains the supervised algorithm to improve its detection performance over time. The methods are comprehensively analyzed in terms of computational complexity, asymptotic optimality, and false alarm rate. The performances of the proposed algorithms are also evaluated using real-world cybersecurity datasets, that show a significant improvement over the state-of-the-art results.

Asymptotic Optimality of Constant-Order Policies in Joint Pricing and Inventory Models

We consider a classic joint pricing and inventory control problem with lead times, which is extensively studied in the literature but is notoriously difficult to solve because of the complex structure of the optimal policy. In this work, rather than analyzing the optimal policy, we propose a class of constant-order dynamic pricing policies, which are fundamentally different from base-stock list price policies, the primary emphasis in the existing literature. Under such a policy, a constant-order amount of new inventory is ordered every period, and a pricing decision is made based on the inventory level. The policy is independent of the lead time. We prove that the best constant-order dynamic pricing policy is asymptotically optimal as the lead time grows large, which is exactly the setting in which the problem becomes computationally intractable because of the curse of dimensionality. As our main methodological contributions, we establish the convergence to a long-run average random yield inventory model with zero lead time and ordering capacities by its discounted counterpart as the discount factor goes to one, nontrivially extending the previous results in Federgruen and Yang that analyze a similar model but without capacity constraints. Funding: Research of X. Chen and L. Xin was partly supported by the National Science Foundation [Grant CMMI-1635160].

Optimal Model Averaging Estimation for the Varying-Coefficient Partially Linear Models with Missing Responses

In this paper, we propose a model averaging estimation for the varying-coefficient partially linear models with missing responses. Within this context, we construct a HRCp weight choice criterion that exhibits asymptotic optimality under certain assumptions. Our model averaging procedure can simultaneously address the uncertainty on which covariates to include and the uncertainty on whether a covariate should enter the linear or nonlinear component of the model. The simulation results in comparison with some related strategies strongly favor our proposal. A real dataset is analyzed to illustrate the practical application as well.