Asymptotic Optimality Research Articles

Estimating conditional average treatment effects with heteroscedasticity by model averaging and matching

We propose a model averaging approach, combined with a partition and matching method to estimate the conditional average treatment effects under heteroskedastic error settings. The proposed approach has asymptotic optimality and consistency of weights and estimator. Numerical studies show that our method has good finite-sample performances.

Projected Inventory-Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

Inventory Projection and Asymptotic Optimality van Jaarsveld and Arts propose a new policy for the canonical periodic review lost sales inventory problem. Under this policy, orders are placed in each period such that the expected inventory level at the moment of replenishment order arrival is at a fixed level. This single-parameter policy is called the projected inventory-level (PIL) policy. The PIL inherits asymptotic optimality properties from the constant-order policy for long lead time and from the base-stock policy when the cost of losing a sale is large. The PIL policy has lower cost than competing heuristics in a numerical study. The PIL seems to be a promising approach for other inventory systems with a high-dimensional state space, such as the perishable item inventory management problem.

Asymptotic SEP Analysis and Optimization of Linear-Quantized Precoding in Massive MIMO Systems

Asymptotic SEP Analysis and Optimization of Linear-Quantized Precoding in Massive MIMO Systems

Asymptotic optimality of the square-root transformation on the gamma distribution using the Kullback–Leibler information number criterion

The Kullback–Leibler information number criterion is a useful criterion for determining an optimal power transformation to approximate the target distribution. When the normal distribution with a fixed variance is set as the target distribution, this criterion may be used to show the asymptotic optimality of the square-root transformation. In particular, when the Box–Cox transformation corresponding to the square-root transformation is applied, the Kullback–Leibler information number of the transformed gamma distribution with the scale parameter equal to the fixed variance converges to zero asymptotically as the shape parameter approaches infinity. That provides a stronger result on the mode of convergence of the variance-stabilizing power transformation on the gamma distribution. Moreover, an analogous result may be obtained by setting the Laplace distribution with a fixed scale parameter as the target distribution. This result utilizes a novel application of the asymptotic expansion for the normalized upper incomplete gamma function at the transition point.

The Choice of Noninformative Priors for Thompson Sampling in Multiparameter Bandit Models

Thompson sampling (TS) has been known for its outstanding empirical performance supported by theoretical guarantees across various reward models in the classical stochastic multi-armed bandit problems. Nonetheless, its optimality is often restricted to specific priors due to the common observation that TS is fairly insensitive to the choice of the prior when it comes to asymptotic regret bounds. However, when the model contains multiple parameters, the optimality of TS highly depends on the choice of priors, which casts doubt on the generalizability of previous findings to other models. To address this gap, this study explores the impact of selecting noninformative priors, offering insights into the performance of TS when dealing with new models that lack theoretical understanding. We first extend the regret analysis of TS to the model of uniform distributions with unknown supports, which would be the simplest non-regular model. Our findings reveal that changing noninformative priors can significantly affect the expected regret, aligning with previously known results in other multiparameter bandit models. Although the uniform prior is shown to be optimal, we highlight the inherent limitation of its optimality, which is limited to specific parameterizations and emphasizes the significance of the invariance property of priors. In light of this limitation, we propose a slightly modified TS-based policy, called TS with Truncation (TS-T), which can achieve the asymptotic optimality for the Gaussian models and the uniform models by using the reference prior and the Jeffreys prior that are invariant under one-to-one reparameterizations. This policy provides an alternative approach to achieving optimality by employing fine-tuned truncation, which would be much easier than hunting for optimal priors in practice.

Simultaneous optimal control of directional missed discovery rates in data stream diagnosis

High-dimensional data streams are ubiquitous in modern manufacturing, due to their ability to provide valuable information about the industrial system’s performance on a real-time basis. If a shift occurs in a production process, fault diagnosis based on the data streams is of critical importance for identifying the root cause. Existing methods have largely focused on controlling the total missed discovery rate without distinguishing missed signals for positive versus negative components of the shift vector. In practice, however, losses incurred from the two directional shifts can differ substantially, so it is desirable to constrain the proportions of missed signals for positive and negative components at two distinctive levels. In this article, we propose a fault classification procedure that controls the two proportions separately. By formulating the problem as Lagrangian multiplier optimization, we show that the proposed procedure is optimal in the sense that it minimizes the expected number of false discoveries. We also suggest an iterative adjustment algorithm that converges to the optimal Lagrangian parameters. The asymptotic optimality for the data-driven version of our procedure is also established. Theoretical justification and numerical comparison with state-of-the-art methods show that the proposed procedure works well in applications.

More Quickly-RRT*: Improved Quick Rapidly-exploring Random Tree Star algorithm based on optimized sampling point with better initial solution and convergence rate

RRT* (Rapidly-exploring Random Tree Star), as a variant of RRT (Rapidly-exploring Random Tree), is widely used to solve path planning problems because of its asymptotic optimality. However, the algorithm is inefficient due to the high initial path cost and the slow convergence rate. In this paper, we propose a More Quickly-RRT* (MQ-RRT*) path planning algorithm based on optimized sampling points to solve the problems. A sparse sampling mechanism is proposed in MQ-RRT* to improve the global search efficiency by reducing repetitive sampling. To make the random tree oriented when expanding, a dynamic goal-biased strategy is proposed, which can reduce the sampling time. Like Q-RRT* (Quick-RRT*), MQ-RRT* expands the set of possible parent nodes in the ChooseParent and Rewire phases, which reduces the path cost. On this basis, a method for creating a new parent node close to the obstacle is proposed. The creation process can be divided into two steps: Remove-tips and CreateNodes, which further reduces the cost of path generation and makes the path smoother by using the triangle inequality principle. Finally, numerical simulations are used to compare the proposed algorithm with RRT*, Q-RRT*, GuILD (Guided Incremental Local Densification), and F-RRT* (Fast-RRT*), which verifies that the proposed algorithm has certain advantages in path cost, convergence rate, and path smoothness.

Predicting the multivariate zero-inflated counts: A novel model averaging method under Pearson loss.

Excessive zeros in multivariate count data are often observed in scenarios of biomedicine and public health. To provide a better analysis on this type of data, we first develop a marginalized multivariate zero-inflated Poisson (MZIP) regression model to directly interpret the overall exposure effects on marginal means. Then, we define a multiple Pearson residual for our newly developed MZIP regression model by simultaneously taking heterogeneity and correlation into consideration. Furthermore, a new model averaging prediction method is introduced based on the multiple Pearson residual, and the asymptotical optimality of this model averaging prediction is proved. Simulations and two empirical applications in medicine are used to illustrate the effectiveness of the proposed method.

Model averaging for right censored data with measurement error.

This paper studies a novel model averaging estimation issue for linear regression models when the responses are right censored and the covariates are measured with error. A novel weighted Mallows-type criterion is proposed for the considered issue by introducing multiple candidate models. The weight vector for model averaging is selected by minimizing the proposed criterion. Under some regularity conditions, the asymptotic optimality of the selected weight vector is established in terms of its ability to achieve the lowest squared loss asymptotically. Simulation results show that the proposed method is superior to the other existing related methods. A real data example is provided to supplement the actual performance.

Hybrid learning of predictive mobile-edge computation offloading under differently-aged network states

By offloading computationally demanding applications to edge servers, mobile edge computing (MEC) can alleviate the stringent hardware requirements and save energy consumption of resource-restrained devices. Mobile edge computation offloading (MECO, i.e., optimizing computation offloading and resource allocation) is critical to the performance of MEC. However, the existing study typically assumed the availability of all the network states for the upcoming time slot, not applicable in practical MECO with differently-aged network states (e.g., fast-changing practical wireless channels and instantly observable local task arrivals). This paper proposes a novel hybrid learning approach that optimizes the instantaneous local processing and predictive computation offloading decisions. We establish a new hybrid learning approach for instantaneous local processing and predictive computation offloading decisions by integrating the learning techniques of stochastic gradient descent (SGD) and online convex optimization (OCO). By decomposing the primal problems, the proposed approach can be implemented in a decentralized manner. We prove that the optimality loss resulting from the differently-aged network states can diminish with the decreasing stepsizes of SGD and OCO. Simulation results validate the asymptotic optimality and superiority of the proposed hybrid learning approach, as compared to state-of-the-art (e.g., based on OCO techniques).

Model-Free Approximate Bayesian Learning for Large-Scale Conversion Funnel Optimization

The flexibility of choosing the ad action as a function of the consumer state is critical for modern-day marketing campaigns. We study the problem of identifying the optimal sequential personalized interventions that maximize the adoption probability for a new product. We model consumer behavior by a conversion funnel that captures the state of each consumer (e.g., interaction history with the firm) and allows the consumer behavior to vary as a function of both her state and firm’s sequential interventions. We show our model captures consumer behavior with very high accuracy (out-of-sample area under the curve of over 0.95) in a real-world email marketing dataset. However, it results in a very large-scale learning problem, where the firm must learn the state-specific effects of various interventions from consumer interactions. We propose a novel attribution-based decision-making algorithm for this problem that we call model-free approximate Bayesian learning. Our algorithm inherits the interpretability and scalability of Thompson sampling for bandits and maintains an approximate belief over the value of each state-specific intervention. The belief is updated as the algorithm interacts with the consumers. Despite being an approximation to the Bayes update, we prove the asymptotic optimality of our algorithm and analyze its convergence rate. We show that our algorithm significantly outperforms traditional approaches on extensive simulations calibrated to a real-world email marketing dataset.

DBVS-APF-RRT*: A global path planning algorithm with ultra-high speed generation of initial paths and high optimal path quality

During the last decade, RRT* algorithm has been widely concerned by researchers because of its asymptotic optimality. However, the slow convergence rate of the RRT* algorithm leads to poor quality of the optimal paths at small number of iterations. Meanwhile, searching the initial path slowly limits its application scenarios. To overcome these problems, a directionally biased variable step APF-RRT* (DBVS-APF-RRT*) algorithm is proposed in this paper. Firstly, a novel directionally biased variable step sampling strategy is used in RRT* algorithm to quickly generate the initial path. Then, global random sampling and key region sampling strategies are added to improve global search ability and optimal path quality. At the same time, an artificial potential field (APF) method is introduced to improve the ability of obstacle avoidance. In addition, we propose a pruning strategy based on triangular inequality with direct connection of goal points to reduce the number of redundant nodes and shorten the path length. Finally, DBVS-APF-RRT* algorithm is compared with RRT*, Bias-RRT*, Informed-RRT* and Bias-P-RRT* algorithms to verify its superiority of optimal path quality, stability of the algorithm and its rapidity of finding the initial path.

Model Averaging for Accelerated Failure Time Models with Missing Censoring Indicators

Model averaging has become a crucial statistical methodology, especially in situations where numerous models vie to elucidate a phenomenon. Over the past two decades, there has been substantial advancement in the theory of model averaging. However, a gap remains in the field regarding model averaging in the presence of missing censoring indicators. Therefore, in this paper, we present a new model-averaging method for accelerated failure time models with right censored data when censoring indicators are missing. The model-averaging weights are determined by minimizing the Mallows criterion. Under mild conditions, the calculated weights exhibit asymptotic optimality, leading to the model-averaging estimator achieving the lowest squared error asymptotically. Monte Carlo simulations demonstrate that the method proposed in this paper has lower mean squared errors compared to other model-selection and model-averaging methods. Finally, we conducted an empirical analysis using the real-world Acute Myeloid Leukemia (AML) dataset. The results of the empirical analysis demonstrate that the method proposed in this paper outperforms existing approaches in terms of predictive performance.

Martingale-residual-based greedy model averaging for high-dimensional current status data.

Current status data are a type of failure time data that arise when the failure time of study subject cannot be determined precisely but is known only to occur before or after a random monitoring time. Variable selection methods for the failure time data have been discussed extensively in the literature. However, the statistical inference of the model selected based on the variable selection method ignores the uncertainty caused by model selection. To enhance the prediction accuracy for risk quantities such as survival probability, we propose two optimal model averaging methods under semiparametric additive hazards models. Specifically, based on martingale residuals processes, a delete-one cross-validation (CV) process is defined, and two new CV functional criteria are derived for choosing model weights. Furthermore, we present a greedy algorithm for the implementation of the techniques, and the asymptotic optimality of the proposed model averaging approaches is established, along with the convergence of the greedy averaging algorithms. A series of simulation experiments demonstrate the effectiveness and superiority of the proposed methods. Finally, a real-data example is provided as an illustration.

BI-RRT*: An improved path planning algorithm for secure and trustworthy mobile robots systems

The optimal RRT in elliptic space sampling (Informed-RRT*) is an extension of RRT that provides asymptotic optimality, however, it experiences gradual progress and close to obstacles. In the paper, we propose a novel path planning algorithm guided bidirectional Informed-RRT* (BI-RRT*), that introduces extension range, dual-direction exploration, and refinement in trajectory design. The growth range refers to maintaining an additional area from the obstacle to enhance the dependability of the path through preventing impacts. Bidirectional search is a search strategy using both start and target points for a initial solution. Smoothing improves path robustness by using cubic spline. Furthermore, simulation tests for the BI-RRT* algorithm are executed, and the efficacy of the suggested algorithm is confirmed through its application in a robot operating system (ROS). Simulations and experimental tests verify that the proposed algorithm improves the path planning capability. We emphasize the importance of safety, privacy, and reliability in the deployment of AI systems. Our algorithm ensures that the planned paths maintain a safe distance from obstacles, reducing the risk of collisions. Additionally, we prioritize privacy by adhering to data protection regulations and implementing secure communication protocols within the AI system. Moreover, we have applied rigorous testing and validation processes to enhance the reliability of our algorithm, ensuring consistent and accurate path planning outcomes.

Nonparametric Additive Models for Billion Observations

The nonparametric additive model (NAM) is a widely used nonparametric regression method. Nevertheless, due to the high computational burden, classic statistical techniques for fitting NAMs are not well-equipped to handle massive data with billions of observations. To address this challenge, we develop a scalable element-wise subset selection method, referred to as Core-NAM, for fitting penalized regression spline based NAMs. Specifically, we first propose an approximation of the penalized least squares estimation, based on which we develop an efficient variant of generalized cross-validation (GCV) to select the smoothing parameter and approximate the Bayesian confidence intervals for statistical inference. Theoretically, we show that the proposed estimator approximately minimizes an upper bound of the estimation mean squared error. Moreover, we provide a non-asymptotic approximation guarantee for the proposed estimator and establish the asymptotic optimality of the proposed variant of GCV. Extensive simulations demonstrate the superior accuracy and efficiency of the Core-NAM method. We also apply the proposed method to a total column ozone dataset containing nearly one billion observations, and the results indicate a speed-up by almost a thousand times with comparable performance compared to the full data approach. Supplementary materials for this article are available online.

Survival tree averaging by functional martingale-based residuals

A large literature has been established for random survival forest (RSF), a popular tool developed to analyze right-censored failure time data, under various situations. However, its prediction performance may not be optimal sometimes. To address this issue, we propose two optimal model averaging methods based on martingale residual processes. In particular, an in-of-bag and out-of-bag (IBOB) data process is defined, and two new IBOB functionals criteria are derived for the selection of weights. Furthermore, for their implementation, a greedy algorithm is presented, and the asymptotic optimality of the proposed model averaging approaches is established along with the convergence of the greedy averaging algorithms. Finally, an extensive simulation study is conducted, which indicates that the proposed methods work well, and an illustration is provided.

A Specified-Time Convergent Multiagent System for Distributed Optimization With a Time-Varying Objective Function

This technical note presents a specified-time convergent multi-agent system for distributed optimization with a time-varying objective function subject to equality constraints. Different from the static optimal solutions to most existing distributed optimization problems, the optimal solutions are time-varying due to the time-varying objective function in this problem. A distributed protocol law is designed to ensure all the agents' convergence to feasible and suboptimal solutions within a specified settling time and keep tracking the time-dependent optimal solutions. The specified-time convergence of the system and the asymptotic optimality of the solution generated by the system are proved based on the Lyapunov theory. A salient feature of the multi-agent system is that its upper bound of settling time can be specified in advance. Two examples are presented to illustrate the theoretical results.

Game-theoretic policy computing and simulation for blockchained buffering system via diffusion approximation

AbstractWe study 2-stage game-theoretic problem oriented 3-stage service policy computing, convolutional neural network (CNN) based algorithm design, and simulation for a blockchained buffering system with federated learning. More precisely, based on the game-theoretic problem consisting of both “win-lose” and “win-win” 2-stage competitions, we derive a 3-stage dynamical service policy via a saddle point to a zero-sum game problem and a Nash equilibrium point to a non-zero-sum game problem. This policy is concerning users-selection, dynamic pricing, and online rate resource allocation via stable digital currency for the system. The main focus is on the design and analysis of the joint 3-stage service policy for given queue/environment state dependent pricing and utility functions. The asymptotic optimality and fairness of this dynamic service policy is justified by diffusion modeling with approximation theory. A general CNN based policy computing algorithm flow chart along the line of the so-called big model framework is presented. Simulation case studies are conducted for the system with three users, where only two of the three users can be selected into the service by a zero-sum dual cost game competition policy at a time point. Then, the selected two users get into service and share the system rate service resource through a non-zero-sum dual cost game competition policy. Applications of our policy in the future blockchain based Internet (e.g., metaverse and web3.0) and supply chain finance are also briefly illustrated.

Linear quadratic mean-field game with volatility uncertainty

In this paper, we consider a linear quadratic (LQ) mean field game (MFG) with uncertain volatility. Applying a robust optimization approach and based on mean-field approximation, we firstly obtain the worst uncertainty by dealing with an LQ MFG with so-called soft-constraint. Then with the help of convex analysis theory, we derive the auxiliary optimal control problems for the agents and construct the decentralized strategies. Finally, by virtue of estimates of (forward) backward stochastic differential equations, the asymptotical optimality of the decentralized strategy is verified. A numerical example is given to illustrate the above theoretical results.