Minimax Estimation Research Articles

Bicriteria multi-machine scheduling with equal processing times subject to release dates

<abstract><p>This paper addresses the problem of scheduling $ n $ equal-processing-time jobs with release dates non-preemptively on identical machines to optimize two criteria simultaneously or hierarchically. For simultaneous optimization of total completion time (and makespan) and maximum cost, an algorithm is presented which can produce all Pareto-optimal points together with the corresponding schedules. The algorithm is then adapted to solve the hierarchical optimization of two min-max criteria, and the final schedule has a minimum total completion time and minimum makespan among the hierarchical optimal schedules. The two algorithms provided in this paper run in $ O(n^3) $ time.</p></abstract>

Learning Minimax-Optimal Terminal State Estimators and Smoothers

We develop the first model-free policy gradient (PG) algorithm for the minimax state estimation of discrete-time linear dynamical systems, where adversarial disturbances could corrupt both dynamics and measurements. Specifically, the proposed algorithm learns a minimax-optimal solution for three fundamental tasks in robust (minimax) estimation, namely terminal state filtering, terminal state prediction, and smoothing, in a unified fashion. We further establish convergence and finite sample complexity guarantees for the proposed PG algorithm. Additionally, we propose a model-free algorithm to evaluate the attenuation (robustness) level of any estimator or smoother, which serves as a model-free solution to identify the maximum size of the disturbance under which the estimator will still be robust. We demonstrate the effectiveness of the proposed algorithms through extensive numerical experiments.

Intelligent decision support in the optimization of irrigation systems in agriculture

The issues of determining the optimal values of the regulatory parameters of irrigation systems engaged in the cultivation of agricultural crops are considered. Following the requirements of a market economy, the main emphasis is placed on taking into account two types of criteria: maximizing the yield of agricultural crops and minimizing monetary costs. The proposed method for solving the multi-criteria optimization problem is based on the combination of the minimax criterion and the medium-step convolution, which makes it possible to scalarize the vector optimality criterion with access to smooth optimization methods. Concerning the case of priority uncertainty according to particular optimality criteria, an intelligent algorithm is proposed based on the approximation of the preference function of the decision-maker by the fuzzy Mamdani model. The multi-criteria optimization of the irrigation system used for growing cotton results differ favorably from the average values. The one hectare yield in the republic- increased by 2%, monetary costs - reduced by 4.5%. It could be concluded that the developed methodology makes it possible to bypass the computational difficulties that arise when solving problems of multi-criteria optimization of irrigation systems engaged in the cultivation of agricultural crops and to obtain real results in conditions of certainty and uncertainty goals.

Learner-Private Convex Optimization

Convex optimization with feedback is a framework where a learner relies on iterative queries and feedback to arrive at the minimizer of a convex function. It has gained considerable popularity thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversaries that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner’s queries in convex optimization with first-order feedback, so that their learned optimal value is provably difficult to estimate for an eavesdropping adversary. We consider two formulations of learner privacy: a Bayesian formulation in which the convex function is drawn randomly, and a maximin formulation in which the function is fixed and the adversary’s probability of error is measured with respect to a minimax criterion. Suppose that the learner wishes to ensure the adversary cannot estimate accurately with probability greater than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/L$ </tex-math></inline-formula> for some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L &gt; 0$ </tex-math></inline-formula> . Our main results show that the query complexity overhead is additive in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> in the maximin formulation, but multiplicative in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> in the Bayesian formulation. Compared to existing learner-private sequential learning models with binary feedback, our results apply to the significantly richer family of general convex functions with full-gradient feedback. Our proofs rely on tools from the theory of Dirichlet processes, as well as a novel strategy designed for measuring information leakage under a full-gradient oracle.

IoT Edge-Computing-Enabled Efficient Localization via Robust Optimal Estimation

Source localization within wireless sensor networks (WSNs) is one of the critical technologies in the Internet of Things (IoT). As the number of network nodes increases, so does the amount of data and computational requirement. It is imperative to introduce edge computing. However, there are still two issues when running existing wireless location algorithms on edge nodes: 1) conventional low-complexity approaches are easily affected by the bias generated in complex environments, leading to low locating accuracy and 2) the optimization algorithms considering the bias have good performances, but they are calculation-efficiency low on edge nodes. This study proposes a computationally efficient and high-precision location method to tackle the troubles. Precisely, we first introduce our previous research to construct a bias-considered nonconvex problem with a linear objective. Then, we propose an angle-assisted Taylor series with zero truncation error to linearize the second-order cone (SOC) constraint in the established problem. Next, we resort to the mini-max criterion to eliminate the angular uncertainty and get a robust linear programming (LP) problem with an optimal solution. So far, we have obtained a convex problem of low complexity. To ensure the calculated efficiency of the proposed problem on edge nodes, we proceed to give the solving process of the problem. Moreover, we provide a constraints tracking mechanism to reduce the number of iterations in the solution procedure, improving computational efficiency. Simulations and experiments demonstrate that the proposed method with similar locating accuracy to state-of-the-art optimization algorithms exhibits much higher computing efficiency on edge nodes.

Joint replenishment meets scheduling

In this paper, we consider a combination of the joint replenishment problem (JRP) and single-machine scheduling with release dates. There is a single machine and one or more item types. Each job has a release date, a positive processing time, and it requires a subset of items. A job can be started at time t only if all the required item types were replenished between the release date of the job and time point t. The ordering of item types for distinct jobs can be combined. The objective is to minimize the total ordering cost plus a scheduling criterion, such as total weighted completion time or maximum flow time, where the cost of ordering a subset of items simultaneously is the sum of a joint ordering cost, and an additional item ordering cost for each item type in the subset. We provide several complexity results for the offline problem, and competitive analysis for online variants with min–sum and min–max criteria, respectively.

Outage Performance Enhancement for NOMA-Based Cooperative Relay Sharing Networks

This letter considers a non-orthogonal multiple access (NOMA) based cooperative relay sharing (CRS) network, where two sources communicate with their corresponding users over the same time and frequency via a shared decode-and-forward relay. A novel transmission scheme using max-min criterion based dynamic decoding order strategy is proposed to minimize the outage probability of the network at the cost of lower complexity and overhead. The closed-form expression of the overall outage probability for the proposed scheme is derived. Both analytical and simulation results show that the proposed scheme can achieve non-zero diversity order and almost the same outage performance as the dynamic power allocation based transmission scheme for NOMA based CRS networks.

Efficient Representation of Large-Alphabet Probability Distributions

A number of engineering and scientific problems require representing and manipulating probability distributions over large alphabets, which we may think of as long vectors of reals summing to 1. In some cases it is required to represent such a vector with only b bits per entry. A natural choice is to partition the interval 0,1 into 2b uniform bins and quantize entries to each bin independently. We show that a minor modification of this procedure – applying an entrywise non-linear function (compander) f(x) prior to quantization – yields an extremely effective quantization method. For example, for b=8(16) and 105-sized alphabets, the quality of representation improves from a loss (under KL divergence) of 0.5(0.1) bits/entry to 10-4(10-9) bits/entry. Compared to floating point representations, our compander method improves the loss from 10-1(10-6) to 10-4(10-9) bits/entry. These numbers hold for both real-world data (word frequencies in books and DNA k-mer counts) and for synthetic randomly generated distributions. Theoretically, we analyze a minimax optimality criterion and show that the closed-form compander f(x)ArcSinh(cK(KlogK)x) is (asymptotically as b∞) optimal for quantizing probability distributions over a -letter alphabet. Non-asymptotically, such a compander (substituting 1/2 for cfor simplicity) has KL-quantization loss bounded by ≤8∙2-2blog2. Interestingly, a similar minimax criterion for the quadratic loss on the hypercube shows optimality of the standard uniform quantizer. This suggests that the ArcSinh quantizer is as fundamental for KL-distortion as the uniform quantizer for quadratic distortion.

Algorithm to support decision-making regarding the placement of reserve units and logistical support of military formations in the area according to the minimax criterion

The scientific issues of the article are aimed at the development of an algorithm for solving one of the important in practice of the class of problems of computational geometry - optimization. This task is related to the optimal placement of reserve units and logistical support of the military formation in the area according to the minimax criterion, which will be the basis for making a decision on the formation of a battle order and will ensure the minimization of the maximum response time to a change in the situation and the time of evacuation (carriage) in the conditions of increasing dynamic and spatial indicators of hostilities.&#x0D; The essence of the study is to choose a method of optimizing the function of the set of concentric circles based on its majorization by a paraboloid to solve the problem of placing a reserve unit and logistical support according to the minimax criterion and developing, based on it, an algorithm to support decision-making regarding the formation of the order of battle of a military formation in the area.&#x0D; Methods of optimization, computational geometry and algorithmization were used to solve the problem.&#x0D; The result of the work is a developed algorithm for finding the minimax point on the terrain in the area of hostilities relative to the units of the military formation and the defined boundaries.&#x0D; The implementation of the algorithm in the decision-making support system will ensure an increase in the efficiency and reasonableness of decision-making regarding the positioning of reserve units and logistical support in the overall construction of the order of battle of the military formation.

Minimax estimation in multi-task regression under low-rank structures

This study investigates the minimaxity of a multi-task nonparametric regression problem. We formulate a simultaneous function estimation problem based on information pooling across multiple experiments under a low-dimensional structure. A nonparametric reduced rank regression estimator based on the nuclear norm penalisation scheme is proposed to incorporate the low-dimensional structure in the estimation process. A rank of a set of functions is defined in terms of their Fourier coefficients to formally characterise the dependence structure among functions. Minimax upper and lower bounds are established under various asymptotic scenarios to examine the role of the low-rank structure in determining optimal rates of convergence. The results confirm that exploiting the low-rank structure can significantly improve the convergence rate for the simultaneous estimation of multiple functions. The results also imply that the proposed estimator is rate optimal in the minimax sense for the rank-constraint Sobolev class of vector-valued functions.

Optimal High Pass FIR Filter Based on Adaptive Systematic Cuckoo Search Algorithm

Abstract This paper presents the design of a desired linear phase digital Finite Impulse Response (FIR) High Pass (HP) filter based on Adaptive Systematic Cuckoo Search Algorithm (ACSA). The deviation, or error from the desired response, is assessed along with the stop-band and pass-band attenuation of the filter. The Cuckoo Search algorithm (CS) is used to avoid local minima because the error surface is typically non-differentiable, nonlinear, and multimodal. The ACSA is applied to the minimax criterion (L∞-norm) based error fitness function, which offers a better equiripple response for passband and stopband, high stopband attenuation, and rapid convergence for the developed optimal HP FIR filter algorithm. The simulation findings demonstrate that when compared to the Parks McClellan (PM), Particle Swarm Optimization (PSO), CRazy Particle Swarm Optimization (CRPSO), and Cuckoo Search algorithms, the proposed HP FIR filter employing ACSA leads to better solutions.

Estimating Optimal Infinite Horizon Dynamic Treatment Regimes via pT-Learning

Recent advances in mobile health (mHealth) technology provide an effective way to monitor individuals’ health statuses and deliver just-in-time personalized interventions. However, the practical use of mHealth technology raises unique challenges to existing methodologies on learning an optimal dynamic treatment regime. Many mHealth applications involve decision-making with large numbers of intervention options and under an infinite time horizon setting where the number of decision stages diverges to infinity. In addition, temporary medication shortages may cause optimal treatments to be unavailable, while it is unclear what alternatives can be used. To address these challenges, we propose a Proximal Temporal consistency Learning (pT-Learning) framework to estimate an optimal regime that is adaptively adjusted between deterministic and stochastic sparse policy models. The resulting minimax estimator avoids the double sampling issue in the existing algorithms. It can be further simplified and can easily incorporate off-policy data without mismatched distribution corrections. We study theoretical properties of the sparse policy and establish finite-sample bounds on the excess risk and performance error. The proposed method is provided in our proximalDTR package and is evaluated through extensive simulation studies and the OhioT1DM mHealth dataset. Supplementary materials for this article are available online.

Nonparametric bivariate density estimation for censored lifetimes

It is well known that estimation of a bivariate cumulative distribution function of a pair of right censored lifetimes presents challenges unparalleled to the univariate case where a product-limit Kaplan–Meyer’s methodology typically yields optimal estimation, and the literature on optimal estimation of the joint probability density is next to none. The paper, for the first time in the survival analysis literature, develops the theory and methodology of sharp minimax and adaptive nonparametric estimation of the joint density under the mean integrated squared error (MISE) criterion. The theory shows how an underlying joint density, together with the bivariate distribution of censoring variables, affect the estimation, and what and how may or may not be estimated in the presence of censoring. Practical example illustrates the problem.

Minimax Registration for Point Cloud Alignment

The alignment, or rigid registration, of three-dimensional (3D) point clouds plays an important role in many applications, such as robotics and computer vision. Recently, with the improvement in high precision and automated 3D scanners, the registration algorithm has become critical in a manufacturing setting for tolerance analysis, quality inspection, or reverse engineering purposes. Most of the currently developed registration algorithms focus on aligning the point clouds by minimizing the average squared deviations. However, in manufacturing practices, especially those involving the assembly of multiple parts, an envelope principle is widely used, which is based on minimax criteria. Our present work models the registration as a minimization problem of the maximum deviation between two point clouds, which can be recast as a second-order cone program. Variants for both pairwise and multiple point clouds registrations are discussed. We compared the performance of the proposed algorithm with other well-known registration algorithms, such as iterative closest point and partial Procrustes registration, on a variety of simulation studies and scanned data. Case studies in both additive manufacturing and reverse engineering applications are presented to demonstrate the usage of the proposed method.

Прогнозирование и минимаксное оценивание состояний производственной системы при наличии рисков

The solution of the problem of forecasting the state of complex socio-economic systems is possible only on the basis of appropriate dynamic economic and mathematical models that describe their main parameters, the presence of control actions and risks. In this paper, it is proposed to use a deterministic minimax approach for modeling and solving the problem of estimating the predicted states of a production system in the presence of risks. To make managerial decisions at a manufacturing enterprise aimed at improving the efficiency of its functioning, it is necessary to have high-quality information support, the basis of which is the solution of the corresponding problem of predicting the states of its basic parameters. In this article, to describe the functioning of a production system, it is proposed to use a discrete-time controlled dynamical system in the presence of risks. It is assumed that the values of the control action (admissible control scenarios) are realized from a finite set of admissible elements of the corresponding finite-dimensional vector space, and the realizations of the values of the phase vector of the model and the risk vector are limited by the given compact polyhedrons in the corresponding finite-dimensional vector spaces. Application of the developed discrete-time controlled dynamical model that describes the output products of an enterprise in the presence of risks, and the developed methodology for the formation and minimax estimation of the predictive set of its phase states in a given period of time, allow us to develop appropriate numerical algorithms that can be used in the development and creation of computer intelligent information systems that provide support for making effective management decisions at manufacturing enterprises. The main results of this work is the development of a new economic-mathematical model that describes the dynamics of the output products of an enterprise in the presence of risks and the creation on its basis of a methodology for constructing and minimax estimation of the predictive set of its phase states in the form of implementing a finite number of one-step operations that allow their algorithmization. The results obtained in this work can serve as a basis for developing methods for optimizing the management of enterprise production processes and creating computer intelligent information systems to support managerial decision-making.

Nonregular and minimax estimation of individualized thresholds in high dimension with binary responses

Given a large number of covariates Z, we consider the estimation of a high-dimensional parameter θ in an individualized linear threshold θTZ for a continuous variable X, which minimizes the disagreement between sign(X−θTZ) and a binary response Y. While the problem can be formulated into the M-estimation framework, minimizing the corresponding empirical risk function is computationally intractable due to discontinuity of the sign function. Moreover, estimating θ even in the fixed-dimensional setting is known as a nonregular problem leading to nonstandard asymptotic theory. To tackle the computational and theoretical challenges in the estimation of the high-dimensional parameter θ, we propose an empirical risk minimization approach based on a regularized smoothed non-convex loss function. The Fisher consistency of the proposed method is guaranteed as the bandwidth of the smoothed loss is shrunk to 0. Statistically, we show that the finite sample error bound for estimating θ in ℓ2 norm is (slogd/n)β/(2β+1), where d is the dimension of θ, s is the sparsity level, n is the sample size and β is the smoothness of the conditional density of X given the response Y and the covariates Z. The convergence rate is nonstandard and slower than that in the classical Lasso problems. Furthermore, we prove that the resulting estimator is minimax rate optimal up to a logarithmic factor. The Lepski’s method is developed to achieve the adaption to the unknown sparsity s or smoothness β. Computationally, an efficient path-following algorithm is proposed to compute the solution path. We show that this algorithm achieves geometric rate of convergence for computing the whole path. Finally, we evaluate the finite sample performance of the proposed estimator in simulation studies and a real data analysis from the ChAMP (Chondral Lesions And Meniscus Procedures) Trial.

Minimax optimal estimation of high-dimensional sparse covariance matrices with missing data

This paper considers the minimax estimation of the high-dimensional sparse covariance matrices in the presence of missing observations. Based on random missing data, the upper bounds of convergence rate about the data-driven thresholding estimator are constructed over a large class of sparse covariance matrices under the [Formula: see text] norm and the Frobenius norm. In addition, we use Le Cam’s lemma and the relation between the total variation affinity and the Kullback–Leibler divergence to establish the lower bounds which illustrate the desired upper bounds cannot be improved. It is worth mentioning that the approach we adopt to get the lower bounds is simpler than the existing ones.

Higher‐order asymptotics of minimax estimators for time series

We consider the minimax estimation of time series in view of higher‐order asymptotic theory. Under the framework of Bayesian inference, we focus on the Bayes estimator and the Bayesian Whittle estimator for parameter estimation. It is shown that these estimators are minimax with respect to the Bayes risk of higher‐order bias appeared in their asymptotic expansion. The minimax problem in the boundary issue with parameter on the boundary of parameter space is also discussed. Our theoretical discovery is justified by simulation studies even when the sample size is small.

Algorithms for distributing DSSS codes in industrial WSNs with real transmitters and narrow-band interference

The versatility of simple electronic nodes and debugging of communication protocols in unlicensed frequency bands constitute the main reasons for the adoption of industrial wireless sensor networks (IWSNs). However, the industrial environments, in which IWSNs function, are characterized by a harsh impact of physical and electromagnetic factors that destabilize the operation of equipment and worsen the noise immunity of channels. To consider the effect of these two groups of factors on communication quality, we develop a novel model of an IWSN channel as the dependence of bit error probability (BEP) on signal distortions and interference impact. We found that direct sequence spread spectrum (DSSS) codes of a signal have different effects on the compensation of nonlinear signal distortions and external interference. Further, we propose an algorithm for the distribution of DSSS codes among real transmitters by BEP values in channels with known additive and multiplicative interference. Here, the distribution of codes is based on the minimax criterion. Finally, for cases when a signal is characterized by a-priori uncertainties of fading, we suggest an alternative algorithm for selecting a DSSS code based on Pareto bicriteria optimization.

Cybersecurity strategy under uncertainties for an IoE environment

Internet of Everything (IoE) technology is increasingly being used by companies to modernize their activities. The very specific characteristics of such environments, in particular, their vulnerable exchanged data and the weak nature of the connected things, expose these companies to risks and security breaches. The principal objective of our work is to provide a cybersecurity strategy capable to consider all types of attacks that can affect an IoE environment while respecting the specified budget. For this purpose, a financial approach based on portfolio management is exploited by allowing to select a portfolio of security controls that minimizes the direct costs and maximizes the security level control. To be solved, the considered problem is assimilated to a combinatorial optimization technique and more precisely to that of the knapsack. We start by modeling the cybersecurity problem under cardinality and budget constraints that should be respected. To tackle the uncertainty of the problem, a robust optimization is used by considering the min-max criterion al-lowing to consider all the possible threats that can be generated by an attacker over the IoE environment. To solve the considered problem, we use a new iterative method under constraints and we compare it to the Non-dominated Sorting Genetic Algorithm (NSGA-II) meta-heuristic to evaluate its performances. The obtained numeric results when evaluating the performances of the proposed strategy have shown its efficiency by finding efficient Pareto fronts for the two considered objective functions. Based on the iterative method, our strategy greatly outperforms the genetic algorithm by allowing good results for different problem sizes and respecting cardinality constraints in a reasonable time.