Theoretical Guarantees Research Articles

Distributed Nonconvex Optimization for Control of Water Networks with Time-coupling Constraints

AbstractIn this paper, we present a new control model for optimizing pressure and water quality operations in water distribution networks. Our formulation imposes a set of time-coupling constraints to manage temporal pressure variations, which are exacerbated by the transition between pressure and water quality controls. The resulting optimization problem is a nonconvex, nonlinear program with nonseparable structure across time steps. This problem proves challenging for state-of-the-art nonlinear solvers, often precluding their direct use for near real-time control in large-scale networks. To overcome this computational burden, we investigate a distributed optimization approach based on the alternating direction method of multipliers (ADMM). In particular, we implement and evaluate two algorithms: a standard ADMM scheme and a two-level variant that provides theoretical convergence guarantees for our nonconvex problem. We use a benchmarking water network and a large-scale operational network in the UK for our numerical experiments. The results demonstrate good convergence behavior across all problem instances for the two-level algorithm, whereas the standard ADMM approach struggles to converge in some instances. With an appropriately tuned penalty parameter, however, both distributed algorithms yield good quality solutions and computational times compatible with near real-time (e.g. hourly) control requirements for large-scale water networks.

Identity Diffuser: Preserving Abnormal Region of Interests While Diffusing Identity

To release medical images that can be freely used in downstream processes while maintaining their utility, it is necessary to remove personal features from the images while preserving the lesion structures. Unlike previous studies that focused on removing lesion structures while preserving the individuality of medical images, this study proposes and validates a new framework that maintains the lesion structures while diffusing individual characteristics. In this framework, we apply local differential privacy techniques to provide theoretical guarantees of privacy protection. Additionally, to enhance the utility of protected medical images, we perform denoising using a diffusion model on the noise-contaminated medical images. Numerous chest X-rays generated by the proposed method were evaluated by physicians, revealing a trade-off between the level of privacy protection and utility. In other words, it was confirmed that increasing the level of personal information protection tends to result in relatively lower utility. This study potentially enables the release of certain types of medical images that were previously difficult to share.

Simple fixes that accommodate switching costs in multi-armed bandits

When switching costs are added to the multi-armed bandit (MAB) problem where the arms’ random reward distributions are previously unknown, usually quite different techniques than those for pure MAB are required. We find that two simple fixes on the existing upper-confidence-bound (UCB) policy can work well for MAB with switching costs (MAB-SC). Two cases should be distinguished. One is with positive-gap ambiguity where the performance gap between the leading and lagging arms is known to be at least some δ>0. For this, our fix is to erect barriers that discourage frivolous arm switchings. The other is with zero-gap ambiguity where absolutely nothing is known. We remedy this by forcing the same arms to be pulled in increasingly prolonged intervals. As usual, the effectivenesses of our fixes are measured by the worst average regrets over long time horizons T. When the barriers are fixed at δ/2, we can accomplish a ln(T)-sized regret bound for the positive-gap case. When intervals are such that n of them occupy n2 periods, we can achieve the best possible T1/2-sized regret bound for the zero-gap case. Other than UCB, these fixes can be applied to a learning while doing (LWD) heuristic to reach satisfactory results as well. While not yet with the best theoretical guarantees, the LWD-based policies have empirically outperformed those based on UCB and other known alternatives. Numerically competitive policies still include ones resulting from interval-based fixes on Thompson sampling (TS).

Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting

We consider the problem of sampling from a high-dimensional target distribution πβ on Rd with density proportional to θ↦e−βU(θ) using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [2] to sample from such target distributions with the gradient of the potential U being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential U. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential U and super-linearly growing gradient of U, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution πβ. We obtain respective rates of convergence O(λ) and O(λ1/2) in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size λ. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm.

Automatic Test Pattern Generation for Robust Quantum Circuit Testing

Quantum circuit testing is essential for detecting potential faults in realistic quantum devices, while the testing process itself also suffers from the inexactness and unreliability of quantum operations. This article alleviates the issue by proposing a novel framework of automatic test pattern generation (ATPG) for robust testing of logical quantum circuits. We introduce the stabilizer projector decomposition (SPD) for representing the quantum test pattern and construct the test application (i.e., state preparation and measurement) using Clifford-only circuits, which are rather robust and efficient as evidenced in the fault-tolerant quantum computation. However, it is generally hard to generate SPDs due to the exponentially growing number of the stabilizer projectors. To circumvent this difficulty, we develop an SPD generation algorithm, as well as several acceleration techniques that can exploit both locality and sparsity in generating SPDs. The effectiveness of our algorithms are validated by (1) theoretical guarantees under reasonable conditions and (2) experimental results on commonly used benchmark circuits, such as Quantum Fourier Transform (QFT), Quantum Volume (QV), and Bernstein-Vazirani (BV) in IBM Qiskit.

Neyman-Pearson Multi-class Classification via Cost-sensitive Learning

Most existing classification methods aim to minimize the overall misclassification error rate. However, in applications such as loan default prediction, different types of errors can have varying consequences. To address this asymmetry issue, two popular paradigms have been developed: the Neyman-Pearson (NP) paradigm and the cost-sensitive (CS) paradigm. Previous studies on the NP paradigm have primarily focused on the binary case, while the multi-class NP problem poses a greater challenge due to its unknown feasibility. In this work, we tackle the multi-class NP problem by establishing a connection with the CS problem via strong duality and propose two algorithms. We extend the concept of NP oracle inequalities, crucial in binary classifications, to NP oracle properties in the multi-class context. Our algorithms satisfy these NP oracle properties under certain conditions. Furthermore, we develop practical algorithms to assess the feasibility and strong duality in multi-class NP problems, which can offer practitioners the landscape of a multi-class NP problem with various target error levels. Simulations and real data studies validate the effectiveness of our algorithms. To our knowledge, this is the first study to address the multi-class NP problem with theoretical guarantees. The proposed algorithms have been implemented in the R package npcs, which is available on CRAN.

Image-Based Visual Servoing of Manipulators With Unknown Depth: A Recurrent Neural Network Approach.

The image-based visual servoing (IBVS) of manipulators is important for intelligent manipulation using visual feedbacks. While the traditional IBVS methods for manipulators require the knowledge of the depth information in the interaction matrix, in this article, we propose a novel IBVS method for manipulators without depth estimation by leveraging the property of the associated image Jacobian. Because of a novel transformation, the IBVS problem is converted into a convex optimization problem subject to the kinematic constraint, joint constraints, and other constraints that are not explicitly related to the depth information. The problem is then solved by developing a recurrent neural network of global asymptotic convergence, and a dynamic neural control law without depth estimation emerges for the IBVS of manipulators. The theoretical guarantee and simulation results are provided to show the efficacy of the proposed method.

Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application

We develop a data-driven optimal shrinkage algorithm, named extended OptShrink (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.

Another Evidence to not Employ Customized Masked Hardware

As a well-studied countermeasure against side-channel analysis attacks, there is a general interest in applying masking to different cryptographic functions executed on different platforms. On the one hand, despite their high performance, masked hardware implementations are dedicated to specific algorithms, making them inflexible. On the other hand, applying masking on software involves serious challenges including significant overhead in terms of efficiency and difficulties to maintain theoretical security guarantees in practice. As a result, a line of research has been devoted to enable masked operations in flexible platforms (i.e., microprocessors) by including some masked modules in their hardware, hence a combination of flexibility and performance. In such scenarios, RISC-V is a natural choice as hardware can be adjusted to the extended instruction set. One such attempt presented at CHES 2021 is known as SCARV, which extends the Instruction Set Architecture (ISA) of a RISC-V core with a rich number of first-order masked operations on both Boolean and arithmetic masked operands. In this work, we conduct a comprehensive analysis of SCARV. Instead of relying on empirical measurements to demonstrate security, we utilize tool-assisted evaluations. Through these evaluations, we identified a couple of design flaws that lead to leakage in the masked implementations hosted by the corresponding processor. These flaws are primarily due to the lack of composability of cascaded components. While heuristic and ad-hoc design principles can result in secure, small, and efficient designs, they lack formal security proofs, which may lead to security flaws, like those we report here. Consequently, this work serves as a motivation for using composable masked modules and tool-assisted evaluations when constructing complex circuits.

Decision Making Under Cumulative Prospect Theory: An Alternating Direction Method of Multipliers

This paper proposes a novel numerical method for solving the problem of decision making under cumulative prospect theory (CPT), where the goal is to maximize utility subject to practical constraints, assuming only finite realizations of the associated distribution are available. Existing methods for CPT optimization rely on particular assumptions that may not hold in practice. To overcome this limitation, we present the first numerical method with a theoretical guarantee for solving CPT optimization using an alternating direction method of multipliers (ADMM). One of its subproblems involves optimization with the CPT utility subject to a chain constraint, which presents a significant challenge. To address this, we develop two methods for solving this subproblem. The first method uses dynamic programming, whereas the second method is a modified version of the pooling-adjacent-violators algorithm that incorporates the CPT utility function. Moreover, we prove the theoretical convergence of our proposed ADMM method and the two subproblem-solving methods. Finally, we conduct numerical experiments to validate our proposed approach and demonstrate how CPT’s parameters influence investor behavior, using real-world data. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms: Continuous. Funding: This research was supported by the National Natural Science Foundation of China [Grants 12171100, 71971083, and 72171138], the Natural Science Foundation of Shanghai [Grant 22ZR1405100], the Major Program of the National Natural Science Foundation of China [Grants 72394360, 72394364], the Program for Innovative Research Team of Shanghai University of Finance and Economics [Grant 2020110930], Fundamental Research Funds for the Central Universities, and the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science, Ministry of Education, East China Normal University. 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.2023.0243 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0243 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost O(ϵ−2) for the required target accuracy ϵ. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

Virtually coupled train set control subject to space-time separation: A distributed economic MPC approach with emergency braking configuration

The emerging virtual coupling technology aims to operate multiple train units in a Virtually Coupled Train Set (VCTS) at a minimal but safe distance. To guarantee collision avoidance, the safety distance should be calculated using the state-of-the-art space-time separation principle that separates the Emergency Braking (EB) trajectories of two successive units during the whole EB process. In this case, the minimal safety distance is usually numerically calculated without an analytic formulation. Thus, the constrained VCTS control problem is hard to address with space-time separation, which is still a gap in the existing literature. To solve this problem, we propose a Distributed Economic Model Predictive Control (DEMPC) approach with computation efficiency and theoretical guarantee. Specifically, to alleviate the computation burden, we transform implicit safety constraints into explicitly linear ones, such that the optimal control problem in DEMPC is a quadratic programming problem that can be solved efficiently. For theoretical analysis, sufficient conditions are derived to guarantee the recursive feasibility and stability of DEMPC, employing compatibility constraints, tube techniques and terminal ingredient tuning. Moreover, we extend our approach with globally optimal and distributed online EB configuration methods to shorten the minimal distance among VCTS. Finally, experimental results demonstrate the performance and advantages of the proposed approaches.

Inclusion Depth for Contour Ensembles.

Ensembles of contours arise in various applications like simulation, computer-aided design, and semantic segmentation. Uncovering ensemble patterns and analyzing individual members is a challenging task that suffers from clutter. Ensemble statistical summarization can alleviate this issue by permitting analyzing ensembles' distributional components like the mean and median, confidence intervals, and outliers. Contour boxplots, powered by Contour Band Depth (CBD), are a popular non-parametric ensemble summarization method that benefits from CBD's generality, robustness, and theoretical properties. In this work, we introduce Inclusion Depth (ID), a new notion of contour depth with three defining characteristics. First, ID is a generalization of functional Half-Region Depth, which offers several theoretical guarantees. Second, ID relies on a simple principle: the inside/outside relationships between contours. This facilitates implementing ID and understanding its results. Third, the computational complexity of ID scales quadratically in the number of members of the ensemble, improving CBD's cubic complexity. This also in practice speeds up the computation enabling the use of ID for exploring large contour ensembles or in contexts requiring multiple depth evaluations like clustering. In a series of experiments on synthetic data and case studies with meteorological and segmentation data, we evaluate ID's performance and demonstrate its capabilities for the visual analysis of contour ensembles.

Harnessing the Power of Sample Abundance: Theoretical Guarantees and Algorithms for Accelerated One-Bit Sensing

Harnessing the Power of Sample Abundance: Theoretical Guarantees and Algorithms for Accelerated One-Bit Sensing

Decentralized Counterfactual Value with Threat Detection for Multi-Agent Reinforcement Learning in mixed cooperative and competitive environments

This paper proposes a fully decentralized approach to address the challenge of general mixed cooperation and competition within the domain of Multi-Agent Reinforcement Learning (MARL). Conventional MARL approaches do not achieve full decentralization as they necessitate either the communication of implicit information or the retention of a centralized critic, rendering them impractical in mixed cooperative and competitive environments. To address these challenges, this paper proposes a Decentralized Counterfactual Value (DCV) to model the behaviors of other agents and mitigate the non-stationary problem, accompanied by a Threat Detection (TD) mechanism to discern latent competitive or cooperative relationships. In addition, DCVTD is incorporated into both value-based and policy-based RL paradigms with theoretical convergence guarantee. Finally, empirical validation across four representative environments demonstrates the superior performance of DCVTD in terms of collective returns, computational efficiency, and agent scalability over other fully decentralized approaches, centralized training with decentralized execution approaches, and alternative approaches involving agent modeling or reward shaping in comprehensive experiments.

UAdam: Unified Adam-Type Algorithmic Framework for Nonconvex Optimization.

Adam-type algorithms have become a preferred choice for optimization in the deep learning setting; however, despite their success, their convergence is still not well understood. To this end, we introduce a unified framework for Adam-type algorithms, termed UAdam. It is equipped with a general form of the second-order moment, which makes it possible to include Adam and its existing and future variants as special cases, such as NAdam, AMSGrad, AdaBound, AdaFom, and Adan. The approach is supported by a rigorous convergence analysis of UAdam in the general nonconvex stochastic setting, showing that UAdam converges to the neighborhood of stationary points with a rate of O(1/T). Furthermore, the size of the neighborhood decreases as the parameter β1 increases. Importantly, our analysis only requires the first-order momentum factor to be close enough to 1, without any restrictions on the second-order momentum factor. Theoretical results also reveal the convergence conditions of vanilla Adam, together with the selection of appropriate hyperparameters. This provides a theoretical guarantee for the analysis, applications, and further developments of the whole general class of Adam-type algorithms. Finally, several numerical experiments are provided to support our theoretical findings.

Euclidean-Distance-Preserved Feature Reduction for efficient person re-identification

Person Re-identification (Re-ID) aims to match person images across non-overlapping cameras. The existing approaches formulate this task as fine-grained representation learning with deep neural networks, which involves extracting image features using a deep convolutional network, followed by mapping the features into a discriminative space through another smaller network, in order to make full use of all possible cues. However, recent Re-ID methods that strive to capture every cue and make the space more discriminative have resulted in longer features, ranging from 1024 to 14336, leading to higher time (distance computation) and space (feature storage) complexities. There are two potential solutions: reduction-after-training methods (such as Principal Component Analysis and Linear Discriminant Analysis) and reduction-during-training methods (such as 1 × 1 Convolution). The former utilizes a statistical approach aiming for a global optimum but lacking end-to-end optimization of large data and deep neural networks. The latter lacks theoretical guarantees and may be vulnerable to training noise such as dataset noise or initialization seed.To address these limitations, we propose a method called Euclidean-Distance-Preserving Feature Reduction (EDPFR) that combines the strengths of both reduction-after-training and reduction-during-training methods. EDPFR first formulates the feature reduction process as a matrix decomposition and derives a condition to preserve the Euclidean distance between features, thus ensuring accuracy in theory. Furthermore, the method integrates the matrix decomposition process into a deep neural network to enable end-to-end optimization and batch training, while maintaining the theoretical guarantee. The result of the EDPFR is a reduction of the feature dimensions from fa and fb to fa′ and fb′, while preserving their Euclidean distance, i.e.L2(fa,fb)=L2(fa′,fb′). In addition to its Euclidean-Distance-Preserving capability, EDPFR also features a novel feature-level distillation loss. One of the main challenges in knowledge distillation is dimension mismatch. While previous distillation losses, usually project the mismatched features to matched class-level, spatial-level, or similarity-level spaces, this can result in a loss of information and decrease the flexibility and efficiency of distillation. Our proposed feature-level distillation leverages the benefits of the Euclidean-Distance-Preserving property and performs distillation directly in the feature space, resulting in a more flexible and efficient approach. Extensive on three Re-ID datasets, Market-1501, DukeMTMC-reID and MSMT demonstrate the effectiveness of our proposed Euclidean-Distance-Preserving Feature Reduction.

Robustness and exploration of variational and machine learning approaches to inverse problems: An overview

AbstractThis paper provides an overview of current approaches for solving inverse problems in imaging using variational methods and machine learning. A special focus lies on point estimators and their robustness against adversarial perturbations. In this context results of numerical experiments for a one‐dimensional toy problem are provided, showing the robustness of different approaches and empirically verifying theoretical guarantees. Another focus of this review is the exploration of the subspace of data‐consistent solutions through explicit guidance to satisfy specific semantic or textural properties.

Continuous-Time Reinforcement Learning Control: A Review of Theoretical Results, Insights on Performance, and Needs for New Designs.

This exposition discusses continuous-time reinforcement learning (CT-RL) for the control of affine nonlinear systems. We review four seminal methods that are the centerpieces of the most recent results on CT-RL control. We survey the theoretical results of the four methods, highlighting their fundamental importance and successes by including discussions on problem formulation, key assumptions, algorithm procedures, and theoretical guarantees. Subsequently, we evaluate the performance of the control designs to provide analyses and insights on the feasibility of these design methods for applications from a control designer's point of view. Through systematic evaluations, we point out when theory diverges from practical controller synthesis. We, furthermore, introduce a new quantitative analytical framework to diagnose the observed discrepancies. Based on the analyses and the insights gained through quantitative evaluations, we point out potential future research directions to unleash the potential of CT-RL control algorithms in addressing the identified challenges.

Reinforcement Learning Control of Hypersonic Vehicles and Performance Evaluations

This work presents a new framework for model-based continuous-time reinforcement learning (CT-RL) control of hypersonic vehicles (HSVs). The predominant classes of CT-RL methods for general nonlinear systems in adaptive dynamic programming (ADP) and deep RL tend to either present substantial theoretical results but lack practical synthesis capability (ADP) or show empirical promise without offering theoretical guarantees (deep RL). Meanwhile, RL control frameworks developed directly for HSVs tend to require a simplified model and complicated control structure, and they lack the substantial numerical evaluations essential for real-world flight implementation. To directly address these challenges, we propose a new decentralized excitable integral reinforcement learning framework within which the reference input-based exploration improves persistence of excitation. Together with new insights on prescaling and established decentralized control structure for HSVs, we demonstrate the resulting controller for significant performance improvement over classical Linear Quadratic Regulator (LQR) and feedback linearization methods. Additionally, we provide convergence, optimality, and closed-loop stability guarantees of the proposed method. We demonstrate these performance guarantees over a set of substantial and systematic numerical evaluations on an unstable, nonminimum phase HSV model subject to varying modeling errors and initial conditions.