Convergence Guarantees Research Articles

An implicit GNN solver for Poisson-like problems

This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems on general unstructured meshes with mixed boundary conditions. By leveraging the Implicit Layer Theory, Ψ-GNN models an “infinitely” deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical pre-requisite for physical applications, and is able to adapt to any initially provided solution. Ψ-GNN is trained using a physics-informed loss, and the training process is stable by design. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, Ψ-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

On compressive self-calibration with structural constraints via alternating minimization

This paper considers the compressive bilinear self-calibration problem with explicit low-dimensional structural constraints. The celebrated proximal alternating linearized minimization (PALM) framework is adapted to simultaneously allow general sub-sampling schemes and structure-promoting regularizers. For the first time in literature, we refine the conditional convergence guarantees of PALM and show that the parameter commonly adopted to remove the scaling ambiguity as well as the structural penalties can ensure the unconditional convergence independent of strict assumptions on the statistical properties of the measurements, subspaces, number of snapshots, or initial iterates. In particular, we impose sparse and small total variation structures on the target signals and provide detailed numerical procedures for efficient computations. The extension to the complex-valued case is also made and extensive numerical experiments are carried out to corroborate the theoretical claims. Different choices of sub-sampling schemes and compression rates are simulated to support the effectiveness of the proposed algorithm under various settings. We also make comparisons with the state-of-art competing methods and the superiority of our proposed algorithm is empirically verified.

Q‐learning‐based H∞ control for LPV systems

AbstractIn this paper, a model‐free H∞ control method is developed for discrete‐time linear parameter‐varying (LPV) systems by leveraging Q‐learning, which employs the data encompassing system states, control inputs, exogenous disturbance, and time‐varying convex hull rather than the dynamical model known a priori. A policy iteration algorithm presented via linear matrix inequality formed by such collected data is developed with convergence guarantee. Our analysis demonstrates that, in the presence of sufficiently rich disturbances, the attenuation level converges to a lower value than the traditional H∞ control solution derived from an exact LPV model. A numerical example is demonstrated to verify the stabilization, disturbance attenuation, adaptivity, and convergence of the H∞ attenuation level. Moreover, a continuous stirred tank reactor (CSTR) approximated by a discrete‐time LPV system is employed to show the practical potential of the developed model‐free approach.

Streaming First‐Order Optimization Methods With Momentum for Dominant Eigenspace Estimation

ABSTRACTIn this article, we consider the dominant eigenspace recovery for PCA in the streaming scenario, and design several new algorithms for this problem by incorporating various momentum schemes into two classical gradient based streaming eigensolvers, which emerges as a frequently‐used trick in improving gradient type methods. Theoretically, we establish the convergence guarantees of the proposed first‐order momentum methods, and prove their decreasing variance under favorable conditions. Extensive numerical experiments are conducted to corroborate the result that momentum does yield variance reduction, and also highlight the advantages of lowering the stepsize sensitivity. In addition, our methods with appropriately chosen momentum yield a dramatically enhanced performance in terms of both the convergence speed and accuracy.

An Integrated Approach to Improving Itinerary Completion in Coordinated Care Networks

Problem definition: Coordinated care network (CCN) is a burgeoning paradigm where patients’ diagnosis and treatment plans are developed based on collaboration between multiple, colocated medical specialties to holistically address patients’ health needs. A primary performance metric for CCNs is how quickly patients can complete their itinerary of appointments at multiple medical services in the network. Rapid completion is critical to care delivery but also presents a major operational challenge. Because information about a patient’s condition and treatment options evolves over the course of the itinerary, care paths are not known a priori. Thus, appointments (except for the first one) cannot be reserved in advance, which may result in significant delays if capacity is not allocated properly. Methodology/results: We study capacity allocation for the patient’s first (root) appointment as the primary operational lever to achieve rapid itinerary completion in CCNs. We develop a novel queueing-based analytical framework to optimize this root appointment allocation, maximizing the proportion of patients completing care by prespecified deadlines. Our framework accounts for the complex interactions among all patients in the network through the blocking process, which contrasts with conventional siloed planning. We provide an exact characterization of the itinerary time and develop a mean-field approximation with convergence guarantees that permits tractable solutions for large-scale network problems. In a simulation case study of Mayo Clinic, our solution improves on-time completion from 60% under the current plan to more than 93%. Managerial implications: We demonstrate that root appointment allocation is a multifaceted problem and that ignoring any of those facets can lead to poor performance. Simultaneously accounting for all of these complexities makes manual template design or traditional optimization methods inadequate, highlighting the significance of our integrated approach. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0649 .

Scalable Clustering: Large Scale Unsupervised Learning of Gaussian Mixture Models with Outliers

Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. This paper introduces a provably robust clustering algorithm based on loss minimization that performs well on Gaussian mixture models with outliers. It provides theoretical guarantees that the algorithm obtains high accuracy with high probability under certain assumptions. Moreover, it can also be used as an initialization strategy for k-means clustering. Experiments on real-world large-scale datasets demonstrate the effectiveness of the algorithm when clustering a large number of clusters, and a k-means algorithm initialized by the algorithm outperforms many of the classic clustering methods in both speed and accuracy, while scaling well to large datasets such as ImageNet.

Dynamic event-triggered integrated task and motion planning for process-aware source seeking

The process-aware source seeking (PASS) problem in flow fields aims to find an informative trajectory to reach an unknown source location while taking the energy consumption in the flow fields into consideration. Taking advantage of the dynamic flow field partition technique, this paper formulates this problem as a task and motion planning (TAMP) problem and proposes a bi-level hierarchical planning framework to decouple the planning of inter-region transition and inner-region trajectory by introducing inter-region junctions. An integrated strategy is developed to enable efficient upper-level planning by investigating the optimal solution of the lower-level planner. In order to leverage the information acquisition and computational burden, a dynamic event-triggered mechanism is introduced to enable asynchronized estimation, region partitioning and re-plans. The proposed algorithm provides guaranteed convergence of the trajectory, and achieves automatic trade-offs of both exploration-exploitation and accuracy-efficiency. Simulations in a highly complicated and realistic ocean surface flow field validate the merits of the proposed algorithm, which demonstrates a significant reduction in computational burden without compromising planning optimality.

Convergence Analysis of Mean Shift.

The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the Łojasiewicz inequality. Our findings extend existing ones covering analytic kernels and the Epanechnikov kernel. Those are significant in that they cover the biweight kernel, which is optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation.

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.

Hitting the High-dimensional notes: an ODE for SGD learning dynamics on GLMs and multi-index models

Abstract We analyze the dynamics of streaming stochastic gradient descent (SGD) in the high-dimensional limit when applied to generalized linear models and multi-index models (e.g. logistic regression, phase retrieval) with general data-covariance. In particular, we demonstrate a deterministic equivalent of SGD in the form of a system of ordinary differential equations that describes a wide class of statistics, such as the risk and other measures of sub-optimality. This equivalence holds with overwhelming probability when the model parameter count grows proportionally to the number of data. This framework allows us to obtain learning rate thresholds for the stability of SGD as well as convergence guarantees. In addition to the deterministic equivalent, we introduce an SDE with a simplified diffusion coefficient (homogenized SGD), which allows us to analyze the dynamics of general statistics of SGD iterates. Finally, we illustrate this theory on some standard examples and show numerical simulations, which give an excellent match to the theory.

Fast convergence to non-isolated minima: four equivalent conditions for $${\textrm{C}^{2}}$$ functions

AbstractOptimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. These singularities are inescapable when the optima are non-isolated. Yet, under the right circumstances, several algorithms preserve their favorable rates even when optima form a continuum (e.g., due to over-parameterization). This has been explained under various structural assumptions, including the Polyak–Łojasiewicz condition, Quadratic Growth and the Error Bound. We show that, for cost functions which are twice continuously differentiable ($$\textrm{C}^2$$ C 2 ), those three (local) properties are equivalent. Moreover, we show they are equivalent to the Morse–Bott property, that is, local minima form differentiable submanifolds, and the Hessian of the cost function is positive definite along its normal directions. We leverage this insight to improve local convergence guarantees for safe-guarded Newton-type methods under any (hence all) of the above assumptions. First, for adaptive cubic regularization, we secure quadratic convergence even with inexact subproblem solvers. Second, for trust-region methods, we argue capture can fail with an exact subproblem solver, then proceed to show linear convergence with an inexact one (Cauchy steps).

Bayesian Federated Learning with Hamiltonian Monte Carlo: Algorithm and Theory

This work introduces a novel and efficient Bayesian federated learning algorithm, namely, the Federated Averaging stochastic Hamiltonian Monte Carlo (FA-HMC), for parameter estimation and uncertainty quantification. We establish rigorous convergence guarantees of FA-HMC on non-iid distributed datasets, under the strong convexity and Hessian smoothness assumptions. Our analysis investigates the effects of parameter space dimension, noise on gradients and momentum, and the frequency of communication (between the central node and local nodes) on the convergence and communication costs of FA-HMC. Beyond that, we establish the tightness of our analysis by showing that the convergence rate cannot be improved even for continuous FA-HMC process. Moreover, extensive empirical studies demonstrate that FA-HMC outperforms the existing Federated Averaging-Langevin Monte Carlo (FA-LD) algorithm. Supplementary materials for this article are available online.

Globally convergent path-aware optimization with mobile robots

Consider a mobile robot that must navigate as quickly as possible to the global maxima of a function (e.g. density of seabed litter, pollutant concentration, wireless signal strength) defined over its operating area. This objective function is initially unknown and is assumed to be Lipschitz continuous. The limited velocity of the robot restricts the next samples to neighboring positions, and to avoid wasting time and energy, the robot’s path must be adapted as new information becomes available. The paper proposes two methods that use an upper bound on the objective to iteratively change the position targeted by the robot as new samples are acquired. The first method is FTW, which Turns When the best value seen so far of the objective Function is larger than the bound of the current target position. The second is FTWD, an extension of FTW that takes into account the Distance to the target. Convergence guarantees are provided for both methods, and a convergence rate is proven to characterize how fast the FTW suboptimality decreases as the number of samples grows. In a numerical study, FTWD greatly improves performance compared to FTW, outperforms two representative source-seeking baselines, and obtains results similar to a much more computationally intensive method that does not guarantee convergence. The relationship between FTW and FTWD is also confirmed in real-robot experiments, where a TurtleBot3 seeks the darkest point on a 2D grayscale map.

On market clearing of day ahead auctions for European power markets: Consumer payment minimisation versus social welfare maximisation

For inflexible demand and considering network constraints, we introduce a Consumer Payment Minimisation (CPM) market clearing mechanism and compare it with the standard Social Welfare Maximisation (SWM) used in European Day Ahead Electricity Markets. CPM presents a challenging optimisation problem, which we addressed by four numerical algorithms with varying computational costs and convergence guarantees, evaluated on synthetic data. We derive strong, partial analytical results and the concept of optimal zonal stack curves to facilitate efficient CPM solutions. Evaluations using 2019–2020 data from the Central Western European Day Ahead Power Market under the assumption of truthful bidding show that SWM models historical prices well, while CPM reduces producers’ market power and decreases zonal prices and total electricity procurement costs compared to SWM.

Optimizing pulmonary carcinoma detection through image segmentation using evolutionary algorithms

<span lang="EN-US">This paper’s goal is to suggest an image segmentation technique for use with medical images, specifically computer tomography scan images, to aid doctors in understanding the images. To address a variety of picture segmentation issues, it is necessary to investigate and apply novel evolutionary algorithms. The study focuses on pulmonary carcinoma, which is the cancer that affects males the most frequently across the globe. For proper treatment and life-saving measures, early identification of lung cancer is essential. To identify lung cancer, doctors frequently employ the computed tomography imaging technique. In order to extract tumours from lung scans, the study analyses the effectiveness of three optimization algorithms: k-means clustering, particle swarm optimization, and modified guaranteed convergence particle swarm optimization. The study also examines the pre-processing performance of four filters, namely the mean, bilateral, gaussian, and laplacian filters, shows that the bilateral filter is best suited for CT scans of the body. To test the proposed technique on 30 examples of lung scans. The proposed algorithm is tested on 30 sample lung images. The results show that the modified guaranteed convergence particle swarm optimization algorithm has the highest accuracy of 96.01%.</span>

Elastic reweighted sparsity regularized sparse unmixing for hyperspectral image analysis

Sparse unmixing is a crucial component in the analysis of hyperspectral images because of its ability to sparsely estimate abundances with a spectral library of considerable scale. The existing sparse unmixing techniques tend to employ the sparsity from a single perspective, i.e., ℓ1 or ℓ2,1. In this paper, we accurately estimate abundances from element-wise and channel-wise sparsity perspectives, which forms a new elastic reweighted sparse regularized sparse unmixing method, termed ElSpaSU. The proposed ElSpaSU method leverages the abundance matrix simultaneously using ℓ1 and ℓ2,1 sparsity regularizations. To further promote sparsity in the abundances, the proposed ElSpaSU incorporates two types of iterative reweighting weights. We utilize the alternating direction method of multipliers (ADMM) framework to ensure guaranteed convergence in solving the optimization problem. Experimental outcomes from simulated and real hyperspectral images highlight the remarkable efficacy of the suggested ElSpaSU in contrast to competing approaches.

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.

Tensor-based statistical learning methods for diagnosing product quality defects in multistage manufacturing processes

Multistage manufacturing processes with identical stages provide three-dimensional process data in which the first dimension represents the process (control/sensing) variable, the second is the stage, and the third is the measurement/sampling/data acquisition time point. Diagnosing quality faults in such processes often requires the simultaneous identification of crucial process variables and stages associated with product quality anomalies. Most existing diagnosis methods convert 3D data into a 2D matrix, resulting in loss of information and reduced diagnostic accuracy and stability. To address this challenge, we propose a penalized tensor regression model that regresses the product quality index against its 3D process data. For the estimation of high-dimensional regression coefficients with the limited amount of historical data, we apply the CANDECOMP/PARAFAC and Tucker decompositions to the coefficient tensor, which significantly reduces the number of parameters to be estimated. Based on the decompositions, a new regularization term is designed to enable the joint identification of critical process variables and stages. To estimate the parameters, we develop the block coordinate proximal descent algorithm and provide its convergence guarantee. Numerical studies demonstrate that the proposed methods can enhance diagnostic stability and on average improve the diagnostic accuracy by around 20% over existing benchmarks.

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.

Information Relaxation and a Duality-Driven Algorithm for Stochastic Dynamic Programs

The curse of dimensionality significantly restricts the use of dynamic programming methods in solving complex problems. Consequently, researchers and practitioners often resort to approximate (suboptimal) control policies that strike a balance between ease of implementation and satisfactory performance. Information relaxation-based duality techniques generate both upper and lower bounds for the true values of stochastic dynamic programming problems, allowing us to evaluate the optimality of approximate policies through the dual gap. However, the literature still lacks guidance on handling cases where the gaps are excessively loose. In “Information Relaxation and a Duality-Driven Algorithm for Stochastic Dynamic Programs,” Chen, Ma, Liu, and Yu develop a novel DDP framework to obtain and tighten confidence interval estimates for the true value functions of SDP problems. Leveraging a new finding that the dual operation yields subsolutions, they establish convergence guarantees for DDP. Additionally, a regression-based Monte Carlo method is introduced, aimed at high-dimensional applications. Numerical examples demonstrate that DDP effectively improves dual gaps for various heuristics that are commonly used in the literature.