Nonconvex Sets Research Articles

Monitor-based synthesis in enforcing disjunctive generalised mutual exclusion constraints

This paper focuses on enforcing disjunctive generalised mutual exclusion constraints, referred to as OR-GMEC, on Petri net models. In this case, the legal marking set is non-convex, while the admissible marking set can either be convex or non-convex. The study presents a necessary condition for the existence of a monitor-based solution for a non-convex admissible marking set. Subsequently, the disjunctive constraints are developed to derive an optimal supervisory controller, and an algorithm is proposed for solution computation. The design procedure includes off-line computation mechanisms, yielding an optimal (maximally permissive) supervisor with only a few monitors. The key advantage of the suggested approach is that, unlike the current techniques, the number of monitors in the supervisor’s structure does not rise as the number of constraints increases.

A Unified Analysis of AdaGrad With Weighted Aggregation and Momentum Acceleration.

Integrating adaptive learning rate and momentum techniques into stochastic gradient descent (SGD) leads to a large class of efficiently accelerated adaptive stochastic algorithms, such as AdaGrad, RMSProp, Adam, AccAdaGrad, and so on. In spite of their effectiveness in practice, there is still a large gap in their theories of convergences, especially in the difficult nonconvex stochastic setting. To fill this gap, we propose weighted AdaGrad with unified momentum and dubbed AdaUSM, which has the main characteristics that: 1) it incorporates a unified momentum scheme that covers both the heavy ball (HB) momentum and the Nesterov accelerated gradient (NAG) momentum and 2) it adopts a novel weighted adaptive learning rate that can unify the learning rates of AdaGrad, AccAdaGrad, Adam, and RMSProp. Moreover, when we take polynomially growing weights in AdaUSM, we obtain its O(log(T)/√T) convergence rate in the nonconvex stochastic setting. We also show that the adaptive learning rates of Adam and RMSProp correspond to taking exponentially growing weights in AdaUSM, thereby providing a new perspective for understanding Adam and RMSProp. Finally, comparative experiments of AdaUSM against SGD with momentum, AdaGrad, AdaEMA, Adam, and AMSGrad on various deep learning models and datasets are also carried out.

Painless Stochastic Conjugate Gradient for Large-Scale Machine Learning.

Conjugate gradient (CG), as an effective technique to speed up gradient descent algorithms, has shown great potential and has widely been used for large-scale machine-learning problems. However, CG and its variants have not been devised for the stochastic setting, which makes them extremely unstable, and even leads to divergence when using noisy gradients. This article develops a novel class of stable stochastic CG (SCG) algorithms with a faster convergence rate via the variance-reduced technique and an adaptive step size rule in the mini-batch setting. Actually, replacing the use of a line search in the CG-type approaches which is time-consuming, or even fails for SCG, this article considers using the random stabilized Barzilai-Borwein (RSBB) method to obtain an online step size. We rigorously analyze the convergence properties of the proposed algorithms and show that the proposed algorithms attain a linear convergence rate for both the strongly convex and nonconvex settings. Also, we show that the total complexity of the proposed algorithms matches that of modern stochastic optimization algorithms under different cases. Scores of numerical experiments on machine-learning problems demonstrate that the proposed algorithms outperform state-of-the-art stochastic optimization algorithms.

Federated deep reinforcement learning based trajectory design for UAV-assisted networks with mobile ground devices

Exploiting highly maneuverable unmanned aerial vehicles (UAVs) has been considered as an efficient way to assist wireless systems, e.g., for applications of data collection. However, several challenges remain to be addressed in the design of such UAV-assisted networks, including multi-UAV joint trajectory determination, data privacy protection, and adaption to the complex channel environment particularly with mobile ground devices (GDs). In this paper, we study a multi-UAV assisted data collection system where UAVs collect data locally from mobile GDs. The aim is to minimize the whole operation time cost via jointly optimizing the UAVs’ three-dimensional (3D) trajectory together with the GDs’ communication scheduling, while satisfying the constraints of no-fly zones (NFZs) and collision avoidance. With a nonconvex feasible set (due to the NFZs), the established problem is nonconvex. Moreover, the randomness of GDs movements significantly reduces the performance of a typical redesign, i.e., determining the UAVs’ trajectory and users’ scheduling before starting the data collection task. To tackle these issues, we first transform the established problem into a Markov decision one, and then propose a multi-agent federated reinforcement learning (MAFRL)-based approach to optimize the dynamic long-term objective via jointly determining UAVs’ 3D trajectory and GD’s communication scheduling. A multi-step propagation technique and a dueling network architecture are adopted to enhance the neural network utilized to train agents, i.e., to accelerate the convergence rate of the proposed method and improve its overall stability. Finally, experimental results reveal the effectiveness of our proposed method in the considered practical scenario.

Distributed learning with compressed gradient differences*

Training large machine learning models requires a distributed computing approach, with communication of the model updates being the bottleneck. For this reason, several methods based on the compression of updates were recently proposed using sparsification or quantization. However, none of the prior methods are able to learn the gradients, which renders them incapable of converging to the true optimum in the batch mode. In this work we propose a new method – DIANA – which resolves this issue via compression of gradient differences. We provide theory in the strongly convex and nonconvex settings that shows improved convergence rates, and use it to obtain the first convergence rate for the previously proposed method TernGrad. Finally, we provide theory to support non-smooth regularizers.

Stabilizing and Accelerating Federated Learning on Heterogeneous Data With Partial Client Participation.

Federated learning (FL) commonly encourages the clients to perform multiple local updates before the global aggregation, thus avoiding frequent model exchanges and relieving the communication bottleneck between the server and clients. Though empirically effective, the negative impact of multiple local updates on the stability of FL is not thoroughly studied, which may result in a globally unstable and slow convergence. Based on sensitivity analysis, we define in this paper a local-update stability index for the general FL, as measured by the maximum inter-client model discrepancy after the multiple local updates that mainly stems from the data heterogeneity. It enables to determine how much the variation of client's models with multiple local updates may influence the global model, and can also be linked with the convergence and generalization. We theoretically derive the proposed local-update stability for current state-of-the-art FL methods, providing possible insight to understanding their motivation and limitation from a new perspective of stability. For example, naively executing the parallel acceleration locally at clients would harm the local-update stability. Motivated by this, we then propose a novel accelerated yet stabilized FL algorithm (named FedANAG) based on the server- and client-level Nesterov accelerated gradient (NAG). In FedANAG, the global and local momenta are elaborately designed and alternatively updated, while the stability of local update is enhanced with help of the global momentum. We prove the convergence of FedANAG for strongly convex, general convex and non-convex settings. We then conduct evaluations on both the synthetic and real-world datasets to first validate our proposed local-update stability. The results further show that across various data heterogeneity and client participation ratios, FedANAG not only accelerates the global convergence by reducing the required number of communication rounds to a target accuracy, but converges to an eventually higher accuracy.

Analysis of regularized federated learning

Federated learning is an efficient machine learning tool for dealing with heterogeneous big data and privacy protection. Federated learning methods with regularization can control the level of communications between the central and local machines. Stochastic gradient descent is often used for implementing such methods on heterogeneous big data, to reduce the communication costs. In this paper, we consider such an algorithm called Loopless Local Gradient Descent which has advantages in reducing the expected communications by controlling a probability level. We improve the method by allowing flexible step sizes and carry out novel analysis for the convergence of the algorithm in a non-convex setting in addition to the standard strongly convex setting. In the non-convex setting, we derive rates of convergence when the smooth objective function satisfies a Polyak-Łojasiewicz condition. When the objective function is strongly convex, a sufficient and necessary condition for the convergence in expectation is presented.

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.

Geometric interpretation of efficient weight vectors

Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.

On the critical points of solutions of PDE in non-convex settings: The case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely{−Δu=λeu in Ωu=0 on ∂Ω, where Ω⊂R2 is a bounded smooth domain and λ>0 is a small parameter.

Alternating generalized projection method applied to phase-only synthesis process of satellite reflectarray antennas

Promising algorithm of alternating generalized projection method (AGP) is proposed for phase-only synthesis process of satellite reflectarray antennas under intersection approach. This promising algorithm is a hybridized algorithm of two specialized algorithms for non-convex sets rescued in the literature: algorithm based on separating hyperplanes and algorithm based on decomposition method in polar cones. Since the sets involved in phase-only synthesis process of satellite reflectarray antennas are non-convex, the conventional von Neumann alternating projection method proposed in the literature does not guarantee convergence to the point in the intersection of the involved sets. In addition, the results of the phase-only synthesis obtained by the different algorithms were compared and promising improvements produced by the proposed hybridized algorithm were shown.

Unmanned aerial vehicle variable radius set covering problem for emergency wireless network

The utilization of unmanned aerial vehicles (UAVs) has garnered increasing interest across various disciplines. The idea of smart cities led to the exploration of using UAVs in the public service industry, with a focus on integrating all aspects of the system through information and physical items. In this paper, we introduce and model the operational problem of using UAVs to construct an emergency wireless network in a disaster situation using the set covering approach. Unique to our study is the consideration that UAVs do not require predetermined positioning or altitude specifications. Consequently, the problem described in this paper can be classified as a planar variable radius covering problem, which involves a nonconvex continuous relaxation feasible set. We introduce the application of the Dantzig–Wolfe decomposition alongside a branch-and-price algorithm to tackle this problem. Additionally, the pricing subproblem’s solvable equivalent formulation is ingeniously derived from existing theories on the minimum covering circle. For large-scale instances, we propose a heuristic derived from an extended formulation. Comparative computational experiments demonstrate that our proposed algorithms significantly surpass the efficiency of the benchmark genetic algorithm previously reported in the literature.

On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it

Abstract We propose iterative algorithms to solve adversarial training problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adversarial learning problems. We present results for non-convex non-concave settings, as well as for non-convex concave ones. Numerical experiments illustrate our results.

Minimization over Nonconvex Sets

Minimization over Nonconvex Sets

Non-degenerate necessary conditions for non-convex differential inclusions

This paper concerns dynamic optimization problems with pathwise state constraints, in which the dynamics are formulated as a differential inclusion. First order necessary conditions of optimality are available for these problems. Foremost among these conditions is the Euler Lagrange inclusion (combined with a transversality and Weierstrass conditions), which improves on earlier conditions such as Clarke's Hamiltonian inclusion. The most recent version of the Euler Lagrange inclusion (which we call the standard Euler Lagrange condition) has been validated under unrestrictive conditions, which allow the velocity sets of the differential inclusion to be non-convex. If the initial state is fixed and located in the boundary of the state constraint set then the Euler Lagrange inclusion conveys no information about minimizers. This is the degeneracy phenomenon for state constrained dynamic optimization. The non-degenerate necessary conditions literature provides additional conditions, to supplement the standard necessary conditions. These typically identify some properties of the boundary values of the maximized Hamiltonian which ensure non-triviality of the necessary conditions, under a simple controllability hypothesis. Non-degenerate conditions, incorporating the Euler Lagrange inclusion are currently available only under the hypothesis that the velocity sets are convex. Using new perturbation techniques, we derive a non-degenerate Euler Lagrange condition that is valid for non-convex velocity sets. These conditions bring into line non-degenerate optimality conditions for differential inclusion problems with those for problems involving controlled differential equations which have previously been validated for non-convex velocity sets.

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

A new stable method to compute mean value coordinates

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments.

Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization

This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm. Funding: This work was supported by Norway Grants 2014-2021 [Grant ELO-Hyp 24/2020]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii [Grants PN-III-P4-PCE-2021-0720, L2O-MOC, nr 70/2022]; and the ITN-ETN project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement [Grant 861137].

Communication-Efficient Zeroth-Order Adaptive Optimization for Federated Learning

Federated learning has become a prevalent distributed training paradigm, in which local devices collaboratively train learning models without exchanging local data. One of the most dominant frameworks of federated learning (FL) is FedAvg, since it is efficient and simple to implement; here, the first-order information is generally utilized to train the parameters of learning models. In practice, however, the gradient information may be unavailable or infeasible in some applications, such as federated black-box optimization problems. To solve the issue, we propose an innovative zeroth-order adaptive federated learning algorithm without using the gradient information, referred to as ZO-AdaFL, which integrates the zeroth-order optimization algorithm into the adaptive gradient method. Moreover, we also rigorously analyze the convergence behavior of ZO-AdaFL in a non-convex setting, i.e., where ZO-AdaFL achieves convergence to a region close to a stationary point at a speed of O(1/T) (T represents the total iteration number). Finally, to verify the performance of ZO-AdaFL, simulation experiments are performed using the MNIST and FMNIST datasets. Our experimental findings demonstrate that ZO-AdaFL outperforms other state-of-the-art zeroth-order FL approaches in terms of both effectiveness and efficiency.

Every Quantum Helps: Operational Advantage of Quantum Resources beyond Convexity.

Identifying what quantum-mechanical properties are useful to untap a superior performance in quantum technologies is a pivotal question. Quantum resource theories provide a unified framework to analyze and understand such properties, as successfully demonstrated for entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a nonconvex set of free states and their interpretation has so far remained elusive. Here we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness measure in general resource theories. First, we characterize the generalized robustness in terms of a nonlinear resource witness and reveal that any state is more advantageous than a free one in some multicopy channel discrimination task. Next, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any specification on the structure of the free states.