arXiv subjects

Angelia Nedić

Publications and source records attributed to Angelia Nedić.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Solutions for Non-monotone Variational Inequalities and Implications for Games

In this paper, we study the existence of solutions in non-monotone variational inequalities (VIs) through the normal mapping properties. In particular, we provide sufficient conditions for the existence of solutions assuming that the normal mapping associated with the VI is norm coercive and its generalized Jacobian has certain properties, such as a full rank at points where the normal mapping is not zero. Then, we investigate sufficient conditions for the VI mapping and its Jacobian that ensure the generalized Jacobian of the normal mapping has full rank, such as the uniform P-function and the uniform P-matrix condition. Subsequently, we focus on VIs arising from games and interpret our results in a game setting and provide a sufficient condition for a game to have a Nash equilibrium. Through examples, we show that our sufficient conditions can be used to assert the existence of a solution to a VI, or a quasi-Nash in a game, while the existing results relying on the uniform P-function property or the P$_Υ$-matrix condition cannot be employed.

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arXiv2026

On Fixed-Time Stability of Continuous Dynamics for Non-Monotone Variational Inequalities

Non-monotone variational inequalities (NMVI) are an important class of problems that generalize non-convex optimization and have various applications in optimization theory, machine learning, game theory, and economics, among others. Most existing work on NMVIs focuses on the asymptotic convergence of algorithms proposed to solve these problems. In this paper, we tackle the problems of exponential and fixed-time stability of the solution set of a class of NMVIs for both unconstrained and constrained problems. We first present novel conditions guaranteeing exponential stability of solutions to unconstrained NMVIs for a uniquely constructed dynamical system under mild assumptions on the gradient of the non-monotone map. Then, under similar assumptions, we construct another novel dynamical system whose equilibrium point is fixed-time stable, i.e., the trajectories reach the equilibrium within a fixed time, independent of the initial conditions. For the case of constrained NMVIs, we employ a continuous-time variant of the Korpelevich method for exponential stability of the solution set, and provide a novel scaling factor in the dynamics to achieve fixed-time stability. We illustrate the efficacy of the proposed modified dynamical systems through numerical simulations and conclude the paper with a brief note on the behavior of the discretized variant of the proposed dynamics and on further work that remains to be done.

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arXiv2026

Randomized Feasibility Methods for Constrained Optimization with Adaptive Step Sizes

We consider minimizing an objective function subject to constraints defined by the intersection of lower-level sets of convex functions. We study two cases: (i) strongly convex and Lipschitz-smooth objective function and (ii) convex but possibly nonsmooth objective function. To deal with the constraints that are not easy to project on, we use a randomized feasibility algorithm with Polyak steps and a random number of sampled constraints per iteration, while taking (sub)gradient steps to minimize the objective function. For case (i), we prove linear convergence in expectation of the objective function values to any prescribed tolerance using an adaptive stepsize. For case (ii), we develop a fully problem parameter-free and adaptive stepsize scheme that yields an $O(1/\sqrt{T})$ worst-case rate in expectation. The infeasibility of the iterates decreases geometrically with the number of feasibility updates almost surely, while for the averaged iterates, we establish an expected lower bound on the function values relative to the optimal value that depends on the distribution for the random number of sampled constraints. For certain choices of sample-size growth, optimal rates are achieved. Finally, simulations on a Quadratically Constrained Quadratic Programming (QCQP) problem, Support Vector Machines (SVM), and logistic regression with group fairness constraints demonstrate the computational efficiency of our algorithm compared to other state-of-the-art methods.

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arXiv2026

Distance-Aware Muon: Adaptive Step Scaling for Normalized Optimization

Muon and related normalized optimizers decouple the choice of update direction from the choice of step scale, but their practical performance remains sensitive to the scale of the normalized step. We study adaptive scaling rules for Muon in general norm geometries and develop three complementary algorithms. For smooth non-convex objectives, we introduce Distance-Adaptive Muon, whose trust-region radius is set from the radius explored by the trajectory, and prove a stationarity guarantee under a bounded-trajectory assumption. We then turn to star-convex objectives, a tractable model of the favorable global geometry often used to reason about the empirical loss landscapes of deep neural networks, where objective-gap guarantees are possible. In this setting, we first introduce Scale-Calibrated Muon, which keeps Muon's exponential moving average but sets the step length from a local descent certificate computed from the current gradient and momentum. For this method, we prove a last-iterate O(1/T) objective-gap bound under a bounded initial sublevel-set assumption, where the corresponding radius parameter appears only in the analysis and not in the algorithm. Finally, we develop Distance-Free Muon, a recentered trust-region method that uses a scalar distance certificate and a majorized one-dimensional search to select the trust-region radius without requiring the unknown distance from the initialization to a global minimizer. Experiments on Transformer language modeling (GPT-124M/WikiText-103) and image classification (ViT-Tiny/CIFAR-100) show that the proposed adaptive scaling rules reduce sensitivity to manual scale tuning and match or improve tuned fixed-scale Muon baselines under the tested budgets.

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arXiv2026

Distributed Normal Map-based Stochastic Proximal Gradient Methods over Networks

Consider $n$ agents connected over a network collaborating to minimize the average of their local cost functions combined with a common nonsmooth function. This paper introduces a unified algorithmic framework for solving such a problem through distributed stochastic proximal gradient methods, leveraging the normal map update scheme. Within this framework, we propose two new algorithms, termed Normal Map-based Distributed Stochastic Gradient Tracking (norM-DSGT) and Normal Map-based Exact Diffusion (norM-ED). We demonstrate that both methods can asymptotically achieve comparable convergence rates to the centralized stochastic proximal gradient descent method under a general variance condition on the stochastic gradients. Additionally, the number of iterations required for norM-ED to achieve such a rate (i.e., the transient time) behaves as $\mathcal{O}(n^{3}/(1-λ)^2)$ for minimizing composite objective functions, matching the performance of the non-proximal ED algorithm. Here $1-λ$ denotes the spectral gap of the mixing matrix related to the underlying network topology. To our knowledge, such a convergence result is state-of-the-art for the considered composite problem. Under the same condition, norM-DSGT enjoys a transient time of $\mathcal{O}(\max\{n^3/(1-λ)^2, n/(1-λ)^3\})$, which matches that of the non-proximal DSGT algorithm and norM-ED under the condition $(1-λ)^{-1}=\mathcal{O}(n^{2})$.

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arXiv2025

Popov Mirror-Prox Method for Variational Inequalities

This paper establishes the convergence properties of the Popov mirror-prox algorithm for solving stochastic and deterministic variational inequalities (VIs) under a polynomial growth condition on the mapping variation. Unlike existing methods that require prior knowledge of problem-specific parameters, we propose step-size schemes that are entirely parameter-free in both constant and diminishing forms. For stochastic and deterministic monotone VIs, we establish optimal convergence rates in terms of the dual gap function over a bounded constraint set. Additionally, for deterministic VIs with Hölder continuous mapping, we prove convergence in terms of the residual function without requiring a bounded set or a monotone mapping, provided a Minty solution exists. This allows our method to address certain classes of non-monotone VIs. However, knowledge of the Hölder exponent is necessary to achieve the best convergence rates in this case. By extending mirror-prox techniques to mappings with arbitrary polynomial growth, our work bridges an existing gap in the literature. We validate our theoretical findings with empirical results on matrix games, piecewise quadratic functions, and image classification tasks using ResNet-18.

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arXiv2025

Randomized Feasibility-Update Algorithms for Stochastic Variational Inequality Problems

This paper considers stochastic monotone variational inequalities whose feasible region is the intersection of a (possibly infinite) number of convex functional level sets. A projection-based approach or direct Lagrangian-based techniques for such problems can be computationally expensive if not impossible to implement. To deal with the problem, we consider randomized methods that avoid the projection step on the whole constraint set by employing random feasibility updates. In particular, we propose and analyze modified stochastic Korpelevich and Popov methods for solving monotone stochastic VIs. We introduce a modified dual gap function and prove the convergence rates with respect to this function. We illustrate the performance of the methods in simulations on a zero-sum matrix game.

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arXiv2025

On Non-Monotone Variational Inequalities

In this paper, we provide some sufficient conditions for the existence of solutions to non-monotone Variational Inequalities (VIs) based on inverse mapping theory and degree theory. We have obtained several applicable sufficient conditions for this problem and have introduced a sufficient condition for the existence of a Minty solution. We have shown that the Korpelevich and Popov methods converge to a solution of a non-monotone VI, provided that a Minty solution exists.

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arXiv2025

Characterizing Trust and Resilience in Distributed Consensus for Cyberphysical Systems

This work considers the problem of resilient consensus where stochastic values of trust between agents are available. Specifically, we derive a unified mathematical framework to characterize convergence, deviation of the consensus from the true consensus value, and expected convergence rate, when there exists additional information of trust between agents. We show that under certain conditions on the stochastic trust values and consensus protocol: 1) almost sure convergence to a common limit value is possible even when malicious agents constitute more than half of the network connectivity, 2) the deviation of the converged limit, from the case where there is no attack, i.e., the true consensus value, can be bounded with probability that approaches 1 exponentially, and 3) correct classification of malicious and legitimate agents can be attained in finite time almost surely. Further, the expected convergence rate decays exponentially as a function of the quality of the trust observations between agents.

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arXiv2025

Multi-Agent Trustworthy Consensus under Random Dynamic Attacks

In this work, we study the consensus problem in which legitimate agents send their values over an undirected communication network in the presence of an unknown subset of malicious or faulty agents. In contrast to former works, we generalize and characterize the properties of consensus dynamics with dependent sequences of malicious transmissions with dynamic (time-varying) rates, based on not necessarily independent trust observations. We consider a detection algorithm utilizing stochastic trust observations available to legitimate agents. Under these conditions, legitimate agents almost surely classify their neighbors and form their trusted neighborhoods correctly with decaying misclassification probabilities. We further prove that the consensus process converges almost surely despite the existence of malicious agents. For a given value of failure probability, we characterize the deviation from the nominal consensus value ideally occurring when there are no malicious agents in the system. We also examine the convergence rate of the process in finite time. Numerical simulations show the convergence among agents and indicate the deviation under different attack scenarios.

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arXiv2025

An Accelerated Distributed Stochastic Gradient Method with Momentum

In this paper, we introduce an accelerated distributed stochastic gradient method with momentum for solving the distributed optimization problem, where a group of $n$ agents collaboratively minimize the average of the local objective functions over a connected network. The method, termed ``Distributed Stochastic Momentum Tracking (DSMT)'', is a single-loop algorithm that utilizes the momentum tracking technique as well as the Loopless Chebyshev Acceleration (LCA) method. We show that DSMT can asymptotically achieve comparable convergence rates as centralized stochastic gradient descent (SGD) method under a general variance condition regarding the stochastic gradients. Moreover, the number of iterations (transient times) required for DSMT to achieve such rates behaves as $\mathcal{O}(n^{5/3}/(1-λ))$ for minimizing general smooth objective functions, and $\mathcal{O}(\sqrt{n/(1-λ)})$ under the Polyak-Łojasiewicz (PL) condition. Here, the term $1-λ$ denotes the spectral gap of the mixing matrix related to the underlying network topology. Notably, the obtained results do not rely on multiple inter-node communications or stochastic gradient accumulation per iteration, and the transient times are the shortest under the setting to the best of our knowledge.

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arXiv2025

Resilient Distributed Optimization for Multi-Agent Cyberphysical Systems

This work focuses on the problem of distributed optimization in multi-agent cyberphysical systems, where a legitimate agent's iterates are influenced both by the values it receives from potentially malicious neighboring agents, and by its own self-serving target function. We develop a new algorithmic and analytical framework to achieve resilience for the class of problems where stochastic values of trust between agents exist and can be exploited. In this case, we show that convergence to the true global optimal point can be recovered, both in mean and almost surely, even in the presence of malicious agents. Furthermore, we provide expected convergence rate guarantees in the form of upper bounds on the expected squared distance to the optimal value. Finally, numerical results are presented that validate our analytical convergence guarantees even when the malicious agents compose the majority of agents in the network and where existing methods fail to converge to the optimal nominal points.

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arXiv2024

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks

This paper proposes a distributed algorithm to find the Nash equilibrium in a class of non-cooperative convex games with partial-decision information. Our method employs a distributed projected gradient play approach alongside consensus dynamics, with individual agents minimizing their local costs through gradient steps and local information exchange with neighbors via a time-varying directed communication network. Addressing time-varying directed graphs presents significant challenges. Existing methods often circumvent this by focusing on static graphs or specific types of directed graphs or by requiring the stepsizes to scale with the Perron-Frobenius eigenvectors. In contrast, we establish novel results that provide a contraction property for the mixing terms associated with time-varying row-stochastic weight matrices. Our approach explicitly expresses the contraction coefficient based on the characteristics of the weight matrices and graph connectivity structures, rather than implicitly through the second-largest singular value of the weight matrix as in prior studies. The established results facilitate proving geometric convergence of the proposed algorithm and advance convergence analysis for distributed algorithms in time-varying directed communication networks. Numerical results on a Nash-Cournot game demonstrate the efficacy of the proposed method.

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arXiv2024

Popov Mirror-Prox for solving Variational Inequalities

We consider the mirror-prox algorithm for solving monotone Variational Inequality (VI) problems. As the mirror-prox algorithm is not practically implementable, except in special instances of VIs (such as affine VIs), we consider its implementation with Popov method updates. We provide convergence rate analysis of our proposed method for a monotone VI with a Lipschitz continuous mapping. We establish a convergence rate of $O(1/t)$, in terms of the number $t$ of iterations, for the dual gap function. Simulations on a two player matrix game corroborate our findings.

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arXiv2024

Randomized Feasibility-Update Algorithms for Variational Inequality Problems

This paper considers a variational inequality (VI) problem arising from a game among multiple agents, where each agent aims to minimize its own cost function subject to its constrained set represented as the intersection of a (possibly infinite) number of convex functional level sets. A direct projection-based approach or Lagrangian-based techniques for such a problem can be computationally expensive if not impossible to implement. To deal with the problem, we consider randomized methods that avoid the projection step on the whole constraint set by employing random feasibility updates. In particular, we propose and analyze such random methods for solving VIs based on the projection method, Korpelevich method, and Popov method. We establish the almost sure convergence of the methods and, also, provide their convergence rate guarantees. We illustrate the performance of the methods in simulations for two-agent games.

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arXiv2024

Learning Trust Over Directed Graphs in Multiagent Systems (extended version)

We address the problem of learning the legitimacy of other agents in a multiagent network when an unknown subset is comprised of malicious actors. We specifically derive results for the case of directed graphs and where stochastic side information, or observations of trust, is available. We refer to this as ``learning trust'' since agents must identify which neighbors in the network are reliable, and we derive a protocol to achieve this. We also provide analytical results showing that under this protocol i) agents can learn the legitimacy of all other agents almost surely, and that ii) the opinions of the agents converge in mean to the true legitimacy of all other agents in the network. Lastly, we provide numerical studies showing that our convergence results hold in practice for various network topologies and variations in the number of malicious agents in the network.

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arXiv2024

Fast Distributed Optimization over Directed Graphs under Malicious Attacks using Trust

In this work, we introduce the Resilient Projected Push-Pull (RP3) algorithm designed for distributed optimization in multi-agent cyber-physical systems with directed communication graphs and the presence of malicious agents. Our algorithm leverages stochastic inter-agent trust values and gradient tracking to achieve geometric convergence rates in expectation even in adversarial environments. We introduce growing constraint sets to limit the impact of the malicious agents without compromising the geometric convergence rate of the algorithm. We prove that RP3 converges to the nominal optimal solution almost surely and in the $r$-th mean for any $r\geq 1$, provided the step sizes are sufficiently small and the constraint sets are appropriately chosen. We validate our approach with numerical studies on average consensus and multi-robot target tracking problems, demonstrating that RP3 effectively mitigates the impact of malicious agents and achieves the desired geometric convergence.

eess.SY↗

arXiv2024

Constrained multi-cluster game: Distributed Nash equilibrium seeking over directed graphs

Motivated by the complex dynamics of cooperative and competitive interactions within networked agent systems, multi-cluster games provide a framework for modeling the interconnected goals of self-interested clusters of agents. For this setup, the existing literature lacks comprehensive gradient-based solutions that simultaneously consider constraint sets and directed communication networks, both of which are crucial for many practical applications. To address this gap, this paper proposes a distributed Nash equilibrium seeking algorithm that integrates consensus-based methods and gradient-tracking techniques, where inter-cluster and intra-cluster communications only use row- and column-stochastic weight matrices, respectively. To handle constraints, we introduce an averaging procedure, which can effectively address the complications associated with projections. In turn, we can show linear convergence of our algorithm, focusing on the contraction property of the optimality gap. We demonstrate the efficacy of the proposed algorithm through a microgrid energy management application.

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