arXiv subjects

Angelia Nedich

Publications and source records attributed to Angelia Nedich.

At least 19 recordsLinked to original sources

arXiv2026

A Certificate of Unboundedness for Polynomial Optimization Problems

Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the existence of a solution prior to undergoing global search. In this manuscript we propose a simple pre-processing algorithm to determine if an arbitrary polynomial optimization problem is unbounded from below thereby providing information about the problem's asymptotic geometry prior to solving the problem if a solution can be found.

math.OC↗

arXiv2025

Fast Distributed Nash Equilibrium Seeking in Monotone Games

This work proposes a novel distributed approach for computing a Nash equilibrium in convex games with merely monotone and restricted strongly monotone pseudo-gradients. By leveraging the idea of the centralized operator extrapolation method presented in [5] to solve variational inequalities, we develop the algorithm converging to Nash equilibria in games, where players have no access to the full information but are able to communicate with neighbors over some communication graph. The convergence rate is demonstrated to be geometric and improves the rates obtained by the previously presented procedures seeking Nash equilibria in the class of games under consideration.

math.OC↗

arXiv2024

Generalized Smooth Stochastic Variational Inequalities: Almost Sure Convergence and Convergence Rates

This paper focuses on solving a stochastic variational inequality (SVI) problem under relaxed smoothness assumption for a class of structured non-monotone operators. The SVI problem has attracted significant interest in the machine learning community due to its immediate application to adversarial training and multi-agent reinforcement learning. In many such applications, the resulting operators do not satisfy the smoothness assumption. To address this issue, we focus on a weaker generalized smoothness assumption called $\alpha$-symmetric. Under $p$-quasi sharpness and $\alpha$-symmetric assumptions on the operator, we study clipped projection (gradient descent-ascent) and clipped Korpelevich (extragradient) methods. For these clipped methods, we provide the first almost-sure convergence results without making any assumptions on the boundedness of either the stochastic operator or the stochastic samples. We also provide the first in-expectation unbiased convergence rate results for these methods under a relaxed smoothness assumption for $\alpha \leq \frac{1}{2}$.

math.OC↗

arXiv2024

Optimizing $(L_0, L_1)$-Smooth Functions by Gradient Methods

We study gradient methods for optimizing $(L_0, L_1)$-smooth functions, a class that generalizes Lipschitz-smooth functions and has gained attention for its relevance in machine learning. We provide new insights into the structure of this function class and develop a principled framework for analyzing optimization methods in this setting. While our convergence rate estimates recover existing results for minimizing the gradient norm in nonconvex problems, our approach significantly improves the best-known complexity bounds for convex objectives. Moreover, we show that the gradient method with Polyak stepsizes and the normalized gradient method achieve nearly the same complexity guarantees as methods that rely on explicit knowledge of~$(L_0, L_1)$. Finally, we demonstrate that a carefully designed accelerated gradient method can be applied to $(L_0, L_1)$-smooth functions, further improving all previous results.

math.OC↗

arXiv2024

Decentralized Federated Learning with Gradient Tracking over Time-Varying Directed Networks

We investigate the problem of agent-to-agent interaction in decentralized (federated) learning over time-varying directed graphs, and, in doing so, propose a consensus-based algorithm called DSGTm-TV. The proposed algorithm incorporates gradient tracking and heavy-ball momentum to distributively optimize a global objective function, while preserving local data privacy. Under DSGTm-TV, agents will update local model parameters and gradient estimates using information exchange with neighboring agents enabled through row- and column-stochastic mixing matrices, which we show guarantee both consensus and optimality. Our analysis establishes that DSGTm-TV exhibits linear convergence to the exact global optimum when exact gradient information is available, and converges in expectation to a neighborhood of the global optimum when employing stochastic gradients. Moreover, in contrast to existing methods, DSGTm-TV preserves convergence for networks with uncoordinated stepsizes and momentum parameters, for which we provide explicit bounds. These results enable agents to operate in a fully decentralized manner, independently optimizing their local hyper-parameters. We demonstrate the efficacy of our approach via comparisons with state-of-the-art baselines on real-world image classification and natural language processing tasks.

math.OC↗

arXiv2024

Non-Monotone Variational Inequalities

In this paper, we focus on deriving some sufficient conditions for the existence of solutions to non-monotone Variational Inequalities (VIs) based on inverse mapping theory. We have obtained several widely applicable sufficient conditions for this problem and have introduced a sufficient condition for the existence of a Minty solution. We have shown that the extra-gradient method converges to a solution of VI in the presence of a Minty solution. Additionally, we have shown that, under some extra assumption, the algorithm is efficient and approaches a particular type of Minty solution. Interpreting these results in an equivalent game theory problem, weak coupling conditions will be obtained, stating that if the players' cost functions are sufficiently weakly coupled, the game has a pure quasi-Nash equilibrium. Moreover, under the additional assumption of the existence of Minty solutions, a pure Nash equilibrium exists for the corresponding game.

math.OC↗

arXiv2024

On Characterizations of Potential and Ordinal Potential Games

This paper investigates some necessary and sufficient conditions for a game to be a potential game. At first, we extend the classical results of Slade and Monderer and Shapley from games with one-dimensional action spaces to games with multi-dimensional action spaces, which require differentiable cost functions. Then, we provide a necessary and sufficient conditions for a game to have a potential function by investigating the structure of a potential function in terms of the players' cost differences, as opposed to differentials. This condition provides a systematic way for construction of a potential function, which is applied to network congestion games, as an example. Finally, we provide some sufficient conditions for a game to be ordinal potential and generalized ordinal potential.

cs.GT↗

arXiv2024

On existence of solutions to non-convex minimization problems

We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and sufficient conditions for the existence of solutions to a general problem of minimizing a proper closed function over a closed, possibly unbounded, set. Towards the result, we introduce cones of retractive directions for a set and a function, establishing some basic properties for them. We also investigate the relationships between the cone of retractive directions of a function and the cone of level sets of the function. Using the cones of retractive directions we provide necessary and sufficient conditions for the existence of solutions that require an asymptotically bounded decay of a function, and a relation between the cones of retractive directions of the constraint set and the asymptotic cone of the objective function. Finally we refine these conditions for more structured problems.

math.OC↗

arXiv2024

Adaptive Methods for Variational Inequalities under Relaxed Smoothness Assumption

Variational Inequality (VI) problems have attracted great interest in the machine learning (ML) community due to their application in adversarial and multi-agent training. Despite its relevance in ML, the oft-used strong-monotonicity and Lipschitz continuity assumptions on VI problems are restrictive and do not hold in practice. To address this, we relax smoothness and monotonicity assumptions and study structured non-monotone generalized smoothness. The key idea of our results is in adaptive stepsizes. We prove the first-known convergence results for solving generalized smooth VIs for the three popular methods, namely, projection, Korpelevich, and Popov methods. Our convergence rate results for generalized smooth VIs match or improve existing results on smooth VIs. We present numerical experiments that support our theoretical guarantees and highlight the efficiency of proposed adaptive stepsizes.

math.OC↗

arXiv2023

Huber Loss-Based Penalty Approach to Problems with Linear Constraints

We consider a convex optimization problem with many linear inequality constraints. To deal with a large number of constraints, we provide a penalty reformulation of the problem, where the penalty is a variant of the one-sided Huber loss function with two penalty parameters. We study the infeasibility properties of the solutions of penalized problems for nonconvex and convex objective functions, as the penalty parameters vary with time. Then, we propose a random incremental penalty method for solving the original problem, and investigate its convergence properties for convex and strongly convex objective functions. We show that the iterates of the method converge to a solution of the original problem almost surely and in expectation for suitable choices of the penalty parameters and the stepsize. Also, we establish convergence rate of the method in terms of the expected function values by utilizing appropriately defined weighted averages of the iterates. We show $O(\ln^{1/2+\epsilon} k/{\sqrt k})$-convergence rate when the objective function is convex and $O(\ln^{\epsilon} k/k)$-convergence rate when the objective function is strongly convex, with $\epsilon>0$ being an arbitrarily small scalar. } To the best of our knowledge, these are the first results on the convergence rate for the penalty-based incremental subgradient method with time-varying penalty parameters.

math.OC↗

arXiv2023

Last Iterate Convergence of Popov Method for Non-monotone Stochastic Variational Inequalities

This paper focuses on non-monotone stochastic variational inequalities (SVIs) that may not have a unique solution. A commonly used efficient algorithm to solve VIs is the Popov method, which is known to have the optimal convergence rate for VIs with Lipschitz continuous and strongly monotone operators. We introduce a broader class of structured non-monotone operators, namely $p$-quasi sharp operators ($p> 0$), which allows tractably analyzing convergence behavior of algorithms. We show that the stochastic Popov method converges almost surely to a solution for all operators from this class under a linear growth. In addition, we obtain the last iterate convergence rate (in expectation) for the method under a linear growth condition for $2$-quasi sharp operators. Based on our analysis, we refine the results for smooth $2$-quasi sharp and $p$-quasi sharp operators (on a compact set), and obtain the optimal convergence rates. We further provide numerical experiments that demonstrate advantages of stochastic Popov method over stochastic projection method for solving SVIs.

math.OC↗

arXiv2023

Accelerating Distributed Nash Equilibrium Seeking

This work proposes a novel distributed approach for computing a Nash equilibrium in convex games with restricted strongly monotone pseudo-gradients. By leveraging the idea of the centralized operator extrapolation method presented in [4] to solve variational inequalities, we develop the algorithm converging to Nash equilibria in games, where players have no access to the full information but are able to communicate with neighbors over some communication graph. The convergence rate is demonstrated to be geometric and improves the rates obtained by the previously presented procedures seeking Nash equilibria in the class of games under consideration.

math.OC↗

arXiv2023

Characterization of Potential Games: Application in Aggregative Games

The main objective of this work is to describe games which fall under title of Potential and simplify the conditions for class of aggregative games. Games classified as aggregative are ones in which, in addition to the player's own action, the payoff for each player depends on an aggregate of all the players' decision variables. In this study, we developed a method based on payoff functions to determine if a given game is potential. Then, in order to identify the Aggregative Games that fall under this class we simplified the criteria for the class of Aggregative Games. A $3$-player Cournot game, also known as an Aggregative Potential Game, is used to test the characterization criteria for Potential Games. A $4$-player Cournot game is also utilized to test the form of potential function we obtained for class of general potential games.

cs.GT↗

arXiv2023

Distributed Stochastic Optimization with Gradient Tracking over Time-Varying Directed Networks

We study a distributed method called SAB-TV, which employs gradient tracking to collaboratively minimize the sum of smooth and strongly-convex local cost functions for networked agents communicating over a time-varying directed graph. Each agent, assumed to have access to a stochastic first-order oracle for obtaining an unbiased estimate of the gradient of its local cost function, maintains an auxiliary variable to asymptotically track the stochastic gradient of the global cost. The optimal decision and gradient tracking are updated over time through limited information exchange with local neighbors using row- and column-stochastic weights, guaranteeing both consensus and optimality. With a sufficiently small constant step-size, we demonstrate that, in expectation, SAB-TV converges linearly to a neighborhood of the optimal solution. Numerical simulations illustrate the effectiveness of the proposed algorithm.

math.OC↗

arXiv2023

CrowdCache: A Decentralized Game-Theoretic Framework for Mobile Edge Content Sharing

Mobile edge computing (MEC) is a promising solution for enhancing the user experience, minimizing content delivery expenses, and reducing backhaul traffic. In this paper, we propose a novel privacy-preserving decentralized game-theoretic framework for resource crowdsourcing in MEC. Our framework models the interactions between a content provider (CP) and multiple mobile edge device users (MEDs) as a non-cooperative game, in which MEDs offer idle storage resources for content caching in exchange for rewards. We introduce efficient decentralized gradient play algorithms for Nash equilibrium (NE) computation by exchanging local information among neighboring MEDs only, thus preventing attackers from learning users' private information. The key challenge in designing such algorithms is that communication among MEDs is not fixed and is facilitated by a sequence of undirected time-varying graphs. Our approach achieves linear convergence to the NE without imposing any assumptions on the values of parameters in the local objective functions, such as requiring strong monotonicity to be stronger than its dependence on other MEDs' actions, which is commonly required in existing literature when the graph is directed time-varying. Extensive simulations demonstrate the effectiveness of our approach in achieving efficient resource outsourcing decisions while preserving the privacy of the edge devices.

cs.GT↗

arXiv2023

Geometric Convergence of Distributed Heavy-Ball Nash Equilibrium Algorithm over Time-Varying Digraphs with Unconstrained Actions

This paper presents a new distributed algorithm that leverages heavy-ball momentum and a consensus-based gradient method to find a Nash equilibrium (NE) in a class of non-cooperative convex games with unconstrained action sets. In this approach, each agent in the game has access to its own smooth local cost function and can exchange information with its neighbors over a communication network. The main novelty of our work is the incorporation of heavy-ball momentum in the context of non-cooperative games that operate on fully-decentralized, directed, and time-varying communication graphs, while also accommodating non-identical step-sizes and momentum parameters. Overcoming technical challenges arising from the dynamic and asymmetric nature of mixing matrices and the presence of an additional momentum term, we provide a rigorous proof of the geometric convergence to the NE. Moreover, we establish explicit bounds for the step-size values and momentum parameters based on the characteristics of the cost functions, mixing matrices, and graph connectivity structures. We perform numerical simulations on a Nash-Cournot game to demonstrate accelerated convergence of the proposed algorithm compared to that of the existing methods.

cs.GT↗

arXiv2023

Accelerated $AB$/Push-Pull Methods for Distributed Optimization over Time-Varying Directed Networks

This paper investigates a novel approach for solving the distributed optimization problem in which multiple agents collaborate to find the global decision that minimizes the sum of their individual cost functions. First, the $AB$/Push-Pull gradient-based algorithm is considered, which employs row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost function, ensuring consensus on the optimal decision. Building on this algorithm, we then develop a general algorithm that incorporates acceleration techniques, such as heavy-ball momentum and Nesterov momentum, as well as their combination with non-identical momentum parameters. Previous literature has established the effectiveness of acceleration methods for various gradient-based distributed algorithms and demonstrated linear convergence for static directed communication networks. In contrast, we focus on time-varying directed communication networks and establish linear convergence of the methods to the optimal solution, when the agents' cost functions are smooth and strongly convex. Additionally, we provide explicit bounds for the step-size value and momentum parameters, based on the properties of the cost functions, the mixing matrices, and the graph connectivity structures. Our numerical results illustrate the benefits of the proposed acceleration techniques on the $AB$/Push-Pull algorithm.

math.OC↗

arXiv2022

AB/Push-Pull Method for Distributed Optimization in Time-Varying Directed Networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in Saadatniaki et al. (2020), where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to Saadatniaki et al. (2020), the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

math.OC↗