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The use of min-max optimization in the adversarial training of deep neural network classifiers, and the training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a generalization of the extragradient algorithm which provably converges to a stationary point. The algorithm applies not only to Euclidean spaces, but also to general ℓ𝑝-normed finite-dimensional real vector spaces. We also discuss its stability under stochastic oracles and provide bounds on its sample complexity. Our iteration complexity and sample complexity bounds either match or improve the best known bounds for the same or less general nonconvex-nonconcave settings, such as those that satisfy variational coherence or in which a weak solution to the associated variational inequality problem is assumed to exist. 

Projection-free conditional gradient (CG) methods are the algorithms of choice for constrained optimization setups in which projections are often computationally prohibitive but linear optimization over the constraint set remains computationally feasible. Unlike in projection-based methods, globally accelerated convergence rates are in general unattainable for CG. However, a very recent work on Locally accelerated CG (LaCG) has demonstrated that local acceleration for CG is possible for many settings of interest. The main downside of LaCG is that it requires knowledge of the smoothness and strong convexity parameters of the objective function. We remove this limitation by introducing a novel, Parameter-Free Locally accelerated CG (PF-LaCG) algorithm, for which we provide rigorous convergence guarantees. Our theoretical results are complemented by numerical experiments, which demonstrate local acceleration and showcase the practical improvements of PF-LaCG over non-accelerated algorithms, both in terms of iteration count and wall-clock time. 

We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods. 

Abstract—Full-duplex (FD) wireless is an attractive communication paradigm with high potential for improving network capacity and reducing delay in wireless networks. Despite significant progress on the physical layer development, the challenges associated with developing medium access control (MAC) protocols for heterogeneous networks composed of both legacy half-duplex (HD) and emerging FD devices have not been fully addressed. Therefore, we focus on the design and performance evaluation of scheduling algorithms for infrastructure-based heterogeneous HD-FD networks (composed of HD and FD users). We first show that centralized Greedy Maximal Scheduling (GMS) is throughput-optimal in heterogeneous HD-FD networks. We propose the Hybrid-GMS (H-GMS) algorithm, a distributed implementation of GMS that combines GMS and a queue-based random-access mechanism. We prove that H-GMS is throughputoptimal. Moreover, we analyze the delay performance of H-GMS by deriving lower bounds on the average queue length. We further demonstrate the benefits of upgrading HD nodes to FD nodes in terms of throughput gains for individual nodes and the whole network. Finally, we evaluate the performance of HGMS and its variants in terms of throughput, delay, and fairness between FD and HD users via extensive simulations. We show that in heterogeneous HD-FD networks, H-GMS achieves 16–30× better delay performance and improves fairness between HD and FD users by up to 50% compared with the fully decentralized Q-CSMA algorithm.