We present a deterministic polynomial-time algorithm for computing @math -approximate (pure) Nash equilibria in weighted congestion games with polynomial cost functions of degree at most @math . This is an exponential improvement of the approximation factor with respect to the previously best algorithm. An appealing additional feature of our algorithm is that it uses only best-improvement steps in the actual game, as opposed to earlier approaches that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by [FOCS'11, TEAC 2015], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of @math for the Price of Anarchy (PoA) of @math -approximate equilibria in weighted congestion games, where @math is the Lambert-W function. More specifically, we show that this PoA is exactly equal to @math , where @math is the unique positive solution of the equation @math . Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, while our lower bound is simple enough to apply even to singleton congestion games.
Since the development of the smoothness method, other approaches to finding tight bounds on the @math have been investigated. Recently, was able to rederive, through the use of a primal-dual framework, the upper bound on the @math of linear unweighted games from @cite_7 . He also provided a simplified lower bound instance. Furthermore, he was able to show the upper bound of @math on the @math of @math -approximate equilibria for the special case of linear weighted games. It turns out to be equal to @math , the special case for @math of the general tight bound @math that we present below. Moreover, he provided matching instances with @math equal to @math , for @math in a certain subset of @math .
@cite_7
2007583146 2070394877 2055400003 2039050532
We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. We resolve the worst-case price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest non-trivial special case of affine cost functions. We make two distinct contributions. On the upper-bound side, we define the framework of "local smoothness", which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worst-case POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worst-case POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. In this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations. In particular, we apply Nesterov's smooth optimization technique [Y. E. Nesterov, Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543-547; Y. E. Nesterov, Math. Programming, 103 (2005), pp. 127-152] to their dual counterparts that are smooth convex problems. It is shown that the resulting approach has @math iteration complexity for finding an @math -optimal solution to both primal and dual problems. We then discuss the application of this approach to sparse covariance selection that is approximately solved as an @math -norm penalized maximum likelihood estimation problem, and also propose a variant of this approach which has substantially outperformed the latter one in our computational experiments. We finally compare the performance of these approaches with other first-order methods, namely, Nesterov's @math smooth approximation scheme and block-coordinate descent method studied in [A. d'Aspremont, O. Banerjee, and L. El Ghaoui, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 56-66; J. Friedman, T. Hastie, and R. Tibshirani, Biostatistics, 9 (2008), pp. 432-441] for sparse covariance selection on a set of randomly generated instances. It shows that our smooth optimization approach substantially outperforms the first method above, and moreover, its variant substantially outperforms both methods above. We consider the problem of minimizing the sum of a smooth function and a separable convex function. This problem includes as special cases bound-constrained optimization and smooth optimization with l1-regularization. We propose a (block) coordinate gradient descent method for solving this class of nonsmooth separable problems. We establish global convergence and, under a local Lipschitzian error bound assumption, linear convergence for this method. The local Lipschitzian error bound holds under assumptions analogous to those for constrained smooth optimization, e.g., the convex function is polyhedral and the smooth function is (nonconvex) quadratic or is the composition of a strongly convex function with a linear mapping. We report numerical experience with solving the l1-regularization of unconstrained optimization problems from in ACM Trans. Math. Softw. 7, 17–41, 1981 and from the CUTEr set (Gould and Orban in ACM Trans. Math. Softw. 29, 373–394, 2003). Comparison with L-BFGS-B and MINOS, applied to a reformulation of the l1-regularized problem as a bound-constrained optimization problem, is also reported.