581 research outputs found
Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems
We propose a novel decomposition framework for the distributed optimization
of general nonconvex sum-utility functions arising naturally in the system
design of wireless multiuser interfering systems. Our main contributions are:
i) the development of the first class of (inexact) Jacobi best-response
algorithms with provable convergence, where all the users simultaneously and
iteratively solve a suitably convexified version of the original sum-utility
optimization problem; ii) the derivation of a general dynamic pricing mechanism
that provides a unified view of existing pricing schemes that are based,
instead, on heuristics; and iii) a framework that can be easily particularized
to well-known applications, giving rise to very efficient practical (Jacobi or
Gauss-Seidel) algorithms that outperform existing adhoc methods proposed for
very specific problems. Interestingly, our framework contains as special cases
well-known gradient algorithms for nonconvex sum-utility problems, and many
blockcoordinate descent schemes for convex functions.Comment: submitted to IEEE Transactions on Signal Processin
Local Convergence of Gradient Methods for Min-Max Games under Partial Curvature
We study the convergence to local Nash equilibria of gradient methods for
two-player zero-sum differentiable games. It is well-known that such dynamics
converge locally when and may diverge when , where is the symmetric part of the Jacobian at equilibrium that accounts for the
"potential" component of the game. We show that these dynamics also converge as
soon as is nonzero (partial curvature) and the eigenvectors of the
antisymmetric part are in general position with respect to the kernel of
. We then study the convergence rates when and prove that they
typically depend on the average of the eigenvalues of , instead of the
minimum as an analogy with minimization problems would suggest. To illustrate
our results, we consider the problem of computing mixed Nash equilibria of
continuous games. We show that, thanks to partial curvature, conic particle
methods -- which optimize over both weights and supports of the mixed
strategies -- generically converge faster than fixed-support methods. For
min-max games, it is thus beneficial to add degrees of freedom "with
curvature": this can be interpreted as yet another benefit of
over-parameterization.Comment: 37 pages, 2 figures, 2 table
Consensus Multiplicative Weights Update: Learning to Learn using Projector-based Game Signatures
Cheung and Piliouras (2020) recently showed that two variants of the
Multiplicative Weights Update method - OMWU and MWU - display opposite
convergence properties depending on whether the game is zero-sum or
cooperative. Inspired by this work and the recent literature on learning to
optimize for single functions, we introduce a new framework for learning
last-iterate convergence to Nash Equilibria in games, where the update rule's
coefficients (learning rates) along a trajectory are learnt by a reinforcement
learning policy that is conditioned on the nature of the game: \textit{the game
signature}. We construct the latter using a new decomposition of two-player
games into eight components corresponding to commutative projection operators,
generalizing and unifying recent game concepts studied in the literature. We
compare the performance of various update rules when their coefficients are
learnt, and show that the RL policy is able to exploit the game signature
across a wide range of game types. In doing so, we introduce CMWU, a new
algorithm that extends consensus optimization to the constrained case, has
local convergence guarantees for zero-sum bimatrix games, and show that it
enjoys competitive performance on both zero-sum games with constant
coefficients and across a spectrum of games when its coefficients are learnt
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