599 research outputs found
Potential games in volatile environments
This papers studies the co-evolution of networks and play in the context of finite population potential games. Action revision, link creation and link destruction are combined in a continuous-time Markov process. I derive the unique invariant distribution of this process in closed form, as well as the marginal distribution over action profiles and the conditional distribution over networks. It is shown that the equilibrium interaction topology is an inhomogeneous random graph. Furthermore, a characterization of the set of stochastically stable states is provided, generalizing existing results to models with endogenous interaction structures.
Stochastic mirror descent dynamics and their convergence in monotone variational inequalities
We examine a class of stochastic mirror descent dynamics in the context of
monotone variational inequalities (including Nash equilibrium and saddle-point
problems). The dynamics under study are formulated as a stochastic differential
equation driven by a (single-valued) monotone operator and perturbed by a
Brownian motion. The system's controllable parameters are two variable weight
sequences that respectively pre- and post-multiply the driver of the process.
By carefully tuning these parameters, we obtain global convergence in the
ergodic sense, and we estimate the average rate of convergence of the process.
We also establish a large deviations principle showing that individual
trajectories exhibit exponential concentration around this average.Comment: 23 pages; updated proofs in Section 3 and Section
Constrained Interactions and Social Coordination
We consider a co-evolutionary model of social coordination and network formation whereagents may decide on an action in a 2 x 2- coordination game and on whom to establish costly links to. We find that a payoff dominant convention is selected for a wider parameter range when agents may only support a limited number of links as compared to a scenario where agents are not constrained in their linking choice. The main reason behind this result is that constrained interactions create a tradeoff between the interactions an agent has and those he would rather have. Further, we discuss convex linking costs and provide suffcient conditions for the payoff dominant convention to be selected in mxm coordination games.
Hessian barrier algorithms for linearly constrained optimization problems
In this paper, we propose an interior-point method for linearly constrained
optimization problems (possibly nonconvex). The method - which we call the
Hessian barrier algorithm (HBA) - combines a forward Euler discretization of
Hessian Riemannian gradient flows with an Armijo backtracking step-size policy.
In this way, HBA can be seen as an alternative to mirror descent (MD), and
contains as special cases the affine scaling algorithm, regularized Newton
processes, and several other iterative solution methods. Our main result is
that, modulo a non-degeneracy condition, the algorithm converges to the
problem's set of critical points; hence, in the convex case, the algorithm
converges globally to the problem's minimum set. In the case of linearly
constrained quadratic programs (not necessarily convex), we also show that the
method's convergence rate is for some
that depends only on the choice of kernel function (i.e., not on the problem's
primitives). These theoretical results are validated by numerical experiments
in standard non-convex test functions and large-scale traffic assignment
problems.Comment: 27 pages, 6 figure
ON THE CONVERGENCE OF GRADIENT-LIKE FLOWS WITH NOISY GRADIENT INPUT
In view of solving convex optimization problems with noisy gradient input, we
analyze the asymptotic behavior of gradient-like flows under stochastic
disturbances. Specifically, we focus on the widely studied class of mirror
descent schemes for convex programs with compact feasible regions, and we
examine the dynamics' convergence and concentration properties in the presence
of noise. In the vanishing noise limit, we show that the dynamics converge to
the solution set of the underlying problem (a.s.). Otherwise, when the noise is
persistent, we show that the dynamics are concentrated around interior
solutions in the long run, and they converge to boundary solutions that are
sufficiently "sharp". Finally, we show that a suitably rectified variant of the
method converges irrespective of the magnitude of the noise (or the structure
of the underlying convex program), and we derive an explicit estimate for its
rate of convergence.Comment: 36 pages, 5 figures; revised proof structure, added numerical case
study in Section
Constrained Interactions and Social Coordination
We consider a co-evolutionary model of social coordination and network formation where agents may decide on an action in a 2x2 - coordination game and on whom to establish costly links to. We find that a payoff domination convention is selected for a wider parameter range when agents may only support a limited number of links as compared to a scenario where agents are not constrained in their linking choice. The main reason behind this result is that whenever there is a small cluster of agents playing the efficient strategy other players want to link up to those layers and choose the efficient action
On a General class of stochastic co-evolutionary dynamics
This paper presents a unified framework to study the co-evolution of networks and play, using the language of evolutionary game theory. We show by examples that the set-up is rich enough to encompass many recent models discussed by the literature. We completely characterize the invariant distribution of such processes and show how to calculate stochastically stable states by means of a treecharacterization algorithm. Moreover, specializing the process a bit further allows us to completely characterize the generated random graph ensemble. This new result demonstrates a new and rather general relation between random graph theory and evolutionary models with endogenous interaction structures.
Random block coordinate methods for inconsistent convex optimisation problems
We develop a novel randomised block coordinate primal-dual algorithm for a
class of non-smooth ill-posed convex programs. Lying in the midway between the
celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated
proximal gradient method, we establish global convergence of the last iterate
as well optimal and complexity rates in the convex and
strongly convex case, respectively, being the iteration count. Motivated by
the increased complexity in the control of distribution level electric power
systems, we test the performance of our method on a second-order cone
relaxation of an AC-OPF problem. Distributed control is achieved via the
distributed locational marginal prices (DLMPs), which are obtained \revise{as}
dual variables in our optimisation framework.Comment: Changed title and revised manuscrip
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