Inertial game dynamics and applications to constrained optimization

Aiming to provide a new class of game dynamics with good long-term rationality properties, we derive a second-order inertial system that builds on the widely studied "heavy ball with friction" optimization method. By exploiting a well-known link between the replicator dynamics and the Shahshahani geometry on the space of mixed strategies, the dynamics are stated in a Riemannian geometric framework where trajectories are accelerated by the players' unilateral payoff gradients and they slow down near Nash equilibria. Surprisingly (and in stark contrast to another second-order variant of the replicator dynamics), the inertial replicator dynamics are not well-posed; on the other hand, it is possible to obtain a well-posed system by endowing the mixed strategy space with a different Hessian-Riemannian (HR) metric structure, and we characterize those HR geometries that do so. In the single-agent version of the dynamics (corresponding to constrained optimization over simplex-like objects), we show that regular maximum points of smooth functions attract all nearby solution orbits with low initial speed. More generally, we establish an inertial variant of the so-called "folk theorem" of evolutionary game theory and we show that strict equilibria are attracting in asymmetric (multi-population) games - provided of course that the dynamics are well-posed. A similar asymptotic stability result is obtained for evolutionarily stable strategies in symmetric (single- population) games.Comment: 30 pages, 4 figures; significantly revised paper structure and added new material on Euclidean embeddings and evolutionarily stable strategie

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

On the robustness of learning in games with stochastically perturbed payoff observations

Motivated by the scarcity of accurate payoff feedback in practical applications of game theory, we examine a class of learning dynamics where players adjust their choices based on past payoff observations that are subject to noise and random disturbances. First, in the single-player case (corresponding to an agent trying to adapt to an arbitrarily changing environment), we show that the stochastic dynamics under study lead to no regret almost surely, irrespective of the noise level in the player's observations. In the multi-player case, we find that dominated strategies become extinct and we show that strict Nash equilibria are stochastically stable and attracting; conversely, if a state is stable or attracting with positive probability, then it is a Nash equilibrium. Finally, we provide an averaging principle for 2-player games, and we show that in zero-sum games with an interior equilibrium, time averages converge to Nash equilibrium for any noise level.Comment: 36 pages, 4 figure

A stochastic approximation algorithm for stochastic semidefinite programming

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continous- time matrix exponential scheme regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an $\varepsilon$-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless multiple-input and multiple-output (MIMO) systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations which illustrate the robustness and scalability of the proposed method in realistic network conditions.Comment: 25 pages, 4 figure

Riemannian game dynamics

We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the derivation of the dynamic

Transmit without regrets: Online optimization in MIMO-OFDM cognitive radio systems

In this paper, we examine cognitive radio systems that evolve dynamically over time due to changing user and environmental conditions. To combine the advantages of orthogonal frequency division multiplexing (OFDM) and multiple-input, multiple-output (MIMO) technologies, we consider a MIMO-OFDM cognitive radio network where wireless users with multiple antennas communicate over several non-interfering frequency bands. As the network's primary users (PUs) come and go in the system, the communication environment changes constantly (and, in many cases, randomly). Accordingly, the network's unlicensed, secondary users (SUs) must adapt their transmit profiles "on the fly" in order to maximize their data rate in a rapidly evolving environment over which they have no control. In this dynamic setting, static solution concepts (such as Nash equilibrium) are no longer relevant, so we focus on dynamic transmit policies that lead to no regret: specifically, we consider policies that perform at least as well as (and typically outperform) even the best fixed transmit profile in hindsight. Drawing on the method of matrix exponential learning and online mirror descent techniques, we derive a no-regret transmit policy for the system's SUs which relies only on local channel state information (CSI). Using this method, the system's SUs are able to track their individually evolving optimum transmit profiles remarkably well, even under rapidly (and randomly) changing conditions. Importantly, the proposed augmented exponential learning (AXL) policy leads to no regret even if the SUs' channel measurements are subject to arbitrarily large observation errors (the imperfect CSI case), thus ensuring the method's robustness in the presence of uncertainties.Comment: 25 pages, 3 figures, to appear in the IEEE Journal on Selected Areas in Communication