Distributed Learning Policies for Power Allocation in Multiple Access Channels

We analyze the problem of distributed power allocation for orthogonal multiple access channels by considering a continuous non-cooperative game whose strategy space represents the users' distribution of transmission power over the network's channels. When the channels are static, we find that this game admits an exact potential function and this allows us to show that it has a unique equilibrium almost surely. Furthermore, using the game's potential property, we derive a modified version of the replicator dynamics of evolutionary game theory which applies to this continuous game, and we show that if the network's users employ a distributed learning scheme based on these dynamics, then they converge to equilibrium exponentially quickly. On the other hand, a major challenge occurs if the channels do not remain static but fluctuate stochastically over time, following a stationary ergodic process. In that case, the associated ergodic game still admits a unique equilibrium, but the learning analysis becomes much more complicated because the replicator dynamics are no longer deterministic. Nonetheless, by employing results from the theory of stochastic approximation, we show that users still converge to the game's unique equilibrium. Our analysis hinges on a game-theoretical result which is of independent interest: in finite player games which admit a (possibly nonlinear) convex potential function, the replicator dynamics (suitably modified to account for nonlinear payoffs) converge to an eps-neighborhood of an equilibrium at time of order O(log(1/eps)).Comment: 11 pages, 8 figures. Revised manuscript structure and added more material and figures for the case of stochastically fluctuating channels. This version will appear in the IEEE Journal on Selected Areas in Communication, Special Issue on Game Theory in Wireless Communication

Resource Allocation in a MAC with and without security via Game Theoretic Learning

In this paper a $K$-user fading multiple access channel with and without security constraints is studied. First we consider a F-MAC without the security constraints. Under the assumption of individual CSI of users, we propose the problem of power allocation as a stochastic game when the receiver sends an ACK or a NACK depending on whether it was able to decode the message or not. We have used Multiplicative weight no-regret algorithm to obtain a Coarse Correlated Equilibrium (CCE). Then we consider the case when the users can decode ACK/NACK of each other. In this scenario we provide an algorithm to maximize the weighted sum-utility of all the users and obtain a Pareto optimal point. PP is socially optimal but may be unfair to individual users. Next we consider the case where the users can cooperate with each other so as to disagree with the policy which will be unfair to individual user. We then obtain a Nash bargaining solution, which in addition to being Pareto optimal, is also fair to each user. Next we study a $K$-user fading multiple access wiretap Channel with CSI of Eve available to the users. We use the previous algorithms to obtain a CCE, PP and a NBS. Next we consider the case where each user does not know the CSI of Eve but only its distribution. In that case we use secrecy outage as the criterion for the receiver to send an ACK or a NACK. Here also we use the previous algorithms to obtain a CCE, PP or a NBS. Finally we show that our algorithms can be extended to the case where a user can transmit at different rates. At the end we provide a few examples to compute different solutions and compare them under different CSI scenarios.Comment: 27 pages, 12 figures. Part of the paper was presented in 2016 IEEE Information theory and applicaitons (ITA) Workshop, San Diego, USA in Feb. 2016. Submitted to journa

Game-theoretical control with continuous action sets

Motivated by the recent applications of game-theoretical learning techniques to the design of distributed control systems, we study a class of control problems that can be formulated as potential games with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems. The method employed is to analyse the learning process under study through a mean-field dynamical system that evolves in an infinite-dimensional function space (the space of probability distributions over the players' continuous controls). To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge to equilibrium in the case of potential games. These results combine to give a provably-convergent learning algorithm in which players do not need to keep track of the controls selected by the other agents.Comment: 19 page

Spectrum Coordination in Energy Efficient Cognitive Radio Networks

Device coordination in open spectrum systems is a challenging problem, particularly since users experience varying spectrum availability over time and location. In this paper, we propose a game theoretical approach that allows cognitive radio pairs, namely the primary user (PU) and the secondary user (SU), to update their transmission powers and frequencies simultaneously. Specifically, we address a Stackelberg game model in which individual users attempt to hierarchically access to the wireless spectrum while maximizing their energy efficiency. A thorough analysis of the existence, uniqueness and characterization of the Stackelberg equilibrium is conducted. In particular, we show that a spectrum coordination naturally occurs when both actors in the system decide sequentially about their powers and their transmitting carriers. As a result, spectrum sensing in such a situation turns out to be a simple detection of the presence/absence of a transmission on each sub-band. We also show that when users experience very different channel gains on their two carriers, they may choose to transmit on the same carrier at the Stackelberg equilibrium as this contributes enough energy efficiency to outweigh the interference degradation caused by the mutual transmission. Then, we provide an algorithmic analysis on how the PU and the SU can reach such a spectrum coordination using an appropriate learning process. We validate our results through extensive simulations and compare the proposed algorithm to some typical scenarios including the non-cooperative case and the throughput-based-utility systems. Typically, it is shown that the proposed Stackelberg decision approach optimizes the energy efficiency while still maximizing the throughput at the equilibrium.Comment: 12 pages, 10 figures, to appear in IEEE Transactions on Vehicular Technolog

Adaptive Power Allocation and Control in Time-Varying Multi-Carrier MIMO Networks

In this paper, we examine the fundamental trade-off between radiated power and achieved throughput in wireless multi-carrier, multiple-input and multiple-output (MIMO) systems that vary with time in an unpredictable fashion (e.g. due to changes in the wireless medium or the users' QoS requirements). Contrary to the static/stationary channel regime, there is no optimal power allocation profile to target (either static or in the mean), so the system's users must adapt to changes in the environment "on the fly", without being able to predict the system's evolution ahead of time. In this dynamic context, we formulate the users' power/throughput trade-off as an online optimization problem and we provide a matrix exponential learning algorithm that leads to no regret - i.e. the proposed transmit policy is asymptotically optimal in hindsight, irrespective of how the system evolves over time. Furthermore, we also examine the robustness of the proposed algorithm under imperfect channel state information (CSI) and we show that it retains its regret minimization properties under very mild conditions on the measurement noise statistics. As a result, users are able to track the evolution of their individually optimum transmit profiles remarkably well, even under rapidly changing network conditions and high uncertainty. Our theoretical analysis is validated by extensive numerical simulations corresponding to a realistic network deployment and providing further insights in the practical implementation aspects of the proposed algorithm.Comment: 25 pages, 4 figure

Boltzmann meets Nash: Energy-efficient routing in optical networks under uncertainty

Motivated by the massive deployment of power-hungry data centers for service provisioning, we examine the problem of routing in optical networks with the aim of minimizing traffic-driven power consumption. To tackle this issue, routing must take into account energy efficiency as well as capacity considerations; moreover, in rapidly-varying network environments, this must be accomplished in a real-time, distributed manner that remains robust in the presence of random disturbances and noise. In view of this, we derive a pricing scheme whose Nash equilibria coincide with the network's socially optimum states, and we propose a distributed learning method based on the Boltzmann distribution of statistical mechanics. Using tools from stochastic calculus, we show that the resulting Boltzmann routing scheme exhibits remarkable convergence properties under uncertainty: specifically, the long-term average of the network's power consumption converges within $\varepsilon$ of its minimum value in time which is at most $\tilde O(1/\varepsilon^2)$, irrespective of the fluctuations' magnitude; additionally, if the network admits a strict, non-mixing optimum state, the algorithm converges to it - again, no matter the noise level. Our analysis is supplemented by extensive numerical simulations which show that Boltzmann routing can lead to a significant decrease in power consumption over basic, shortest-path routing schemes in realistic network conditions.Comment: 24 pages, 4 figure