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

Combining statistical methods with dynamical insight to improve nonlinear estimation

Physical processes such as the weather are usually modelled using nonlinear dynamical systems. Statistical methods are found to be difficult to draw the dynamical information from the observations of nonlinear dynamics. This thesis is focusing on combining statistical methods with dynamical insight to improve the nonlinear estimate of the initial states, parameters and future states. In the perfect model scenario (PMS), method based on the Indistin-guishable States theory is introduced to produce initial conditions that are consistent with both observations and model dynamics. Our meth-ods are demonstrated to outperform the variational method, Four-dimensional Variational Assimilation, and the sequential method, En-semble Kalman Filter. Problem of parameter estimation of deterministic nonlinear models is considered within the perfect model scenario where the mathematical structure of the model equations are correct, but the true parameter values are unknown. Traditional methods like least squares are known to be not optimal as it base on the wrong assumption that the distribu-tion of forecast error is Gaussian IID. We introduce two approaches to address the shortcomings of traditional methods. The first approach forms the cost function based on probabilistic forecasting; the second approach focuses on the geometric properties of trajectories in short term while noting the global behaviour of the model in the long term. Both methods are tested on a variety of nonlinear models, the true parameter values are well identified. Outside perfect model scenario, to estimate the current state of the model one need to account the uncertainty from both observatiOnal noise and model inadequacy. Methods assuming the model is perfect are either inapplicable or unable to produce the optimal results. It is almost certain that no trajectory of the model is consistent with an infinite series of observations. There are pseudo-orbits, however, that are consistent with observations and these can be used to estimate the model states. Applying the Indistinguishable States Gradient De-scent algorithm with certain stopping criteria is introduced to find rel-evant pseudo-orbits. The difference between Weakly Constraint Four-dimensional Variational Assimilation (WC4DVAR) method and Indis-tinguishable States Gradient Descent method is discussed. By testing on two system-model pairs, our method is shown to produce more consistent results than the WC4DVAR method. Ensemble formed from the pseudo-orbit generated by Indistinguishable States Gradient Descent method is shown to outperform the Inverse Noise ensemble in estimating the current states. Outside perfect model scenario, we demonstrate that forecast with relevant adjustment can produce better forecast than ignoring the existence of model error and using the model directly to make fore-casts. Measurement based on probabilistic forecast skill is suggested to measure the predictability outside PMS

Statistical Mechanics of Recurrent Neural Networks I. Statics

A lecture notes style review of the equilibrium statistical mechanics of recurrent neural networks with discrete and continuous neurons (e.g. Ising, coupled-oscillators). To be published in the Handbook of Biological Physics (North-Holland). Accompanied by a similar review (part II) dealing with the dynamics.Comment: 49 pages, LaTe