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## Section: New Results

### Game Theory and Applications

#### Fair Scheduling in Large Distributed Computing Sytems

Fairly sharing resources of a distributed computing system between users is a critical issue that we have investigated in two ways.

Our first proposal specifically addresses the question of designing a distributed sharing mechanism. A possible answer resorts to Lagrangian optimization and distributed gradient descent. Under certain conditions, the resource sharing problem can be formulated as a global optimization problem, which can be solved by a distributed self-stabilizing demand and response algorithm. In the last decade, this technique has been applied to design network protocols (variants of TCP, multi-path network protocols, wireless network protocols) and even distributed algorithms for smart grids. In [9] , we explain how to use this technique for scheduling Bag-of-Tasks (BoT) applications on a Grid since until now, only simple mechanisms have been used to ensure a fair sharing of resources amongst these applications. Although the resulting algorithm is in essence very similar to previously proposed algorithms in the context of flow control in multi-path networks, we show using carefully designed experiments and a thorough statistical analysis that the grid context is surprisingly more difficult than the multi-path network context. Interestingly, we can show that, in practice, the convergence of the algorithm is hindered by the heterogeneity of application characteristics, which is completely overlooked in related theoretical work. Our careful investigation provides enough insights to understand the true difficulty of this approach and to propose a set of non-trivial adaptations that enable convergence in the grid context. The effectiveness of our proposal is proven through an extensive set of complex and realistic simulations.

Our second proposal is centralized but more fine grain as it does drop the steady-state hypothesis and considers sequences of campaigns. Campaign Scheduling is characterized by multiple job submissions issued from multiple users over time. The work in [18] presents a new fair scheduling algorithm called OStrich whose principle is to maintain a virtual time-sharing schedule in which the same amount of processors is assigned to each user. The completion times in the virtual schedule determine the execution order on the physical processors. Then, campaigns are interleaved in a fair way by OStrich. For independent sequential jobs, we show that OStrich guarantees the stretch of a campaign to be proportional to campaign’s size and the total number of users. The theoretical performance of our solution is assessed by simulating OStrich compared to the classical FCFS algorithm, issued from synthetic workload traces generated by two different user profiles. This is done to demonstrate how OStrich benefits both types of users, in contrast to FCFS.

#### Fundamentals of Continuous Games

We have made the following contributions:

1. Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the players' strategy shares. In [12] , we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g., the acceleration of the players' evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics. We show that strictly dominated strategies become extinct in $n$-th order payoff-monotonic dynamics $n$ orders as fast as in the corresponding first order dynamics; furthermore, in stark contrast to first order, weakly dominated strategies also become extinct for $n\ge 2$. All in all, higher order payoff-monotonic dynamics lead to the elimination of weakly dominated strategies, followed by the iterated deletion of strictly dominated strategies, thus providing a dynamic justification of the well-known epistemic rationalizability process of Dekel and Fudenberg. Finally, we also establish a higher order analogue of the folk theorem of evolutionary game theory, and we show that convergence to strict equilibria in $n$-th order dynamics is $n$ orders as fast as in first order.

2. In [37] we introduce a new class of game dynamics made of a pay-off replicator-like term modulated by an entropy barrier which keeps players away from the boundary of the strategy space. We show that these entropy-driven dynamics are equivalent to players computing a score as their on-going exponentially discounted cumulative payoff and then using a quantal choice model on the scores to pick an action. This dual perspective on entropy-driven dynamics helps us to extend the folk theorem on convergence to quantal response equilibria to this case, for potential games. It also provides the main ingredients to design a discrete time effective learning algorithm that is fully distributed and only requires partial information to converge to QRE. This convergence is resilient to stochastic perturbations and observation errors and does not require any synchronization between the players.

#### Application to Wireless Networks

We have made the following contributions:

1. Starting from an entropy-driven reinforcement learning scheme for multi-agent environments, we develop in [36] a distributed algorithm for robust spectrum management in Gaussian multiple-input, multiple-output (MIMO) uplink channels. In continuous time, our approach to optimizing the transmitters' signal distribution relies on the method of matrix exponential learning, adjusted by an entropy-driven barrier term which generates a distributed, convergent algorithm in discrete time. As opposed to traditional water-filling methods, the algorithm's convergence speed can be controlled by tuning the users' learning rate; accordingly, entropy-driven learning algorithms in MIMO systems converge arbitrarily close to the optimum signal covariance profile within a few iterations (even for large numbers of users and/or antennas per user), and this convergence remains robust even in the presence of imperfect (or delayed) measurements and asynchronous user updates.

2. Consider a wireless network of transmitter-receiver pairs where the transmitters adjust their powers to maintain a target SINR level in the presence of interference. In [46] , we analyze the optimal power vector that achieves this target in large, random networks obtained by "erasing" a finite fraction of nodes from a regular lattice of transmitter-receiver pairs. We show that this problem is equivalent to the so-called Anderson model of electron motion in dirty metals which has been used extensively in the analysis of diffusion in random environments. A standard approximation to this model is the so-called coherent potential approximation (CPA) method which we apply to evaluate the first and second order intra-sample statistics of the optimal power vector in one- and two-dimensional systems. This approach is equivalent to traditional techniques from random matrix theory and free probability, but while generally accurate (and in agreement with numerical simulations), it fails to fully describe the system: in particular, results obtained in this way fail to predict when power control becomes infeasible. In this regard, we find that the infinite system is always unstable beyond a certain value of the target SINR, but any finite system only has a small probability of becoming unstable. This instability probability is proportional to the tails of the eigenvalue distribution of the system which are calculated to exponential accuracy using methodologies developed within the Anderson model and its ties with random walks in random media. Finally, using these techniques, we also calculate the tails of the system's power distribution under power control and the rate of convergence of the Foschini-Miljanic power control algorithm in the presence of random erasures.