The analysis of a set of $n$ stochastic entities interacting with each others can be particularly difficult. The mean field approximation is a very effective technique to characterize the transient probability distribution or steady-state regime of such systems when the number of entities $n$ grows very large. The idea of mean-field approximation is to replace a complex stochastic system by a simpler deterministic dynamical system. This dynamical system is constructed by assuming that the objects are asymptotically independent. Each object is viewed as interacting with an average of the other objects (the mean-field). When each object has a finite or countable state-space, this dynamical system is usually a non-linear ordinary differential equation (ODE). An introduction to these techniques is provided in the book chapter .

Simgrid is a toolkit providing core functionalities for the simulation of distributed applications in heterogeneous distributed environments. Although it was initially designed to study large distributed computing environments such as grids, we have recently applied it to performance prediction of HPC configurations.

The increased penetration of renewable energy sources in existing power systems has led to necessary developments in electricity market mechanisms. Most importantly, renewable energy generation is increasingly made accountable for deviations between scheduled and actual energy generation. However, there is no mechanism to enforce accountability for the additional costs induced by power fluctuations. These costs are socialized and eventually supported by electricity customers. In , we propose some metrics for assessing the contribution of all market participants to power regulation needs, as well as an attribution mechanism for fairly redistributing related power regulation costs. We discuss the effect of various metrics used by the attribution mechanisms, and we illustrate, in a game-theoretical framework, their consequences on the strategic behavior of market participants. We also illustrate, by using the case of Western Denmark, how these mechanisms may affect revenues and the various market participants.

Ever since the early development stages of wireless networks, the importance of radiated power has made power control an essential component of network design. In , we analyzed the problem of power control in large, random wireless networks that are obtained by “erasing” a finite fraction of nodes from a regular $d$-dimensional lattice of $N$ transmit-receive pairs. Drawing on tools and ideas from statistical physics, we showed that this problem can be mapped to the Anderson impurity model for diffusion in random media; in this way, by employing the so-called coherent potential approximation (CPA) method, we calculated the average power in the system (and its variance) for 1-D and 2-D networks. In this regard, even though infinitely large systems are always unstable beyond a critical value of the users' SINR target, finite systems remain stable with high probability even beyond this critical SINR threshold. We calculated this probability by analyzing the density of low lying eigenvalues of an associated random Schrödinger operator, and we showed that the network can exceed this critical SINR threshold by a factor of at least $O\left({(logN)}^{-2/d}\right)$ before undergoing a phase transition to the unstable regime.

The recent increase in the use of wireless networks for video transmission has led to the increase in the use of rate-adaptive protocols to maximize the resource utilization and increase the efficiency in the transmission. However, a number of these protocols lead to interactions among the users that are subjective in nature and affect the overall performance. In , we analyzed the interplay between the wireless network and video transmission dynamics in the light of subjective perceptions of the end users in their interactions – specifically, the trade-off between maximizing the quality of service (QoS) or quality of experience (QoE) and minimizing the transmission cost. By using methods from game theory, we derived an optimized transmission scheme that allows the efficient use of traditional protocols by taking into account the subjective interactions that occur in practical scenarios.

In cognitive radio networks, secondary (unlicensed) users (SUs) can access the spectrum opportunistically, whenever they sense an opening by the network's primary (licensed) users (PUs). In , we analyzed the minimization of overall power consumption over several orthogonal frequency bands under constraints on the minimum quality of service (QoS) and maximum peak and average interference to the network's PUs. To that end, we proposed a projected sub-gradient algorithm which quickly converges to an optimal configuration if the users' channels are fast fading.

Despite such benefits, the conventional cognitive radio network (CCRN) paradgim is not particularly attractive for opportunistic spectrum access because the network's PUs can recapture SU channels at will, thus interrupting the transmission of the latter. To address this crucial limitation, we proposed in a semi-cognitive radio network (SCRN) paradigm where PUs are constrained to first use any free channels before being allowed to capture channels that are in use by SUs. These constraints slightly degrade the performance of the network's PUs, but a) they offer remarkable performance improvements to the network's SUs; and b) they can be compensated by imposing a monetary (or other) penalty to the network's secondary owners. In , we provided a game-theoretic analysis of the performance trade-offs involved for both the PUs and the SUs, and we derived both centralized and distributed learning algorithms that allow the system control process to converge to a stable state.

The vast majority of works on wireless resource allocation (spectrum, power, etc.) has focused on two limit cases: In the static regime, the attributes of the network are assumed effectively static and the system's optimality analysis relies on techniques from (static) optimization. On the other hand, in the so-called stochastic regime, the network is assumed to evolve randomly following some fixed probability law, and the allocation of wireless resources is optimized using tools from stochastic optimization and control. In practical wireless networks however, both assumptions fail because of factors that introduce an unpredictable variability to the system (such as user mobility, users going arbitrarily on- and off-line, etc.).

The works , , treat this problem by providing no-regret learning algorithms for single-user rate maximization and power control in multi-carrier cognitive radio and Internet of Things networks. The extension of these works to multi-antenna systems was carried out in , where we derived a matrix exponential learning algorithm for dynamic power allocation and control in time-varying MIMO systems. Building on this, we also showed in that regret minimization techniques can also be applied to the much more challenging problem of energy efficiency maximization in dynamic networks – i.e. the maximization of successfully received bits per Watt of transmitted power in environments that fluctuate unpredictably over time. Finally, as was shown in , , , these unilateral performance gains also extend to large networks comprising hundreds (or even thousands) of users: there, the proposed matrix exponential learning algorithm converges to a stable state within a few iterations, even for very large of antennas and subcarriers.

Routing plays a crucial part in the efficient operation of packet-switched data networks, especially with regard to latency reduction and energy efficiency. However, in addition to being distributed (so as to cope with the prolific size of today's networks), optimized routing schemes must also be able to adapt to changes in the underlying network (e.g. due to variations in traffic demands, link quality, etc.).

First, to address the issue of latency reduction, we provided in an adaptive multi-flow routing algorithm to select end-to-end paths in packet-switched networks. The algorithm is based only on local information, so it is suitable for distributed implementation; furthermore, it provides guarantees that the network configuration converges to a stable state and exhibits several robustness properties that make it suitable for use in dynamic real-life networks (such as robustness to measurement errors, outdated information and update desynchronization).

Concerning energy efficiency, examines the problem of routing in optical networks with the aim of minimizing traffic-driven power consumption. To tackle this, proposed a pricing scheme which, combined with a distributed learning method based on the Boltzmann distribution of statistical mechanics, exhibits remarkable operation properties even under uncertainty. Specifically, the long-term average of the network's power consumption converges quickly to its minimum value (in practice, within a few iterations of the algorithm), and this convergence remains robust in the face of uncertainty of arbitrarily high magnitude.

One of the most widely used algorithms for learning in finite games is the so-called best response algorithm (BRA); nonetheless, even though sevaral worst-case bounds are known for its convergence time, the algorithm's performance in typical game-theoretic scenarios seems to be far better than these worst-case bounds suggest. In , , , , we computed the average execution time of the BR algorithm using Markov chain coupling techniques that recast the average execution time of this discrete algorithm as the solution of an ordinary differential equation. In so doing, we showed that the worst-case complexity of the BR algorithm in a potential game with $N$ players and $A$ actions per player is $AN(N-1)$, while its average complexity over random potential games is $O\left(N\right)$, independently of $A$.

In , we also studied the convergence rate of the HEDGE algorithm (which, contrary to the BR algorithm, leads to no regret even in adversarial settings). Motivated by applications to data networks where fast convergence is essential, we analyzed the problem of learning in generic $N$-person games that admit Nash equilibria in pure strategies. Despite the (unbounded) uncertainty in the players’ observations, we show that hedging eliminates dominated strategies (a.s.) and, with high probability, it converges locally to pure Nash equilibria at the exponential rate $O(exp(-c{\sum }_{j=1}^t{\gamma }_j))$, where ${\gamma }_j$ is the algorithm s step size.

These results are strongly related to the long-term rationality properties (elimination of dominated strategies, convergence to pure Nash equilibria and evolutionarily stable states, etc.) of an underlying class of game dynamics based on regularization and Riemannian geometry. Specifically, in , we introduced a class of evolutionary game dynamics whose defining element is a state-dependent geometric structure on the set of population states. When this geometric structure satisfies a certain integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics and are equivalent to a class of reinforcement learning dynamics studied in . Finally, as we showed in , these properties also hold even in the presence of noise, i.e. when the players only have noisy observations of their payoff vectors.

A key limitation of existing game-theoretic learning algorithms is that they invariably revolve around games with a finite number of actions per players. However, this assumption is often unrealistic (especially in network-based applications of game theory), a factor which severely limits the applicability of learning techniques in real-life problems.

To address this issue, we studied in a class of control problems that can be formulated as potential games with continuous action sets, and we proposed an actor-critic reinforcement learning algorithm that provably converges to equilibrium in said class. 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 proved 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.

Finally, to address cases where mixing over a continuum of actions is unrealistic, we examined in the convergence properties of a class of learning schemes for $N$-person games with continuous action spaces based on a continuous optimization technique known as “dual averaging”. To study this multi-agent, pure-strategy learning process, we introduced the notion of variational stability (VS), and we showed that stable equilibria are locally attracting with high probability whereas globally stable states are globally attracting with probability 1. Finally, we examined the scheme's convergence speed and we showed that if the game admits a strict equilibrium and the players' mirror maps are surjective, then, with high probability, the process converges to equilibrium in a finite number of steps, no matter the level of uncertainty in the players' observations (or payoffs).

A key feature of modern data networks is their distributed nature and the stochasticity surrounding users and their possible actions. To account for these issues in a general optimization context, we proposed in a distributed, asynchronous algorithm for stochastic semidefinite programming which is a stochastic approximation of the continous-time matrix exponential scheme derived in . This algorithm converges almost surely to an $ϵ$-approximation of an 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.

More generally, in view of solving convex optimization problems with noisy gradient input, we also analyzed in the asymptotic behavior of gradient-like flows that are subject to stochastic disturbances. For concreteness, we focused on the widely studied class of mirror descent methods for constrained convex programming and we examined the dynamics' convergence and concentration properties in the presence of noise. In the small noise limit, we showed that the dynamics converge to the solution set of the underlying problem with probability 1. Otherwise, in the case of persistent noise, we estimated the measure of the dynamics' long-run concentration around interior solutions and their convergence to boundary solutions that are sufficiently “robust”. Finally, we showed that a rectified variant of the method with a decreasing sensitivity parameter converges irrespective of the magnitude of the noise or the structure of the underlying convex program, and we derived an explicit estimate for its rate of convergence.

In modern High Performance Computing architectures, the memory subsystem is a common performance bottleneck. When optimizing an application, the developer has to study its memory access patterns and adapt accordingly the algorithms and data structures it uses. The objective is twofold: on one hand, it is necessary to avoid missuses of the memory hierarchy such as false sharing of cache lines or contention in a NUMA interconnect. On the other hand, it is essential to take advantage of the various cache levels and the memory hardware prefetcher. Still, most profiling tools focus on CPU metrics. The few of them able to provide an overview of the memory patterns involved by the execution rely on hardware instrumentation mechanisms and have two drawbacks. The first one is that they are based on sampling which precision is limited by hardware capabilities. The second one is that they trace a subset of all the memory accesses, usually the most frequent, without information ab out the other ones. In we present Moca, an efficient tool for the collection of complete spatio-temporal memory traces. Moca is based on a Linux kernel module and provides a coarse grained trace of a superset of all the memory accesses performed by an application over its addressing space during the time of its execution. The overhead of Moca is reasonable when taking into account the fact that it is able to collect complete traces which are also more precise than the ones collected by comparable tools.

Benchmarking has proven to be crucial for the investigation of the behavior and performances of a system. However, the choice of relevant benchmarks still remains a challenge. To help the process of comparing and choosing among benchmarks, in we propose a solution for automatic benchmark profiling. It computes unified benchmark profiles reflecting benchmarks' duration, function repartition, stability, CPU efficiency, parallelization and memory usage. It identifies the needed system information for profile computation, collects it from execution traces and produces profiles through efficient and reproducible trace analysis treatments. The paper presents the design, implementation and the evaluation of the approach.