Mahmoud El Chamie

A local average consensus algorithm for wireless sensor networks

In many application scenarios sensors need to calculate the average of some local values, e.g. of local measurements. A possible solution is to rely on consensus algorithms. In this case each sensor maintains a local estimate of the global average, and keeps improving it by performing a weighted sum of the estimates of all its neighbors. The number of iterations needed to reach an accurate estimate depends on the weights used at each sensor. Speeding up the convergence rate is important also to reduce the number of messages exchanged among neighbors and then the energetic cost of these algorithms. While it is possible in principle to calculate the optimal weights, the known algorithm requires a single sensor to discover the topology of the whole network and perform the calculations. This may be unfeasible for large and dynamic sensor networks, because of sensor computational constraints and of the communication overhead due to the need to acquire the new topology after each change. In this paper we propose a new average consensus algorithm, where each sensor selects its own weights on the basis of some local information about its neighborhood. Our algorithm is tailored for networks having cluster structure, like it is common for wireless sensor networks. In realistic sensor network topologies, the algorithm shows faster convergence than other existing consensus protocols.

Design and analysis of distributed averaging with quantized communication

Consider a network whose nodes have some initial values, and it is desired to design an algorithm that builds on neighbor to neighbor interactions with the ultimate goal of convergence to the average of all initial node values or to some value close to that average. Such an algorithm is called generically “distributed averaging”, and our goal in this paper is to study the performance of a subclass of distributed averaging algorithms where the information exchange between neighboring nodes (agents) is subject to deterministic uniform quantization. With such quantization, the precise average cannot be achieved (except in exceptional cases), but some value close to it, called quantized consensus. It is shown in this paper that in finite time, the algorithm will either cause all agents to reach a quantized consensus where the consensus value is the largest integer not greater than the average of their initial values, or will lead all variables to cycle in a small neighborhood around the average, depending on initial conditions. In the latter case, tight bounds for the size of the neighborhood are given, and it is further shown that the error can be made arbitrarily small by adjusting the algorithms parameters in a distributed manner.

Distributed Weight Selection in Consensus Protocols by Schatten Norm Minimization

In this work we study the weight optimization problem for average consensus protocols by reformulating it as a Schatten norm minimization with parameter p. We show that as p approaches infinity, the optimal solution of the Schatten norm induced problem recovers the optimal solution of the original problem. Moreover, by tuning the parameter p in our proposed minimization, we can simply trade-off the quality of the solution (i.e., the speed of convergence) for communication/computation requirements (in terms of number of messages exchanged and volume of data processed). We then propose a distributed algorithm to solve the Schatten norm minimization and we show that it outperforms the other distributed weight selection methods.

On fair network cache allocation to content providers

In-network caching is an important solution for content offloading from content service providers. However despite a rather high maturation in the definition of caching techniques, minor attention has been given to the strategic interaction among the multiple content providers. Situations involving multiple content providers (CPs) and one Internet Service Provider (ISP) having to give them access to its caches are prone to high cache contention, in particular at the appealing topology cross-points. While available cache contention situations from the literature were solved by considering each storage as one autonomous and self managed cache, we propose in this paper to address this contention situation by segmenting the storage on a per-content provider basis (e.g., each CP receives a portion of the storage space depending on its storage demand). We propose a resource allocation and pricing framework to support the network cache provider in the cache allocation to multiple CPs, for situations where CPs have heterogeneous sets of files and untruthful demands need to be avoided. As cache imputations to CPs need to be fair and robust against overclaiming, we evaluate common proportional and max-min fairness (PF, MMF) allocation rules, as well as two coalitional game rules, the Nucleolus and the Shapley value. When comparing our cache allocation algorithm for the different allocation rules with the naive least-recently-used-based cache allocation approach, we find that the latter provides proportional fairness. Moreover, the game-theoretic rules outperform in terms of content access latency the naive cache allocation approach as well as PF and MMF approaches, while sitting in between PF and MMF in terms of fairness. Furthermore, we show that our pricing scheme encourages the CPs to declare their truthful demands by maximizing their utilities for real declarations.

Convex synthesis of randomized policies for controlled Markov chains with density safety upper bound constraints

The main objective of this paper is to synthesize optimal decision-making policies for a finite-horizon Markov Decision Process (MDP) while satisfying a safety constraint that imposes an upper bound on the state probability density function (pdf) of the underlying Markov Chain (MC) for all time steps. The classical approach based on state-action frequencies for constrained MDPs yields decision policies that provide safety constraint satisfaction for stationary distributions (i.e., asymptotically), but not necessarily providing safety during the transient regime. This paper introduces a new synthesis method for randomized Markovian policies for finite-horizon MDPs, where the safety constraint satisfaction is guaranteed for both the transient and the stationary distributions independent from initial state (i.e., providing safe policies for the worst-case analysis). An efficient Linear Programming (LP) based synthesis algorithm is proposed, which produces a convex set of feasible policies and ensures that the expected total reward is above a computable lower-bound. A simulation example of a swarm of autonomous agents is also presented to demonstrate the practical importance of having safe policies.

Optimal strategies for dynamic weight selection in consensus protocols in the presence of an adversary

In this paper, we consider optimal design strategies in consensus protocols for networks vulnerable to adversarial attacks. First we study dynamic (multi-stage) weight selection optimal control for consensus protocols. For the general (multi-stage) case, the solution exists but can rarely be expressed in closed-form. In view of this, we apply optimization techniques to obtain a locally (and possibly globally) optimizing feasible control path. For the one-stage case, however, we obtain a closed-form solution for the optimal control and provide sufficient conditions for the existence of a control that makes the system reach consensus in only one iteration. We then consider a game theoretical model for the problem of a network with an adversary corrupting the control signal with noise. We derive the optimal strategies for both players (the adversary and the network designer) of the resulting game using a saddle point equilibrium (SPE) solution in mixed strategies.

Newton's method for constrained norm minimization and its application to weighted graph problems

Due to increasing computer processing power, Newtons method is receiving again increasing interest for solving optimization problems. In this paper, we provide a methodology for solving smooth norm optimization problems under some linear constraints using the Newtons method. This problem arises in many machine learning and graph optimization applications. We consider as a case study optimal weight selection for average consensus protocols for which we show how Newtons method significantly outperforms gradient methods both in terms of convergence speed and in term of robustness to the step size selection.

The discrete-time Altafini model of opinion dynamics with communication delays and quantization

The discrete-time Altafini model is an opinion dynamics model in which the interactions among a group of agents are described by a time-varying signed digraph. This paper first uses graph theoretic constructions to study modified versions of the Altafini model in which there are communication delays or quantized communication. The condition under which consensus in absolute value or bipartite consensus is achieved proves to be the same as the condition in the delay-free case. The paper also analyzes the performance of the model where the information exchanged between neighboring agents is subject to a certain type of deterministic uniform quantization. We show that in finite time and depending on initial conditions, the model on any static, connected, undirected signed graph will either cause all agents to reach a quantized consensus in absolute value, or will lead all variables to oscillate in a small neighborhood around the absolute value.

Convex synthesis of optimal policies for Markov Decision Processes with sequentially-observed transitions

This paper extends finite state and action space Markov Decision Process (MDP) models by introducing a new type of measurement for the outcomes of actions. The new measurement allows to sequentially observe the next-state transition for taking an action, i.e., the actions are ordered and the next action outcome in the sequence is observed only if the current action is not chosen. The sequentially-observed MDP (SO-MDP) shares some properties with a standard MDP: among history dependent policies, Markovian ones are still optimal. SO-MDP policies have the advantage of producing better rewards than standard optimal MDP policies due to additional measurements. Computing these policies, on the other hand, is more complex and we present a linear programming based synthesis of the optimal decision policies for the finite horizon SO-MDPs. A simulation example of multiple autonomous agents is also provided to demonstrate the SO-MDP model and the proposed policy synthesis method.

Safe Metropolis–Hastings algorithm and its application to swarm control ☆

Abstract This paper presents a new method to synthesize safe reversible Markov chains via extending the classical Metropolis–Hastings (M–H) algorithm. The classical M–H algorithm does not impose safety upper bound constraints on the probability vector, discrete probability density function, that evolves with the resulting Markov chain. This paper presents a new M–H algorithm for Markov chain synthesis that ensures such safety constraints together with reversibility and convergence to a desired stationary (steady-state) distribution. Specifically, we provide a convex synthesis method that incorporates the safety constraints via designing the proposal matrix for the M–H algorithm. It is shown that the M–H algorithm with this proposal matrix, safe M–H algorithm, ensures safety for a well-characterized convex set of stationary probability distributions, i.e., it is robustly safe with respect to this set of stationary distributions. The size of the safe set is then incorporated in the design problem to further enhance the robustness of the synthesized M–H proposal matrix. Numerical simulations are provided to demonstrate that multi-agent systems, swarms, can utilize the safe M–H algorithm to control the swarm density distribution. The controlled swarm density tracks time-varying desired distributions, while satisfying the safety constraints. Numerical simulations suggest that there is insignificant trade-off between the speed of convergence and the robustness.