Cooperative Convex Optimization in Networked Systems: Augmented Lagrangian Algorithms with Directed Gossip Communication

We study distributed optimization in networked systems, where nodes cooperate to find the optimal quantity of common interest, x=x^\star. The objective function of the corresponding optimization problem is the sum of private (known only by a node,) convex, nodes' objectives and each node imposes a private convex constraint on the allowed values of x. We solve this problem for generic connected network topologies with asymmetric random link failures with a novel distributed, decentralized algorithm. We refer to this algorithm as AL-G (augmented Lagrangian gossiping,) and to its variants as AL-MG (augmented Lagrangian multi neighbor gossiping) and AL-BG (augmented Lagrangian broadcast gossiping.) The AL-G algorithm is based on the augmented Lagrangian dual function. Dual variables are updated by the standard method of multipliers, at a slow time scale. To update the primal variables, we propose a novel, Gauss-Seidel type, randomized algorithm, at a fast time scale. AL-G uses unidirectional gossip communication, only between immediate neighbors in the network and is resilient to random link failures. For networks with reliable communication (i.e., no failures,) the simplified, AL-BG (augmented Lagrangian broadcast gossiping) algorithm reduces communication, computation and data storage cost. We prove convergence for all proposed algorithms and demonstrate by simulations the effectiveness on two applications: l_1-regularized logistic regression for classification and cooperative spectrum sensing for cognitive radio networks.Comment: 28 pages, journal; revise

Slotted Aloha for Networked Base Stations

We study multiple base station, multi-access systems in which the user-base station adjacency is induced by geographical proximity. At each slot, each user transmits (is active) with a certain probability, independently of other users, and is heard by all base stations within the distance $r$. Both the users and base stations are placed uniformly at random over the (unit) area. We first consider a non-cooperative decoding where base stations work in isolation, but a user is decoded as soon as one of its nearby base stations reads a clean signal from it. We find the decoding probability and quantify the gains introduced by multiple base stations. Specifically, the peak throughput increases linearly with the number of base stations $m$ and is roughly $m/4$ larger than the throughput of a single-base station that uses standard slotted Aloha. Next, we propose a cooperative decoding, where the mutually close base stations inform each other whenever they decode a user inside their coverage overlap. At each base station, the messages received from the nearby stations help resolve collisions by the interference cancellation mechanism. Building from our exact formulas for the non-cooperative case, we provide a heuristic formula for the cooperative decoding probability that reflects well the actual performance. Finally, we demonstrate by simulation significant gains of cooperation with respect to the non-cooperative decoding.Comment: conference; submitted on Dec 15, 201