Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.Comment: 40 pages, 4 figure

A joint time-invariant filtering approach to the linear Gaussian relay problem

In this paper, the linear Gaussian relay problem is considered. Under the linear time-invariant (LTI) model the problem is formulated in the frequency domain based on the Toeplitz distribution theorem. Under the further assumption of realizable input spectra, the LTI Gaussian relay problem is converted to a joint design problem of source and relay filters under two power constraints, one at the source and the other at the relay, and a practical solution to this problem is proposed based on the projected subgradient method. Numerical results show that the proposed method yields a noticeable gain over the instantaneous amplify-and-forward (AF) scheme in inter-symbol interference (ISI) channels. Also, the optimality of the AF scheme within the class of one-tap relay filters is established in flat-fading channels.Comment: 30 pages, 10 figure

Jointly Optimal Channel and Power Assignment for Dual-Hop Multi-channel Multi-user Relaying

We consider the problem of jointly optimizing channel pairing, channel-user assignment, and power allocation, to maximize the weighted sum-rate, in a single-relay cooperative system with multiple channels and multiple users. Common relaying strategies are considered, and transmission power constraints are imposed on both individual transmitters and the aggregate over all transmitters. The joint optimization problem naturally leads to a mixed-integer program. Despite the general expectation that such problems are intractable, we construct an efficient algorithm to find an optimal solution, which incurs computational complexity that is polynomial in the number of channels and the number of users. We further demonstrate through numerical experiments that the jointly optimal solution can significantly improve system performance over its suboptimal alternatives.Comment: This is the full version of a paper to appear in the IEEE Journal on Selected Areas in Communications, Special Issue on Cooperative Networking - Challenges and Applications (Part II), October 201

Sparse Inverse Covariance Selection via Alternating Linearization Methods

Gaussian graphical models are of great interest in statistical learning. Because the conditional independencies between different nodes correspond to zero entries in the inverse covariance matrix of the Gaussian distribution, one can learn the structure of the graph by estimating a sparse inverse covariance matrix from sample data, by solving a convex maximum likelihood problem with an $\ell_1$-regularization term. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem's special structure; in particular, the subproblems solved in each iteration have closed-form solutions. Moreover, our algorithm obtains an $\epsilon$-optimal solution in $O(1/\epsilon)$ iterations. Numerical experiments on both synthetic and real data from gene association networks show that a practical version of this algorithm outperforms other competitive algorithms