Decentralized Sparse Linear Regression via Gradient-Tracking

We study sparse linear regression over a network of agents, modeled as an undirected graph without a center node. The estimation of the $s$-sparse parameter is formulated as a constrained LASSO problem wherein each agent owns a subset of the $N$ total observations. We analyze the convergence rate and statistical guarantees of a distributed projected gradient tracking-based algorithm under high-dimensional scaling, allowing the ambient dimension $d$ to grow with (and possibly exceed) the sample size $N$. Our theory shows that, under standard notions of restricted strong convexity and smoothness of the average loss functions, suitable conditions on the network connectivity and algorithm tuning, the distributed algorithm converges globally at a linear rate to an estimate that is within the centralized statistical precision of the model, $O(s\log d/N)$. When $s\log d/N=o(1)$, a condition necessary for statistical consistency, an $\varepsilon$-optimal solution is attained after ${O}(\kappa \log (1/\varepsilon))$ gradient computations and $O(\kappa/(1-\rho) \log (1/\varepsilon))$ communication rounds, where $\kappa$ is the restricted condition number of the loss function and $\rho$ measures the network connectivity. The computation cost matches that of the centralized projected gradient algorithm despite having data distributed; whereas the communication rounds reduce as the network connectivity improves. Overall, our study reveals interesting connections between statistical efficiency, network connectivity and topology, and convergence rate in the high dimensional setting.