Double Averaging and Gradient Projection: Convergence Guarantees for Decentralized Constrained Optimization

We consider a generic decentralized constrained optimization problem over static, directed communication networks, where each agent has exclusive access to only one convex, differentiable, local objective term and one convex constraint set. For this setup, we propose a novel decentralized algorithm, called double averaging and gradient projection (DAGP). We achieve global optimality through a novel distributed tracking technique we call distributed null projection. Further, we show that DAGP can be used to solve unconstrained problems with nondifferentiable objective terms with a problem reduction scheme. Assuming only smoothness of the objective terms, we study the convergence of DAGP and establish sublinear rates of convergence in terms of feasibility, consensus, and optimality, with no extra assumption (e.g., strong convexity). For the analysis, we forego the difficulties of selecting Lyapunov functions by proposing a new methodology of convergence analysis, which we refer to as aggregate lower-bounding. To demonstrate the generality of this method, we also provide an alternative convergence proof for the standard gradient descent algorithm with smooth functions.