Parallel Restarted SPIDER -- Communication Efficient Distributed Nonconvex Optimization with Optimal Computation Complexity

In this paper, we propose a distributed algorithm for stochastic smooth, non-convex optimization. We assume a worker-server architecture where N nodes, each having n (potentially infinite) number of samples, collaborate with the help of a central server to perform the optimization task. The global objective is to minimize the average of local cost functions available at individual nodes. The proposed approach is a non-trivial extension of the popular parallel-restarted SGD algorithm, incorporating the optimal variance-reduction based SPIDER gradient estimator into it. We prove convergence of our algorithm to a first-order stationary solution. The proposed approach achieves the best known communication complexity O(ε-1) along with the optimal computation complexity. For finite-sum problems (finite n), we achieve the optimal computation (IFO) complexity O(√Nnε-1). For online problems (n unknown or infinite), we achieve the optimal IFO complexity O(ε-3/2). In both the cases, we maintain the linear speedup achieved by existing methods. This is a massive improvement over the O(ε-2) IFO complexity of the existing approaches. Additionally, our algorithm is general enough to allow non-identical distributions of data across workers, as in the recently proposed federated learning paradigm.