Stochastic Optimization from Distributed, Streaming Data in Rate-limited Networks

Motivated by machine learning applications in networks of sensors, internet-of-things (IoT) devices, and autonomous agents, we propose techniques for distributed stochastic convex learning from high-rate data streams. The setup involves a network of nodes—each one of which has a stream of data arriving at a constant rate—that solve a stochastic convex optimization problem by collaborating with each other over ratelimited communication links. To this end, we present and analyze two algorithms—termed distributed stochastic approximation mirror descent (D-SAMD) and accelerated distributed stochastic approximation mirror descent (AD-SAMD)—that are based on two stochastic variants of mirror descent. The main collaborative step in the proposed algorithms is approximate averaging of the local, noisy subgradients using distributed consensus. While distributed consensus is well suited for collaborative learning, its use in our setup results in perturbed subgradient averages due to rate-limited links, which may slow down or prevent convergence. Our main contributions in this regard are: (i) bounds on the convergence rates of D-SAMD and AD-SAMD in terms of the number of nodes, network topology, and ratio of the data streaming and communication rates, and (ii) sufficient conditions for order-optimum convergence of D-SAMD and ADSAMD. In particular, we show that there exist regimes under which AD-SAMD, when compared to D-SAMD, achieves orderoptimum convergence with slower communications rates. This is in contrast to the centralized setting in which use of accelerated mirror descent results in a modest improvement over regular mirror descent for stochastic composite optimization. Finally, we demonstrate the effectiveness of the proposed algorithms using numerical experiments.