  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 rate-limited 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 and in which nodes collaborate via
approximate averaging of the local, noisy subgradients using distributed
consensus. Our main contributions 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 these algorithms. In particular, we show that
for sufficiently well-connected networks, distributed learning schemes can
obtain order-optimum convergence even if the communications rate is small.
Further we find that the use of accelerated methods significantly enlarges the
regime in which order-optimum convergence is achieved; this is in contrast to
the centralized setting, where accelerated methods usually offer only a modest
improvement. Finally, we demonstrate the effectiveness of the proposed
algorithms using numerical experiments.
