  Distributed adaptive stochastic gradient methods have been widely used for
large-scale nonconvex optimization, such as training deep learning models.
However, their communication complexity on finding $\varepsilon$-stationary
points has rarely been analyzed in the nonconvex setting. In this work, we
present a novel communication-efficient distributed Adam in the
parameter-server model for stochastic nonconvex optimization, dubbed {\em
Efficient-Adam}. Specifically, we incorporate a two-way quantization scheme
into Efficient-Adam to reduce the communication cost between the workers and
server. Simultaneously, we adopt a two-way error feedback strategy to reduce
the biases caused by the two-way quantization on both the server and workers,
respectively. In addition, we establish the iteration complexity for the
proposed Efficient-Adam with a class of quantization operators, and further
characterize its communication complexity between the server and workers when
an $\varepsilon$-stationary point is achieved. Finally, we apply Efficient-Adam
to solve a toy stochastic convex optimization problem and train deep learning
models on real-world vision and language tasks. Extensive experiments together
with a theoretical guarantee justify the merits of Efficient Adam.
