  A new amortized variance-reduced gradient (AVRG) algorithm was developed in
\cite{ying2017convergence}, which has constant storage requirement in
comparison to SAGA and balanced gradient computations in comparison to SVRG.
One key advantage of the AVRG strategy is its amenability to decentralized
implementations. In this work, we show how AVRG can be extended to the network
case where multiple learning agents are assumed to be connected by a graph
topology. In this scenario, each agent observes data that is spatially
distributed and all agents are only allowed to communicate with direct
neighbors. Moreover, the amount of data observed by the individual agents may
differ drastically. For such situations, the balanced gradient computation
property of AVRG becomes a real advantage in reducing idle time caused by
unbalanced local data storage requirements, which is characteristic of other
reduced-variance gradient algorithms. The resulting diffusion-AVRG algorithm is
shown to have linear convergence to the exact solution, and is much more memory
efficient than other alternative algorithms. In addition, we propose a
mini-batch strategy to balance the communication and computation efficiency for
diffusion-AVRG. When a proper batch size is employed, it is observed in
simulations that diffusion-AVRG is more computationally efficient than exact
diffusion or EXTRA while maintaining almost the same communication efficiency.
