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    D2^2: Decentralized Training over Decentralized Data

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    While training a machine learning model using multiple workers, each of which collects data from their own data sources, it would be most useful when the data collected from different workers can be {\em unique} and {\em different}. Ironically, recent analysis of decentralized parallel stochastic gradient descent (D-PSGD) relies on the assumption that the data hosted on different workers are {\em not too different}. In this paper, we ask the question: {\em Can we design a decentralized parallel stochastic gradient descent algorithm that is less sensitive to the data variance across workers?} In this paper, we present D2^2, a novel decentralized parallel stochastic gradient descent algorithm designed for large data variance \xr{among workers} (imprecisely, "decentralized" data). The core of D2^2 is a variance blackuction extension of the standard D-PSGD algorithm, which improves the convergence rate from O(σnT+(nζ2)13T2/3)O\left({\sigma \over \sqrt{nT}} + {(n\zeta^2)^{\frac{1}{3}} \over T^{2/3}}\right) to O(σnT)O\left({\sigma \over \sqrt{nT}}\right) where ζ2\zeta^{2} denotes the variance among data on different workers. As a result, D2^2 is robust to data variance among workers. We empirically evaluated D2^2 on image classification tasks where each worker has access to only the data of a limited set of labels, and find that D2^2 significantly outperforms D-PSGD
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