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    Unified Analysis of Stochastic Gradient Methods for Composite Convex and Smooth Optimization

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    Type
    Preprint
    Authors
    Khaled, Ahmed
    Sebbouh, Othmane
    Loizou, Nicolas
    Gower, Robert M.
    Richtarik, Peter cc
    KAUST Department
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2020-06-20
    Permanent link to this record
    http://hdl.handle.net/10754/663909
    
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    Abstract
    We present a unified theorem for the convergence analysis of stochastic gradient algorithms for minimizing a smooth and convex loss plus a convex regularizer. We do this by extending the unified analysis of Gorbunov, Hanzely \& Richt\'arik (2020) and dropping the requirement that the loss function be strongly convex. Instead, we only rely on convexity of the loss function. Our unified analysis applies to a host of existing algorithms such as proximal SGD, variance reduced methods, quantization and some coordinate descent type methods. For the variance reduced methods, we recover the best known convergence rates as special cases. For proximal SGD, the quantization and coordinate type methods, we uncover new state-of-the-art convergence rates. Our analysis also includes any form of sampling and minibatching. As such, we are able to determine the minibatch size that optimizes the total complexity of variance reduced methods. We showcase this by obtaining a simple formula for the optimal minibatch size of two variance reduced methods (\textit{L-SVRG} and \textit{SAGA}). This optimal minibatch size not only improves the theoretical total complexity of the methods but also improves their convergence in practice, as we show in several experiments.
    Publisher
    arXiv
    arXiv
    2006.11573
    Additional Links
    https://arxiv.org/pdf/2006.11573
    Collections
    Preprints; Computer Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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