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    A Stochastic Decoupling Method for Minimizing the Sum of Smooth and Non-Smooth Functions

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    Type
    Preprint
    Authors
    Mishchenko, Konstantin
    Richtarik, Peter cc
    KAUST Department
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2019-05-27
    Permanent link to this record
    http://hdl.handle.net/10754/660279
    
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    Abstract
    We consider the problem of minimizing the sum of three convex functions: i) a smooth function $f$ in the form of an expectation or a finite average, ii) a non-smooth function $g$ in the form of a finite average of proximable functions $g_j$, and iii) a proximable regularizer $R$. We design a variance reduced method which is able progressively learn the proximal operator of $g$ via the computation of the proximal operator of a single randomly selected function $g_j$ in each iteration only. Our method can provably and efficiently accommodate many strategies for the estimation of the gradient of $f$, including via standard and variance-reduced stochastic estimation, effectively decoupling the smooth part of the problem from the non-smooth part. We prove a number of iteration complexity results, including a general ${\cal O}(1/t)$ rate, ${\cal O}(1/t^2)$ rate in the case of strongly convex $f$, and several linear rates in special cases, including accelerated linear rate. For example, our method achieves a linear rate for the problem of minimizing a strongly convex function $f$ under linear constraints under no assumption on the constraints beyond consistency. When combined with SGD or SAGA estimators for the gradient of $f$, this leads to a very efficient method for empirical risk minimization with large linear constraints. Our method generalizes several existing algorithms, including forward-backward splitting, Douglas-Rachford splitting, proximal SGD, proximal SAGA, SDCA, randomized Kaczmarz and Point-SAGA. However, our method leads to many new specific methods in special cases; for instance, we obtain the first randomized variant of the Dykstra's method for projection onto the intersection of closed convex sets.
    Publisher
    arXiv
    arXiv
    1905.11535
    Additional Links
    https://arxiv.org/pdf/1905.11535
    Collections
    Preprints; Computer Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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