Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors

1 INTRODUCTION . Imaging inverse problems seek to recover an unknown image x 2 Rn from its noisy measurements y 2 Rm . Such problems arise in many fields , ranging from low-level computer vision to biomedical imaging . Since many imaging inverse problems are ill-posed , it is common to regularize the solution by using prior information on the unknown image . Widely-adopted image priors include total variation , low-rank penalties , and transform-domain sparsity ( Rudin et al. , 1992 ; Figueiredo & Nowak , 2001 ; 2003 ; Hu et al. , 2012 ; Elad & Aharon , 2006 ) . There has been considerable recent interest in plug-and-play priors ( PnP ) ( Venkatakrishnan et al. , 2013 ; Sreehari et al. , 2016 ) and regularization by denoising ( RED ) ( Romano et al. , 2017 ) , as frameworks for exploiting image denoisers as priors for image recovery . The popularity of deep learning has led to a wide adoption of deep denoisers within PnP/RED , leading to their state-of-the-art performance in a variety of applications , including image restoration ( Mataev et al. , 2019 ) , phase retrieval ( Metzler et al. , 2018 ) , and tomographic imaging ( Wu et al. , 2020 ) . Their empirical success has also prompted a follow-up theoretical work clarifying the existence of explicit regularizers ( Reehorst & Schniter , 2019 ) , providing new interpretations based on fixed-point projections ( Cohen et al. , 2020 ) , and analyzing their coordinate/online variants ( Sun et al. , 2019a ; Wu et al. , 2020 ) . Nonetheless , current PnP/RED algorithms are inherently serial . As illustrated in Fig . 1 , this makes them suboptimal on multicore systems that are often required for processing large-scale datasets ( Recht et al. , 2011 ) , such as those involving biomedical ( Ong et al. , 2020 ) and astronomical images ( Akiyama et al. , 2019 ) We address this gap by proposing a novel asynchronous RED ( ASYNC-RED ) algorithm . The algorithm decomposes the inference problem into a sequence of partial ( block-coordinate ) updates on x executed asynchronously in parallel over a multicore system . ASYNC-RED leads to a more efficient usage of available cores by avoiding synchronization of partial updates . ASYNC-RED is also scalable in terms of the number of measurements , since it processes only a small random subset of y at every iteration . We present two new theoretical results on the convergence of ASYNC-RED based on a unified set of explicit assumptions on the data-fidelity and the denoiser . Specifically , we establish its fixed-point convergence in the batch setting and extend this analysis to the randomized minibatch scenario . Our results extend recent work on serial block-coordinate RED ( BC-RED ) ( Sun et al. , Published as a conference paper at ICLR 2021 < latexit sha1_base64= '' +7+mD4Qola2MF8vLZ024UevUrMM= '' > 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< /latexit > 2019a ) and are fully consistent with the traditional asynchronous parallel optimization methods ( Lian et al. , 2015 ; Sun et al. , 2017 ) . We numerically validate ASYNC-RED on image recovery from linear and noisy measurements using pre-trained deep denoisers as image priors . 2 BACKGROUND . Inverse problems . Inverse problems are traditionally formulated as a composite optimization problem bx = arg min x2Rn g ( x ) + h ( x ) , ( 1 ) where g is the data-fidelity term that ensures consistency of x with the measured data y and h is the regularizer that infuses the prior knowledge on x . For example , consider the smooth ` 2-norm data-fidelity term g ( x ) = ky Axk22 , which assumes a linear observation model y = Ax + e , and the nonsmooth TV regularizer h ( x ) = ⌧kDxk1 , where ⌧ > 0 is the regularization parameter and D is the image gradient ( Rudin et al. , 1992 ) . Regularization by denoising ( RED ) . RED is a recent methodology for imaging inverse problems that seeks vectors x⇤ 2 Rn satisfying G ( x⇤ ) = rg ( x⇤ ) + ⌧ ( x⇤ D ( x⇤ ) ) = 0 , x⇤ 2 zer ( G ) : = { x 2 Rn : G ( x ) = 0 } ( 2 ) where rg denotes the gradient of the data-fidelity term and D : Rn ! Rn is an image denoiser parameterized by > 0 . Under additional technical assumptions , the solutions x⇤ 2 zer ( G ) can be associated with an explicit objective function of form ( 1 ) . Specifically , when D is locally homogeneous and has a symmetric Jacobian satisfying strong passivity ( Romano et al. , 2017 ; Reehorst & Schniter , 2019 ) , H ( x ) : = x D ( x ) corresponds to the gradient of a convex regularizer h ( x ) = 1 2 xT ( x D ( x ) ) . ( 3 ) A simple strategy , known as GM-RED , for computing x⇤ 2 zer ( G ) is based on the first-order fixed-point iteration xt = xt 1 G ( xt 1 ) , with G : Rn ! Rn , ( 4 ) where > 0 denotes the stepsize . In this paper , we extend this first-order RED algorithm to design ASYNC-RED . Since many denoisers do not satisfy the assumptions necessary for having an explicit objective ( Reehorst & Schniter , 2019 ) , our theoretical analysis considers a broader setting where D does not necessarily correspond to any explicit regularizer . The benefit of our analysis is that it accommodates powerful deep denoisers ( such as DnCNN ( Zhang et al. , 2017a ) ) that have been shown to achieve the state-of-the-art performance ( Sun et al. , 2019a ; Wu et al. , 2020 ; Cohen et al. , 2020 ) . Published as a conference paper at ICLR 2021 Plug-and-play priors ( PnP ) and other related work . There are other lines of works that combine the iterative methods with advanced denoisers . One closely-related framework is known as the deep mean-shift priors ( Bigdeli et al. , 2017 ) . It develops an implicit regularizer whose gradient is specified by a denoising autoencoder . Another well-known framework is PnP , which generalizes proximal methods by replacing the proximal map with an image denoiser ( Venkatakrishnan et al. , 2013 ) . Applications and theoretical analysis of PnP are widely studied in ( Sreehari et al. , 2016 ; Zhang et al. , 2017b ; Sun et al. , 2019 ; Zhang et al. , 2019 ; Ahmad et al. , 2020 ; Wei et al. , 2020 ) and ( Chan et al. , 2017 ; Meinhardt et al. , 2017 ; Buzzard et al. , 2018 ; Sun et al. , 2019b ; Tirer & Giryes , 2019 ; Teodoro et al. , 2019 ; Ryu et al. , 2019 ; Xu et al. , 2020 ) , respectively . In particular , Buzzard et al . ( 2018 ) proposed a parallel extension of PnP called Consensus Equilibrium ( CE ) , which enables synchronous parallel updates of x . Note that while we developed ASYNC-RED as a variant of RED , our framework and analysis can be also potentially applied to PnP/CE . The plug-in strategy can be also applied to another family of algorithms known as approximate message passing ( AMP ) ( Metzler et al. , 2016a ; b ; Fletcher et al. , 2018 ) . The AMP-based algorithms are known to be nearly-optimal for random measurement matrices , but are generally unstable for general A ( Rangan et al. , 2014 ; 2015 ) . Asynchronous parallel optimization . There are two main lines of work in asynchronous parallel optimization , the one involving the asynchrony in coordinate updates ( Iutzeler et al. , 2013 ; Liu et al. , 2015 ; Peng et al. , 2016 ; Bianchi et al. , 2015 ; Sun et al. , 2017 ; Hannah & Yin , 2018 ; Hannah et al. , 2019 ) , and the other focusing on the study of various asynchronous stochastic gradient methods ( Recht et al. , 2011 ; Lian et al. , 2015 ; Liu et al. , 2018 ; Zhou et al. , 2018 ; Lian et al. , 2018 ) . Our work contributes to the area by developing a novel deep-regularized asynchronous parallel method with provable convergence guarantees .

Summary: The paper describes a novel implementation of RED, regularization by denoising, which better leverages multicore architectures to achieve a significant speedup. The proposed implementation splits the gradient step into smaller components, which can each be executed independently on different cores and then used to update a shared copy. The crucial result is two sets of convergence guarantees showing that this delayed update will not cause too much error, even if the updates from different cores arrive at different times. The speedups achieved range from 6× to 8× on two tasks (compressive sensing and computed tomography reconstruction).