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: This paper proposed for the first time the asynchronous variants of deterministic and stochastic regularization-by-denoising (RED) algorithms which have become popular recently in image recovery and reconstruction applications since they leverage the power of  pretrained deep denoising neural networks into the traditional model-based schemes, and often achieve state-of-the-art recovery results. These new variants are aimed to fully utilize the multi-cores structure of computational devices and to maximize the practicality of PnP/RED methods in large-scale inverse problems. The authors provide gradient-norm convergence analysis for the proposed algorithms under standard assumptions, along with detailed numerical studies demonstrating practical advantageous of proposed methods.