Primal-dual Hybrid Gradient Algorithm Research Articles

On dynamical system modeling of learned primal-dual with a linear operator K : stability and convergence properties

Learned Primal-Dual (LPD) is a deep learning based method for composite optimization problems that is based on unrolling/unfolding the primal-dual hybrid gradient algorithm. While achieving great successes in applications, the mathematical interpretation of LPD as a truncated iterative scheme is not necessarily sufficient to fully understand its properties. In this paper, we study the LPD with a general linear operator. We model the forward propagation of LPD as a system of difference equations and a system of differential equations in discrete- and continuous-time settings (for primal and dual variables/trajectories), which are named discrete-time LPD and continuous-time LPD, respectively. Forward analyses such as stabilities and the convergence of the state variables of the discrete-time LPD to the solution of continuous-time LPD are given. Moreover, we analyze the learning problems with/without regularization terms of both discrete-time and continuous-time LPD from the optimal control viewpoint. We prove convergence results of their optimal solutions with respect to the network state initialization and training data, showing in some sense the topological stability of the learning problems. We also establish convergence from the solution of the discrete-time LPD learning problem to that of the continuous-time LPD learning problem through a piecewise linear extension, under some appropriate assumptions on the space of learnable parameters. This study demonstrates theoretically the robustness of the LPD structure and the associated training process, and can induce some future research and applications.

Robust noise-power estimation via primal–dual algorithms with measurements from a subset of network nodes

Noise power or signal variance is an important performance parameter or indicator used for many applications in signal processing and wireless communications. This paper investigates power estimation of random noises on sensor networks by using measurements taken over a subset of nodes. In many circumstances, the power values are typically smooth on the network, that is, the powers measured by neighboring nodes tend to have similar values. Using this smoothness, the power-estimation problem is formulated as a regularized optimization problem involving minimization of a sum of non-negative log-likelihood and the total variation. To find its optimal solution, a primal–dual hybrid gradient (PDHG) algorithm is developed, which uses primal–dual updates to achieve good estimation results. To further improve the estimation performance, two variants of the PDHG algorithm are also proposed: one has faster convergence by using a variable step-size searching strategy, and the other is robust to outliers and can identify abnormal nodes by using an outlier-rejection scheme. Several numerical experiments confirm the efficiency of our proposed algorithms.

Time Efficient Offloading Optimization in Automotive Multi-Access Edge Computing Networks Using Mean-Field Games

Emerging connected vehicular services, such as intelligent driving and high-definition (HD) map, are gaining increasing interest with the fast development of multi-access edge computing (MEC). For most time-sensitive and computation-intensive vehicular services, the data offloading process significantly influences the capacity and performance of MEC, especially when the number of connected vehicles is enormous. In this work, we consider data offloading optimization for a large-scale automotive MEC network. The problem is challenging due to the large number of connected vehicles and the complicated interaction between vehicles and edge servers. To tackle the scalability problem, we reformulate the original offloading optimization problem into a Mean-Field-Game (MFG) problem by abstracting the interaction among the connected vehicles as a distribution over their state spaces of task sizes, known as the mean-field term. To solve the problem efficiently, we propose a G-prox Primal-Dual-Hybrid-Gradient (PDHG) algorithm that transforms the MFG problem into a saddle-point problem. Based on our developed MFG model and G-prox PDHG algorithm, we propose the first data offloading scheme whose computation time is independent of the number of connected vehicles in automotive MEC systems. Extensive evaluation results corroborate the superior performance of our proposed scheme compared with the state-of-the-art methods.

Understanding the Convergence of the Preconditioned PDHG Method: A View of Indefinite Proximal ADMM

The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework covering PDHG, alternating direction method of multipliers (ADMM), and other methods. We give the optimal convergence condition of PrePDHG in the sense that the key parameters in the condition can not be further improved, which fills the theoretical gap in the-state-of-art convergence results of PrePDHG, and obtain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The theoretical analysis is achieved by establishing the equivalence between PrePDHG and indefinite proximal ADMM. Besides, we discuss various choices of the proximal matrices in PrePDHG and derive some interesting results. For example, the convergence condition of diagonal PrePDHG is improved to be tight, the dual stepsize of the balanced augmented Lagrangian method can be enlarged to 4/3 from 1, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel iterations is also explored. Numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction verify the effectiveness and superiority of PrePDHG.

Distributed Data Offloading in Ultra-Dense LEO Satellite Networks: A Stackelberg Mean-Field Game Approach

The low earth orbit (LEO) satellite network is regarded as a promising technology for delivering seamless service to remote areas such as rural areas. In this paper, we consider a dynamic data offloading problem in ultra-dense LEO satellites networks with large-scale ground users, where each ground user makes the distributed offloading decision based on its state information, the influence from other ground users, and the fees paid to satellites. To investigate the interaction problem between ground users and satellites, we formulate the problem as a Stackelberg game. Specifically, the satellites serve as the leaders, who decide the data service price at each time slot. The ground users, on the contrary, are the followers who decide the power control based on their states, the influence from other ground users and the fees paid to satellites. Since the influence is difficult to estimate due to the large number of ground users, we employ the mean field game algorithm to transform the influence from others and satellites into the mean field term, and reformulate the optimization problem as a Stackelberg mean field game (SMFG). Each ground user makes the data offloading decision by learning the future impact of the whole network distributively. For ground users, we solve the power control optimization problem by utilizing the G-prox primal-dual hybrid gradient (PDHG) algorithm, where the Fokker-Planck-Kolmogorov (FPK) equation is converted into a linear form via the Taylor expansion. For satellites, we address the service pricing optimization problem by using the adjoint algorithm. Finally, the numerical results demonstrate the effectiveness of the proposed algorithm.

Fast and memory-efficient reconstruction of sparse Poisson data in listmode with non-smooth priors with application to time-of-flight PET

Objective. Complete time of flight (TOF) sinograms of state-of-the-art TOF PET scanners have a large memory footprint. Currently, they contain ∼4 · 109 data bins which amount to ∼17 GB in 32 bit floating point precision. Moreover, their size will continue to increase with advances in the achievable detector TOF resolution and increases in the axial field of view. Using iterative algorithms to reconstruct such enormous TOF sinograms becomes increasingly challenging due to the memory requirements and the computation time needed to evaluate the forward model for every data bin. This is especially true for more advanced optimization algorithms such as the stochastic primal-dual hybrid gradient (SPDHG) algorithm which allows for the use of non-smooth priors for regularization using subsets with guaranteed convergence. SPDHG requires the storage of additional sinograms in memory, which severely limits its application to data sets from state-of-the-art TOF PET systems using conventional computing hardware. Approach. Motivated by the generally sparse nature of the TOF sinograms, we propose and analyze a new listmode (LM) extension of the SPDHG algorithm for image reconstruction of sparse data following a Poisson distribution. The new algorithm is evaluated based on realistic 2D and 3D simulationsn, and a real data set acquired on a state-of-the-art TOF PET/CT system. The performance of the newly proposed LM SPDHG algorithm is compared against the conventional sinogram SPDHG and the listmode EM-TV algorithm. Main results. We show that the speed of convergence of the proposed LM-SPDHG is equivalent the original SPDHG operating on binned data (TOF sinograms). However, we find that for a TOF PET system with 400 ps TOF resolution and 25 cm axial FOV, the proposed LM-SPDHG reduces the required memory from approximately 56 to 0.7 GB for a short dynamic frame with 107 prompt coincidences and to 12.4 GB for a long static acquisition with 5·108 prompt coincidences. Significance. In contrast to SPDHG, the reduced memory requirements of LM-SPDHG enables a pure GPU implementation on state-of-the-art GPUs—avoiding memory transfers between host and GPU—which will substantially accelerate reconstruction times. This in turn will allow the application of LM-SPDHG in routine clinical practice where short reconstruction times are crucial.

Computational mean-field information dynamics associated with reaction-diffusion equations

We formulate and compute a class of mean-field information dynamics for reaction-diffusion equations. Given a class of nonlinear reaction-diffusion equations and entropy type Lyapunov functionals, we study their gradient flows formulations with generalized optimal transport metrics and mean-field control problems. We apply the primal-dual hybrid gradient algorithm to compute the mean-field control problems with potential energies. A byproduct of the proposed method contains a new and efficient variational scheme for solving implicit in time schemes of mean-field control problems. Several numerical examples demonstrate the solutions of mean-field control problems.

On the Convergence of Stochastic Primal-Dual Hybrid Gradient

In this paper, we analyze the recently proposed stochastic primal-dual hybrid gradient (SPDHG) algorithm and provide new theoretical results. In particular, we prove almost sure convergence of the iterates to a solution and linear convergence with standard step sizes, independent of strong convexity constants. Our assumption for linear convergence is metric subregularity, which is satisfied for smooth and strongly convex problems in addition to many nonsmooth and/or nonstrongly convex problems, such as linear programs, Lasso, and support vector machines. In the general convex case, we prove optimal sublinear rates for the ergodic sequence and for randomly selected iterate, without bounded domain assumptions. We also provide numerical evidence showing that SPDHG with standard step sizes shows favorable and robust practical performance against its specialized strongly convex variant SPDHG-$\mu$ and other state-of-the-art algorithms including variance reduction methods and stochastic dual coordinate ascent.

Revisiting convexity-preserving signal recovery with the linearly involved GMC penalty

• A new method for setting the matrix parameter in the linearly involved GMC is proposed. • An alternative algorithm is presented to solve the linear involved convexity-preserving model. • Two properties of the solution path are proved to help with tuning parameter selection. The generalized minimax concave (GMC) penalty is a newly proposed regularizer that can maintain the convexity of the objective function. This paper deals with signal recovery with the linearly involved GMC penalty. First, we propose a new method to set the matrix parameter in the penalty via solving a feasibility problem. The new method possesses appealing advantages over the existing method. Second, we recast the linearly involved GMC model as a saddle-point problem and use the primal-dual hybrid gradient (PDHG) algorithm to compute the solution. Another important work in this paper is that we provide guidance on the tuning parameter selection by proving desirable properties of the solution path. Finally, we apply the linearly involved GMC penalty to 1-D signal recovery and matrix regression. Numerical results show that the linearly involved GMC penalty can obtain better recovery performance and preserve the signal structure more successfully in comparison with the total variation (TV) regularizer.

A Deep Primal-Dual Proximal Network for Image Restoration

Image restoration remains a challenging task in image processing. Numerous methods tackle this problem, which is often solved by minimizing a nonsmooth penalized co-log-likelihood function. Although the solution is easily interpretable with theoretic guarantees, its estimation relies on an optimization process that can take time. Considering the research effort in deep learning for image classification and segmentation, this class of methods offers a serious alternative to perform image restoration but stays challenging to solve inverse problems. In this work, we design a deep network, named DeepPDNet, built from primal-dual proximal iterations associated with the minimization of a standard penalized co-log-likelihood with an analysis prior, allowing us to take advantage of both worlds. We reformulate a specific instance of the Condat-Vũ primal-dual hybrid gradient (PDHG) algorithm as a deep network with fixed layers. Each layer corresponds to one iteration of the primal-dual algorithm. The learned parameters are both the PDHG algorithm step-sizes and the analysis linear operator involved in the penalization (including the regularization parameter). These parameters are allowed to vary from a layer to another one. Two different learning strategies: “Full learning” and “Partial learning” are proposed, the first one is the most efficient numerically while the second one relies on standard constraints ensuring convergence of the standard PDHG iterations. Moreover, global and local sparse analysis prior are studied to seek a better feature representation. We apply the proposed methods to image restoration on the MNIST and BSD68 datasets and to single image super-resolution on the BSD100 and SET14 datasets. Extensive results show that the proposed DeepPDNet demonstrates excellent performance on the MNIST dataset compared to other state-of-the-art methods and better or at least comparable performance on the more complex BSD68, BSD100, and SET14 datasets for image restoration and single image super-resolution task.

Joint Sensing Task Assignment and Collision-Free Trajectory Optimization for Mobile Vehicle Networks Using Mean-Field Games

With the increasing popularity of mobile vehicles, such as unmanned aerial vehicles (UAVs) and mobile robots, it is foreseen that they will play an important role in Internet-of-Things (IoT) networks due to their high mobility and rapid deployment. Specifically, mobile vehicles equipped with sensors act as IoT devices and can be dispatched to several sensing regions to perform sensing tasks. In this article, we consider mobile vehicles for sensing applications and investigate the corresponding joint task assignment and collision-free trajectory optimization problem. This problem is challenging as the number of involved vehicles can be very large, and to tackle the problem efficiently, we reformulate the original optimization problem into a mean-field-game (MFG) problem by simplifying the interaction between vehicles as a distribution over their state space, known as the mean-field term. To solve the MFG problem efficiently, we propose a G-prox primal-dual hybrid gradient (PDHG) algorithm that transforms the MFG problem into a saddle-point problem by defining a Lagrangian functional with a proximal operator. The complexity of this algorithm is shown to be linear with the total number of grid points in the proposed MFG problem. We provide a comprehensive theoretical analysis of the proposed model and algorithm. Numerical results together with the practical implementation on real mobile robots show that our proposed system model and algorithm are of significant effectiveness and efficiency.

A deep unrolling network inspired by total variation for compressed sensing MRI

Compressed Sensing theory breaks through the limitation of the Nyquist sampling law and provides theoretical support for accelerating the imaging process of MRI while reconstructing high-quality images. It can use less sampling data through the reconstructed algorithm to restore the original signal. Reconstruction time and reconstruction algorithms play important roles in compressed sensing. However, it is difficult to apply the iterative reconstruction method in clinical applications because of the long reconstruction time. In addition, hand-crafted image prior which is widely used in traditional iterative method lacks of adaptivity for MRI reconstruction. In this paper, inspired by the total variation model, we propose a novel deep network for CSMRI, dubbed as TV-Inspired Network (TVINet), which incorporates the deep priors into the traditional iterative algorithm. We apply a general compressed sensing reconstruction framework inspired by TV regularization and solve it by combines the iterative method with deep learning. The design of TVINet comes from the inference process on the basis of Primal Dual Hybrid Gradient algorithm, which makes the network has preferable interpretability. The linear operator and regularization term are learned from the training dataset by using a convolution network. The proposed approach trains the network end-to-end and reconstructs the desired image directly from the under-sampling data. Experimental results show that the proposed method has better PSNR or SSIM than state-of-the-art methods, and maintains more complex and fine details in the reconstructed image.

Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications

Several Variants of the Primal-Dual Hybrid Gradient Algorithm with Applications

Least-squares reverse time migration via linearized waveform inversion using a Wasserstein metric

Least-squares reverse time migration (LSRTM), an effective tool for imaging the structures of the earth from seismograms, can be characterized as a linearized waveform inversion problem. We have investigated the performance of three minimization functionals as the [Formula: see text] norm, the hybrid [Formula: see text] norm, and the Wasserstein metric ([Formula: see text] metric) for LSRTM. The [Formula: see text] metric used in this study is based on the dynamic formulation of transport problems, and a primal-dual hybrid gradient algorithm is introduced to efficiently compute the [Formula: see text] metric between two seismograms. One-dimensional signal analysis has demonstrated that the [Formula: see text] metric behaves like the [Formula: see text] norm for two amplitude-varied signals. Unlike the [Formula: see text] norm, the [Formula: see text] metric does not suffer from the differentiability issue for null residuals. Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that, compared to the [Formula: see text] norm, the hybrid [Formula: see text] norm and [Formula: see text] metric can accelerate LSRTM and are less sensitive to non-Gaussian noise. For the field data application, the [Formula: see text] metric produces the most reliable imaging results. The hybrid [Formula: see text] norm requires tedious trial-and-error tests for the judicious threshold parameter selection. Hence, the more automatic [Formula: see text] metric is recommended as a robust alternative to the customary [Formula: see text] norm for time-domain LSRTM.

On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective

The proximal point algorithm (PPA) is a fundamental method for convex programming. When applying the PPA to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and sometimes even admit closed form or efficient solutions. This idea results in the so-called customized PPA (also known as preconditioned PPA), and it covers the linearized ALM, the primal-dual hybrid gradient algorithm, ADMM as special cases. Since each customized PPA owes special structures and has popular applications, it is interesting to ask wether we can design a simple relaxation strategy for these algorithms. In this paper we treat these customized PPA algorithms uniformly by a mixed variational inequality approach, and propose a new relaxation strategy for these customized PPA algorithms. Our idea is based on correcting the dual variables individually and does not rely on relaxing the primal variables. This is very different from previous works. From variational inequality perspective, we prove the global convergence and establish a worst-case convergence rate for these relaxed PPA algorithms. Finally, we demonstrate the performance improvements by some numerical results.

An adaptive model combining a total variation filter and a fractional-order filter for image restoration

In this paper, we propose a model that combines a total variation filter with a fractional-order filter, which can unite the advantages of the two filters, and has a remarkable effect in the protection of image edges and texture details; simultaneously, the proposed model can eliminate the staircase effect. In addition, the model improves the PSNR compared with the total variation filter and the fractional-order filter when removing noise. Zhu and Chan presented the primal-dual hybrid gradient algorithm and proved that it is effective for the total variation filter. On the basis of their work, we employ the primal-dual hybrid gradient algorithm to solve the combined model in this article. The final experimental results show that the new model and algorithm are effective for image restoration.

A modified level set algorithm based on point distance shape constraint for lesion and organ segmentation

PurposeThe segmentation of organs and lesions from medical images is a challenging task due to the presents of noise, intensity inhomogeneity, blurry/weak boundaries. In this paper, a point distance shape constraint is proposed and incorporated in the level set framework for the segmentation of objects with various shapes. MethodsThe proposed shape constraint is a linear combination of the Euclidean distance of user selected points. By selecting different numbers of points, it can generate different shape constraints and therefore is more flexible in dealing with different shapes. Then this shape constraint is incorporated into the variational level set framework. A convex relaxation is applied to get a convex model which can be efficiently solved by a primal-dual hybrid gradient algorithm. ResultsThe proposed algorithm is tested on 60 CT images with the nodular type of hepatic cellular cancer (HCC), 100 ultrasound kidney images, 20 prostate MR images, 20 lumbar CT images and 100 transrectal ultrasound prostate images. The algorithms performance is evaluated using a number of metrics by comparison with expert delineations. The validation results show that, for five datasets mentioned previously, the average DSCs of the proposed algorithm are 95.6% ± 1.4%, 94.3% ± 3.1%, 91.3% ± 3.8%, 92.7% ± 1.5% and 94.4% ± 2.2% respectively. Both quantitative and qualitative evaluation confirm that the proposed method can provide more accurate segmentation than four state-of-the-art methods. ConclusionThe proposed point distance shape constraint segmentation model can accurately segment organs and lesions with a number of shapes in medical images.

On the equivalence of the primal-dual hybrid gradient method and Douglas–Rachford splitting

The primal-dual hybrid gradient (PDHG) algorithm proposed by Esser, Zhang, and Chan, and by Pock, Cremers, Bischof, and Chambolle is known to include as a special case the Douglas–Rachford splitting algorithm for minimizing the sum of two convex functions. We show that, conversely, the PDHG algorithm can be viewed as a special case of the Douglas–Rachford splitting algorithm.

A splitting method for overcoming the curse of dimensionality in Hamilton–Jacobi equations arising from nonlinear optimal control and differential games with applications to trajectory generation

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points $(x,t)$ directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.

Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications

We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The anal...