Zheng Xu^{1}, Soham De^{1}, Mário A. T. Figueiredo^{2}, Christoph Studer ^{3}, Tom Goldstein^{1} ^{1}Department of Computer Science, University of Maryland, College Park, MD
^{2}Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, Portugal
^{3}Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY
Abstract
The alternating direction method of multipliers (ADMM) is a common optimization tool for solving constrained and non-differentiable problems. We provide an empirical study of the practical performance of ADMM on several nonconvex applications, including ℓ0 regularized linear regression, ℓ0 regularized image denoising, phase retrieval, and eigenvector computation. Our experiments suggest that ADMM performs well on a broad class of non-convex problems. Moreover, recently proposed adaptive ADMM methods, which automatically tune penalty parameters as the method runs, can improve algorithm efficiency and solution quality compared to ADMM with a non-tuned penalty.
An Empirical Study of ADMM for
Nonconvex Problems
Zheng Xu^{1}, Soham De^{1}, Mário A. T. Figueiredo^{2}, Christoph Studer ^{3}, Tom Goldstein^{1}^{1}Department of Computer Science, University of Maryland, College Park, MD^{2}Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, Portugal^{3}Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY
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^{†}^{†}
ZX, SD, and TG were supported by US NSF grant CCF-1535902 and by US ONR grant N00014-15-1-2676.
CS was supported in part by Xilinx Inc., and by the US NSF under grants ECCS-1408006 and CCF-1535897.
1 Introduction
The alternating direction method of multipliers (ADMM) has been applied to solve a wide range of constrained convex and nonconvex optimization problems.
ADMM decomposes complex optimization problems into sequences of simpler subproblems that are often solvable in closed form. Furthermore, these sub-problems are often amenable to large-scale distributed computing environments Goldstein et al. (2016); Taylor et al. (2016). ADMM solves the problem
minu∈Rn,v∈RmH(u)+G(v),subject to~{}~{}Au+Bv=b,
(1)
where H:Rn→¯R, G:Rm→¯R, A∈Rp×n, B∈Rp×m, and b∈Rp, by the following steps,
uk+1=
argminuH(u)+⟨λk,−Au⟩+τk2∥b−Au−Bvk∥22
(2)
vk+1=
argminvG(v)+⟨λk,−Bv⟩+τk2∥b−Auk+1−Bv∥22
(3)
λk+1=
λk+τk(b−Auk+1−Bvk+1),
(4)
where λ∈Rp is a vector of dual variables (Lagrange multipliers), and τk is a scalar penalty parameter.
The convergence of the algorithm can be monitored using primal and dual “residuals,” both of which approach zero as the iterates become more accurate, and which are defined as
rk=b−Auk−Bvk,anddk=τkATB(vk−vk−1),
(5)
respectively Boyd et al. (2011). The iteration is generally stopped when
ADMM was introduced by Glowinski and Marroco (1975) and Gabay and Mercier (1976), and convergence has been proved under mild conditions for convex problems Gabay (1983); Eckstein and Bertsekas (1992); He and Yuan (2015). The practical performance of ADMM on convex problems has been extensively studied, see Boyd et al. (2011); Goldstein et al. (2014); Xu et al. (2016a) and references therein. For nonconvex problems, the convergence of ADMM under certain assumptions are studied in Wang et al. (2014); Li and Pong (2015); Hong et al. (2016); Wang et al. (2015). The current weakest assumptions are given in Wang et al. (2015), which requires a number of strict conditions on the objective, including a Lipschitz differentiable objective term. In practice, ADMM has been applied on various nonconvex problems, including nonnegative matrix factorization Xu et al. (2012), ℓp-norm regularization (0<p<1)Bouaziz et al. (2013); Chartrand and Wohlberg (2013), tensor factorization Liavas and Sidiropoulos (2015); Xu et al. (2016b), phase retrieval Wen et al. (2012), manifold optimization Lai and Osher (2014); Kovnatsky et al. (2015), random fields Miksik et al. (2014), and deep neural networks Taylor et al. (2016).
The penalty parameter τk is the only free choice in ADMM, and plays an important role in the practical performance of the method. Adaptive methods have been proposed to automatically tune this parameter as the algorithm runs. The residual balancing method He et al. (2000) automatically increase or decrease the penalty so that the primal and dual residuals have approximately similar magnitudes. The more recent AADMM method Xu et al. (2016a) uses a spectral (Barzilai-Borwein) rule for tuning the penalty parameter. These methods achieve impressive practical performance for convex problems
and are guaranteed to converge under moderate conditions (such as when adaptivity is stopped after a finite number of iterations).
In this manuscript, we study the practical performance of ADMM on several nonconvex applications, including ℓ0 regularized linear regression, ℓ0 regularized image denoising, phase retrieval, and eigenvector computation. While the convergence of these applications may (not) be guaranteed by the current theory, ADMM is one of the (popular) choices to solve these nonconvex problems. The following questions are addressed using these model problems: (i) does ADMM converge in practice, (ii) does the update order of H(u) and G(v) matter, (iii) is the local optimal solution good, (iv) does the penalty parameter τk matter, and (v) is an adaptive penalty choice effective?
2 Nonconvex applications
ℓ0 regularized linear regression.
Sparse linear regression can be achieved using the non-convex, ℓ0 regularized problem
minx12∥Dx−c∥22+ρ∥x∥0,
(7)
where D∈Rn×m is the data matrix, c is a measurement vector, and x is the regression coefficients. ADMM is applied to solve problem (7) using the equivalent formulation
minu,v12∥Du−c∥22+ρ∥v∥0subject to~{}~{}u−v=0.
(8)
ℓ0 regularized image denoising.
The ℓ0 regularizer Dong and Zhang (2013) can be substituted for the ℓ1 regularizer when computing total variation for image denoising. This results in the formulation Chartrand (2007)
minx12∥x−c∥22+ρ∥∇x∥0
(9)
where c represents a given noisy image, ∇ is the linear discrete gradient operator, and ∥⋅∥2/∥⋅∥0 is the ℓ2/ℓ0 norm.
We solve the equivalent problem
minu,v12∥u−c∥22+ρ∥v∥0subject to~{}~{}∇u−v=0.
(10)
The resulting ADMM sub-problems can be solved in closed form using fast Fourier transforms Goldstein and Osher (2009).
Phase retrieval.
Ptychographic phase retrieval Yang et al. (2011); Wen et al. (2012) solves the problem
minx12||abs(Dx)−c||22,
(11)
where x∈Cn, D∈Cm×n, and abs(⋅) denotes the elementwise magnitude of a complex vector. ADMM is applied to the equivalent problem
minu,v12||abs(u)−c||22%
subject to~{}~{}u−Dv=0.
(12)
Eigenvector problem.
The eigenvector problem is a fundamental problem in numerical linear algebra. The leading eigenvalue of a matrix D is found by computing
max∥Dx∥22subject to~{}~{}∥x∥2=1.
(13)
ADMM is applied to the equivalent problem
min−∥Du∥22+ι{z:∥z∥2=1}(v)%
subject to~{}~{}u−v=0,
(14)
where ιS is the characteristic function defined by ιS(v)=0, if v∈S, and ιS(v)=∞, otherwise.
3 Experiments & Observations
Experimental setting.
We implemented “vanilla ADMM” (ADMM with constant penalty),
and fast ADMM with Nesterov acceleration and restart Goldstein et al. (2014). We also implemented two methods for automatically selecting penalty parameters: residual balancing He et al. (2000), and the spectral adaptive method Xu et al. (2016a). For ℓ0 regularized linear regression, the synthetic problem in Zou and Hastie (2005); Goldstein et al. (2014); Xu et al. (2016a) and realistic problems in Efron et al. (2004); Zou and Hastie (2005); Xu et al. (2016a) are investigated with ρ=1.
For ℓ0 regularized image denoising, a one-dimensional synthetic problem was created by the process described in Zou and Hastie (2005), and is shown in Fig.3. For the total-variation experiments, the "Barbara" , "Cameraman", and "Lena" images are investigated, where Gaussian noise with zero mean and standard deviation 20 was added to each image (Fig.4). ρ=1 and ρ=500 are used for the synthetic problem and image problems, respectively. For phase retrieval, a synthetic problem is constructed with a random matrix D∈C15000×500, x∈C500, e∈C15000 and c=abs(Dx+e). Three images in Fig.4 are used. Each image is measured with 21 octanary pattern filters as described in Candes et al. (2015).
For the eigenvector problem, a random matrix D∈R20×20 is used.
Does ADMM converge in practice? The convergence of vanilla ADMM is quite sensitive to the choice of penalty parameter. For vanilla ADMM, the iterates may oscillate, and if convergence occurs it may be very slow when the penalty parameter is not properly tuned. The residual balancing method converges more often than vanilla ADMM, and the spectral adaptive ADMM converges the most often. However, none of these methods uniformly beats all others, and it appears that vanilla ADMM with a highly tuned stepsize can sometimes outperform adaptive variants.
Does the update order of H(u) and G(v) matter? In Fig.1, ADMM is performed by first minimizing with respect to the smooth objective term, and then the nonsmooth term. We repeat the experiments with the update order swapped, and report the results in Fig.2 of the appendix. When updating the non-smooth term first, the convergence of ADMM for the phase retrieval problem becomes less reliable. However, for some problems (like image denoising), convergence happened a bit faster than with the original update order. Although the behavior of ADMM changes, there is no predictable difference between the two update orderings.
Is the local optimal solution good? The bottom row of Fig.1 presents the objective/PSNR achieved by the ADMM variants when varying the (initial) penalty parameter. In general, the quality of the solution depends strongly on the penalty parameter chosen. There does not appear to be a predictable relationship between the best penalty for convergence speed and the best penalty for solution quality.
Does the adaptive penalty work? In Table 1, we see that adaptivity not only speeds up convergence, but for most problem instances it also results in better minimizers. This behavior is not uniform across all experiments though, and for some problems a slightly lower objective value can be achieved using a finely tuned constant stepsize.
width × height for image restoration; width × height × filters for phase retrieval
Table 1: Iterations (with runtime in seconds) and objective (or PSNR) for the various algorithms and applications described in the text. Absence of convergence after n iterations is indicated as n+.
4 Conclusion
We provide a detailed discussion of the performance of ADMM on several nonconvex applications, including ℓ0 regularized linear regression, ℓ0 regularized image denoising, phase retrieval, and eigenvector computation. In practice, ADMM usually converges for those applications, and the penalty parameter choice has a significant effect on both convergence speed and solution quality. Adaptive penalty methods such as AADMM Xu et al. (2016a) automatically select the penalty parameter, and perform optimization with little user oversight. For most problems, adaptive stepsize methods result in faster convergence or better minimizers than vanilla ADMM with a constant non-tuned penalty parameter. However, for some difficult non-convex problems, the best results can still be obtained by fine-tuning the penalty parameter.
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Appendix A Appendix: more experimental results
Appendix B Appendix: implementation details
b.1 ℓ0 regularized linear regression
ℓ0 regularized linear regression is a nonconvex problem
minx12∥Dx−c∥22+ρ∥x∥0
(15)
where D∈Rn×m is the data matrix, c is the measurement vector, and x is the regression coefficients. ADMM is applied to solve problem (15) by solving the equivalent problem
minu,v12∥Du−c∥22+ρ∥v∥0subject to~{}~{}u−v=0.
(16)
The proximal operator of the ℓ0 norm is the hard-thresholding,
where ⊙ represents element-wise multiplication, and IS is the indicator function of the set S: IS(v)=1, if v∈S, and IS(v)=0, otherwise.
Then the steps of ADMM can be written
The ℓ0 regularizer Dong and Zhang [2013] is an alternative to the ℓ1 regularizer when computing total variation Goldstein and Osher [2009], Goldstein et al. [2014]. ℓ0 regularized image denoising solves the nonconvex problem
minx12∥x−c∥22+ρ∥∇x∥0
(22)
where c represents a given noisy image, ∇ is the linear gradient operator, and ∥⋅∥2/∥⋅∥0 denotes the ℓ2/ℓ0 norm of vectors. The steps of ADMM for this problem are
where sign(⋅) denotes the elementwise phase of a complex-valued vector.
In the following ADMM steps, notice that the dual variable λ∈Cm is complex, and the penalty parameter τ∈R is a real non-negative scalar,
The eigenvector problem is a fundamental problem in numerical linear algebra. The leading eigenvector of a matrix can be recovered by solving the Rayleigh quotient maximization problem
max∥Dx∥22subject to~{}~{}∥x∥2=1.
(33)
ADMM is applied to the equivalent problem
min−∥Du∥22+ι{z:∥z∥2=1}(v)%
subject to~{}~{}u−v=0,
(34)
where ιS is the characteristic function of the set S: ιS(v)=0, if v∈S, and ιS(v)=∞, otherwise. The ADMM steps are
Appendix C Appendix: synthetic and realistic datasets
We provide the detailed construction of the synthetic dataset for our linear regression experiments. The same synthetic dataset has been used in Zou and Hastie [2005], Goldstein et al. [2014], Xu et al. [2016a]. Based on three random normal vectors νa,νb,νc∈R50, the data matrix D=[d1…d40]∈R50×40 is defined as