Nonconvex and nonsmooth programmings (NNPs) have received wide attentions in recent years [27, 22, 33, 37, 5]. Many problems in vision and learning societies, such as sparse coding and dictionary learning , matrix factorization , image restoration , and image classification  can be (re)formulated as specific NNPs. In this work, we consider a general NNP in following formulation:
s are vectors or matrices throughout the paper.
In the past few years, there have been literatures [2, 8, 33, 34, 7] in optimization and numerical analysis on designing converged algorithms in view of the general NNP (1). These algorithms have been applied to various problems, such as non-negative matrix factorization , SDL with penalty  and non-negative Tucker decomposition ; at the same time, abundant experimental analyses have demonstrated the efficiency and convergence properties of these algorithms. However, in pursuit of convergence, most of the existing algorithms are designed with fixed iteration schemes; those inflexible schemes are tightly constrained and fail to take the model structures of specific problems into consideration.
While on the other hand, specific solvers designed for practical use are far more flexible than the converged algorithms proposed for the general problem (1). Those specific solvers always take advantages of the problem structures and then employ effective numerical methods to solve specific sub-problems. Moreover, we have noticed from previous work [10, 30, 19, 11, 25] that solving sub-problems with inner iterations is a frequently used strategy in numerous efficient solvers. Though few of the inexact solvers are designed with rigid theoretical support and analyses, their efficiencies and convergence properties have been verified from experiments under certain conditions.
Motivated from various inexact solvers, we in this paper propose an unified and flexible algorithm framework named inexact proximal alternating direction method (IPAD). Our IPAD is designed for solving the general NNP problem (1), at the same time, keeping the flexibility when dealing with specific problems. Different from existing solvers in practice, IPAD theoretically give rigid conditions on parameters and stopping criteria to ensure the convergence, thus is more rigid and robust for practical use. The theoretical support provided in this paper can be regarded as a guidance for designing inexact algorithms for solving specific problems in a concise framework. As far as we know, we are the first to incorporate various numerical methods into a general algorithm framework and at the same time give rigorous convergence analyses for NNPs. In summary, our contributions are three folds:
Different from most existing numerical algorithms for NNPs, which always fix their updating schemes during iterations, we provide a novel perspective to incorporate different optimization strategies into a unified and flexible proximal optimization framework for (1).
Even with inexact subproblems and flexible inner iteration schemes, we prove that the convergence property of the resulting hybrid optimization framework can still be guaranteed. Indeed, our theoretical results are the best we can ask for, unless further assumptions are made on the general NNPs in (1).
As an application example, we show implementations of applying IPAD with different inner algorithms to solve the widely concerned regularized SDL model. Numerical evaluations and abundant experimental comparisons demonstrate promising experimental results of the proposed algorithms, which verify the flexibility and efficiency of our optimization framework.
2 Related Work
2.1 Existing Optimization Strategies for NNPs
In past several years, accompanied with the rising popularity of investigating sparsity and low-rankness (naturally with nonconvex and nonsmooth properties) for vision and learning tasks [22, 37, 32, 1, 23, 39]
, developing numerical solvers for different types of NNP models have attracted considerable research interests from computer vision, machine learning and optimization societies.
As a typical nonconvex and nonsmooth measurement, norm has been widely applied for image processing problems [37, 32, 1]; and various methods have been designed for solving its optimization. In , the authors propose a proximal method for finding desirable solution to TV problem, but they only prove a weak convergence result under mild conditions. The rank function [23, 39] has also been well studied: the work in  optimizes a low-rank minimization by an iteratively reweighted algorithm. However, they can only prove that any limit point is a stationary point, which cannot guarantee the correct convergence of the iterations. Another typical problem, non-negative matrix factorization is a powerful technique for various applications; algorithms like ANLS  have been specially designed for solving this specific problem.
On the other hand, some algorithms with rigid theoretical analyses have been proposed for solving the general NNP problem (1) [2, 8, 31]. The authors in  propose an algorithm named GIST for solving a special class of NNP (1
) and apply it to logistic regression problem. A proximal alternating linearized method is proposed in, however, it is time-consuming for solving special problems. Xu et al. in  propose a MM method for the general NNP, but their algorithm is too complicated to be implemented in practice. The authors in  provide their convergence result by assuming subproblems are exactly solved, however, this condition is unattainable in most situations.
In summary, most optimization schemes for specific NNPs are efficient in practice, however, their convergence guarantees are relatively weak. While, algorithms designed for the general NNP are well-converged, but they always have strict conditions and are inefficient in practice.
2.2 Inexact Inner Iterations in Application Fields
, the “improper” iteration skills have been frequently used in solving sparse coding, blind kernel estimation, and medical imaging problems in practice.
One of those inexact schemes is employing numerical methods as inner iteration methods for solving special subproblems. For instance, the authors in  employ half-quadratic splitting and Lagrangian dual method as the inner iteration schemes for estimating clear images in blind deconvolution problem. While in a text image deblurring paper , the authors update variables though alternating direction method, meanwhile, applying half-quadratic splitting for solving subproblems. Although these inexact solvers are effective in practice, they are totally designed in freestyle and are uncontrolled in the performances.
Another kind of inexact schemes is adopting neural networks for approximating exact solutions during iterations. For example, time-unfolded recurrent neural networks can be used to produce the best possible approximations for sparse coding problem[18, 28]. On the other hand, fusing various neural networks in ADMM framework [13, 36] has great performances for approximately optimizing problems for image restoration and compressed sensing. The efficiencies of these inexact solvers and their convergence performances have been verified from experiments, however, few of them are designed with rigid theoretical support.
3 The Proposed Optimization Framework
To simplify subsequent derivations, we propose an IPAD and analyze its properties for problem (1) with variables111For simplicity, we replace the notations in problem (1) with the new ones. We hope this will not cause confusions.:
Though the whole properties are analyzed for problem (2), it is straightforward to extend them to the general one. Then we give assumptions on the objective function:
(1) and are proper, lower semi-continuous functions;
(2) is a function and its gradient is Lipschitz continuous on a bounded set;
(3) is a coercive, Kurdyka-Łojasiewicz (KŁ) function222To be self-contained, the definition of KŁ function is given in the supplemental material due to space limit..
It should be mentioned that the most frequently used -norm is a KŁ function which also satisfies the second assumption. On the other hand, regularizers like penalty, norm, SCAD , MCP  and indicator functions are all KŁ functions and at the same time satisfy the first assumption. Since the finite sums of KŁ functions are also KŁ functions , thus not a few models in image processing and machine learning satisfy these assumptions.
3.1 Inexact Proximal Reformulation
For solving problem (2
), it is natural and common to apply a proximal alternating direction method (PAD)[35, 11] for alternatively updating variables in a cyclic order:
where and are proximal parameters added on the subproblems respectively; the notations and are used to represent exact solutions of the subproblems in Eq. (3). It should be pointed out that adding proximal terms on subproblems is a common skill to improve the stability of algorithms on solving nonconvex problems .
As claimed before, in most cases, it is either impossible or extremely hard for calculating exact solutions of subproblems. Thus not a few work employ inner iteration schemes to compute approximations to the exact solutions, which means, the inexact solutions and calculated by inner methods are approximations to and :
We can see from the above that our IPAD is in a general framework: it alternatively updates variables by approximately solving the subproblems of PAD. While on the other hand, it does not restrict specific formulas for solving the subproblems, hence IPAD has quite flexible inner iteration schemes that can fuse efficient numerical methods into.
3.2 Flexible Inner Iteration Schemes
Our IPAD framework is highly flexible since it allows fusing any efficient algorithms, to compute inexact solutions for specific subproblems. This flexibility is especially welcomed in practice: since the constraints added on variables are always quite different from one another, efficient algorithms for solving specific subproblems should be carefully chosen. For example, when facing the SDL problem with regularization [15, 24], both subproblems are convex. Then various numerical methods designed under convex case like homotopy method , FISTA  and ADMM  can be applied to solving these subproblems.
On the other hand, although many previously used inexact solvers can be presented under our IPAD framework [21, 30, 11], almost all the inner iteration schemes are experimentally designed. For example, a 2-step inner loop is supposed to be “good-enough” in ; the authors of  stop the inner iterations at fixed steps to achieve acceptable solutions. Different from the previous work, we in the following Criterion 1 provide theoretical conditions for stopping the inner iterations.
Since calculating the inexact solutions brings errors and in the first-order optimality conditions:
with and , we in the following Criterion 1 show that these errors should be bounded to a certain extent at every iteration.
The errors and in Eq. (5) must satisfy
where parameters and are two positive integers defined before the iteration starts.
With the stopping criteria (6) for inner iteration schemes, we summarize the whole algorithm in Alg. 1. However, this inexact framework is still on conceptual progress: the stopping criteria can not be directly calculated from (6) and an implementation should be put forward for carrying IPAD for practical use. Before providing a practical implementation in Sec. 3.4, we first analyze the convergence properties of IPAD with the help of the well-designed Criterion 1.
3.3 Convergence Analyses
In this section, we provide the theoretical support for IPAD method333Due to space limit, all the related proofs in this section will be detailedly given in the supplemental material.. The strategic point on analyzing the convergence properties of IPAD is regarding and as the exact solutions on solving the following subproblems:
This equivalent conversion is rigid since the first-order optimality conditions of (7) are exactly the same with Eq. (5). However, it should be emphasized that and are not computed by directly minimizing (7); this equivalent conversion is nothing but assisting in theoretical analyses.
Before proposing the main theorem, we give the requirements on and .
Parameters and satisfy
to ensure the whole IPAD algorithm converges.
Then with the help of Assumption 1, we can obtain the main theorem in Theorem 1: our proposed IPAD has the best convergence property, that is, the global convergence property for general NNP: our IPAD generates a Cauchy sequence that converges to a critical point of the problem.
Under the Assumption 1 and suppose the sequence generated by IPAD is bounded. Then is a Cauchy sequence that converges to a critical point of .
For proving this main convergence theorem, the two assertions in the following key lemma are the cornerstones. From the two assertions, obviously, the objective function is sufficiently descent (Eq. (9) in Lemma 1) during iterations, which together with the second assertion ensures that there is a subsequence of that converges to a critical point of the problem. By further combining the KŁ property, we have the global convergence property for our proposed IPAD method as claimed in Theorem 1.
Suppose that the sequence generated by IPAD is bounded. Then the following two assertions hold under the Assumption 1:
where constants and with Lipschitz constant of on bounded sets.
For the convergence rate, our IPAD method shares the same result with PALM  when the desingularising function of is satisfied with positive constant and Specifically, IPAD converges in a finite number of steps when equals to 1. For and , IPAD converges with a sublinear rate and a linear rate respectively. Though the convergence rate of IPAD is the same with PALM in theory (the convergence rate is not affected by the algorithm but the objective function of NNP ), the experimental results in Section 4 verify the efficiency of our algorithm.
It is natural to extend IPAD to solving the general multi-block NNP (1). Furthermore, all the convergence analyses conducted on problem (2) can be straightforwardly extended to the multi-block case. Since not a few problems can be (re)formulated as the general formulation (1), thus our IPAD method can be applied for solving a wider range of problems in applications.
We can see from the above contents that, arbitrarily incorporating inner numerical methods into IPAD still ensure the global convergence property of the whole algorithm as long as the stopping criteria (6) and the parameter conditions (8) are satisfied. From this perspective, IPAD is not a single algorithm, but a general and flexible algorithm framework with rigid convergence properties. In addition, through blending other numerical methods into our IPAD framework, some previous inexact methods applied to applications [24, 21, 11] can also be proved to achieve the global convergence property.
3.4 Implementable Error Bound
Though we have given criteria of inexactness in the Criterion 1, there appears another problem: the errors and can not be directly calculated. Since and in Eq. (5) are unattainable in practice, and can not be directly calculated by the equalities in Eq. (5). Thus the aforementioned IPAD algorithm, i.e., Alg. 1 is only in the conceptual progress; the criteria (6) are not implementable in practice.
Instead, we in this section provide a truly implementation for carrying IPAD for practical use. After calculating the inexact solutions and of Eq. (4) by some numerical methods, we first compute two intermediate variables and as the solutions of specific proximal mappings444 denotes proximal mapping to proper, lower semi-continuous function .
where variables and are computed as follows:
After getting the intermediate variables and from and , the errors and are calculated by:
We give a proposition as follows to show that these two errors are exactly the implementable versions of Eq. (5). In order to make discussions smoother, we provide the proof of Proposition 1 at the end of this paper in Appendix A.
Thus, the errors in (13) are used for checking whether the stopping criteria (6) are satisfied; once the criteria are satisfied, we re-assign to and to . Then and become the final solutions of their corresponding subproblems at -th step. For clarity, we give the detailed inexact process of inner iteration schemes in Alg. 2.
4 Experimental Results
Though our IPAD algorithm framework can be directly applied to numerous problems in computer vision and machine learning [40, 21, 27, 11], we in this paper just consider to utilize the widely concerned -regularized SDL model  as an example to verify the flexibility and efficiency of our IPAD framework. All the compared algorithms are implemented by Matlab R2013b and tested on a PC with 8 GB of RAM and Intel Core i5-4200M CPU.
4.1 SDL with Penalty
The SDL problem with penalty is formulated as:
where denotes the penalty that counts the number of non-zero elements of . Here the indicator function acts on the set
As for another set , we make it empty for synthetic data but define it as
for real-world data to enhance the stability of the model . Moreover, we denote to simplify the deduction.
It is observed that problem (14) is a special case of problem (2). Thus, we can apply IPAD for inexactly solving the following subproblems since it is extremely hard to get exact solutions of the subproblems.
Moreover, PALM can also be applied to solve (14), but it requires computing Lipschitz constants at every iteration.
4.2 Synthetic Data
We generate synthetic data with different sizes to help analyze the property of IPAD (see Table 1). All the algorithms for the synthetic data stop when satisfying:
where is the objective value at step .
4.2.1 Efficiency of Inexact Strategy
To show the respective effects of using inexact strategies on different subproblems, we propose IPAD-PITH which obtains by PITH but keeps -subproblem the same as PALM555Due to the space limit, we give detailed algorithm implementations, further theoretical analyses and more results in supplemental material.. We also design IPAD-ADMM that computes by ADMM but remains -subproblem the same as PALM.
|Image / /||PALM ||mPALM ||INV ||IPAD-ADMM|
|“Peppers512” / /||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time|
|“Lena512” / /||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time|
|“Barbara512” / /||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time|
|“Hill512” / /||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time||PSNR||Iter||Time|
|KSVD ||PSNR / Time||PSNR / Time||PSNR / Time||PSNR / Time|
|29.21 / 184.88||30.09 / 235.22||31.14 / 318.06||32.49 / 481.48|
|IPAD-ADMM||PSNR / Time||PSNR / Time||PSNR / Time||PSNR / Time|
|28.98 / 20.04||29.85 / 26.75||30.89 / 22.72||32.28 / 22.65|
The comparisons in Table 1 among PALM, IPAD-PITH and IPAD-ADMM show that inexact strategies help reduce the iteration steps: both the IPAD-PITH and IPAD-ADMM converges with less iterations than PALM. However, the performances of IPAD-PITH and IPAD-ADMM are quite different in inner iterations. We can see from Figure 1(d) that IPAD-ADMM uses few inner steps during iterations. However, IPAD-PITH reaches the maximum inner steps (set as 20) at almost every iteration. This from one side shows that ADMM is suitable for solving (16) but PITH is less efficient for solving (14). On the other side it is caused by the problems themselves: (16) has unique solution but problem (15) is a challenging NP hard problem and only sub-optimal solution can be found in polynomial time. Since PITH converges with unexpected time, thus we only test it on the data with relatively low dimension ().
We also apply inexact strategies to both subproblems of and . Since PITH is time-consuming for solving (15), thus we reduce the maximum inner step to 2 for PITH. We name this algorithm as IPAD-P2A. Then we can see from the Figure 1 and Table 1 that IPAD-P2A uses less iteration steps and sometimes converges faster to an optimal solution.
Though Theorem 1 is proved for applying the inexact strategy to both subproblems, the two hybrid forms of IPAD, i.e. IPAD-PITH and IPAD-ADMM, are also converged. Furthermore, a hybrid IPAD optionally combines PALM, PAM and IPAD can be proved to have global convergence property for solving problem (2)
It can be observed that IPAD-P2A can not be seen as a special case of the hybrid IPAD since it contains one inexact strategy for and two-steps prox-linear iterations for . But fortunately, the experimental results verify the convergence of IPAD-P2A and we can also prove the convergence from theoretical analyses.
By comparing the algorithms, all the inexact strategies of our IPAD method perform better than PALM and are verified to be practicable, converged and efficient. However, a less efficient numerical method for solving subproblem indeed reduce the efficiency of the whole algorithm. Therefore, we suggest carefully choosing effective numerical methods for solving subproblems.
4.2.2 Other Comparisons
At last, we compare INV that solves in the same way as PALM but treats as the solution of a linear system first and then project the solution on set . Though this strategy seems to be efficient in practice , it lacks theoretical guarantee. Firstly, calculated by INV is not an exact solution of (16). Secondly, it is computed without measuring the inexactness. So applying INV sometimes creates oscillations during iterations (Figure 1 (f)) and the performances of INV are unstable especially in real-world applications (see the experimental results in Section 4.3). Thus we do not recommend using it in practice.
4.3 Real-world Data
We apply IPAD to real-world data on image denoising problem [15, 12] and compare PALM , mPALM , INV  with our IPAD-ADMM for solving this real-world application. All the algorithms for comparison terminate when reaching and are demonstrated on 7 widely-used images. The patches in each image, of size , are regularly sampled in an overlapping manner. The noisy images are obtained by adding Gaussian randomly noises with level .
As shown in Table 2, PALM seems to converge quickly but get bad recovered results. However, the truth is: the large upper bound of the Lipschitz constant emphasizes the function of the proximal term. So it causes tiny differences between and . Thus, PALM does not converge when reaching the stopping criterion; on the contrary, it converges quite slow. For the failure of using PALM, we adopt the strategy used in , which regards the problem (14) as a multi-block problem: solving separately by PALM. We name their algorithm mPALM and show the results in Table 2. This time, mPALM converges to an acceptable result when reaching the stopping criterion.
We in Fig. 2 present comparison results on convergence performances: the vibration of INV  causes more iterations than IPAD-ADMM. We also select two examples in Fig. 3 to present the inner iteration numbers of our inexact algorithm. It can be seen from the figure that the inner iteration behaviors may be in totally different style; it depends on the input data and algorithm parameters. In addition, we also some visual comparisons in Fig. 4 and 4 to show that our IPAD-ADMM performs more stable and efficient for the real-world applications.
Finally, we give comparisons between the state-of-the-art KSVD technique  and our proposed IPAD-ADMM in Table 3. First we want to point it out that KSVD is designed for solving a quasi- (not the exact penalty) problem with the usage of OMP . We can see that though our PSNR values are slightly less than KSVD, we use less than a tenth of the time of KSVD. Thus, our IPAD is a fast algorithm with flexible inner iterations for solving the problem SDL with penalty in practice.
This paper provided a fast optimization framework to solve the challenging nonconvex and nonsmooth programmings (NNPs) in vision and learning societies. Different from most existing solvers, which always fix their updating schemes during iterations, we showed that under some mild conditions, any numerical algorithms can be incorporated into our general algorithmic framework and the convergence of the hybrid iterations can always be guaranteed. We conjectured that our theoretical results are the best we can ask for unless further assumptions are made on the general NNPs in (1). Numerical evaluations on both synthetic data and real images demonstrated promising experimental results of the proposed algorithms.
6 Appendix A: proof of Proposition 1
From the Alg. 2, at the -th iteration, suppose that an inner iteration scheme is conducted by time, and denote its current iterative solution as . Then from the calculations in Eq. (11), we can deduce the following equalities.
The above deductions can be similarly extended to the case of . Together with Eq. (12), we have
From the definition of mapping, Eq. (19) is equal to
where and . The above equalities are exactly the first-order optimality conditions of (4) by regarding and as the inexactness.
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