DWSR: an architecture optimization framework for adaptive super-resolution neural networks based on meta-heuristics

Vol.:(0123456789) Artificial Intelligence Review (2024) 57:23 https://doi.org/10.1007/s10462-023-10648-4 1 3 DWSR: anarchitecture optimization framework foradaptive super‑resolution neural networks based onmeta‑heuristics Shu‑ChuanChu1· Zhi‑ChaoDou1· Jeng‑ShyangPan1,2· LingpingKong3· VáclavSnášel3· JunzoWatada4 Published online: 30 January 2024 © The Author(s) 2024 Abstract Despite recent advancements in super-resolution neural network optimization, a fundamental challenge remains unresolved: as the number of parameters is reduced, the network’s performance significantly deteriorates. This paper presents a novel framework called the Depthwise Separable Convolution Super-Resolution Neural Network Framework (DWSR) for optimizing super-resolution neural network architectures. The depthwise separable convolutions are introduced to reduce the number of parameters and minimize the impact on the performance of the super-resolution neural network. The proposed framework uses the RUNge Kutta optimizer (RUN) variant (MoBRUN) as the search method. MoBRUN is a multi-objective binary version of RUN, which balances multiple objectives when optimizing the neural network architecture. Experimental results on publicly available datasets indicate that the DWSR framework can reduce the number of parameters of the Residual Dense Network (RDN) model by 22.17% while suffering only a minor decrease of 0.018 in Peak Signal-to-Noise Ratio (PSNR), the framework can reduce the number of parameters of the Enhanced SRGAN (ESRGAN) model by 31.45% while losing only 0.08 PSNR. Additionally, the framework can reduce the number of parameters of the HAT model by 5.38% while losing only 0.02 PSNR. Keywords Neural architecture search· Super-resolution· Swarm intelligence· Multiobjective· Runge Kutta optimizer 1 Introduction Super-resolution reconstruction of images is the technique of restoring a low-resolution image to a high-resolution image that is true, clear, and with as few human traces as possible Hou and Andrews (1978). Compared with low-resolution images, high-resolution images usually contain greater pixel density, richer texture details, and higher trustworthiness. However, we usually cannot directly obtain high-resolution images with sharpened edges and no block blur due to the limitations of recording devices and image degradation models Bulat etal. (2018). There are many image super-resolution methods, such as interpolation-based, degradation model-based, and deep learning-based methods Keys (1981); Extended author information available on the last page of the article S.-C.Chu et al. 1 3 23 Page 2 of 21 Schermelleh etal. (2019). Dong etal. first proposed using convolutional neural networks to deal with the image super-resolution problem in 2014 Dong etal. (2014). A three-layer convolutional neural network (SRCNN) is designed to learn the mapping relationship between low-resolution and high-resolution images directly in this paper. In 2016, Shi etal. considered the Efficient Sub-Pixel Convolutional Neural Network (ESPCN) from a lowresolution image and learned how to scale the image from a sample Shi etal. (2016). In 2017, Christian Ledig etal. proposed a super-resolution image reconstruction by adversarial networks from a photo-aware perspective Ledig etal. (2017). In recent years, researchers have been committed to getting higher accuracy of the network and enhancing the credibility of the generated images Liu etal. (2022). At the same time, they also hope to reduce the number of network parameters and improve the confidence of image generation. Most of the network architectures currently in use have been carefully designed by researchers. The number of parameters usually increases when designing to improve network performance, increasing the generation time of super-resolution images Zhang etal. (2018); Lim etal. (2017); Wang etal. (2018). There is a pressing requirement to design lightweight networks based on how to reduce the number of network parameters effectively. Neural Architecture Search (NAS) Elsken etal. (2019) has made breakthroughs in various applications in recent years. Examples include image recognition, image segmentation, and super-resolution. In the super-resolution domain, chu etal. were the earliest to suggest the NAS technique with the Multi-objective reinforced evolution in mobile neural architecture search (MoreMNAS) to search for super-resolution neural network architecture Chu etal. (2019). Song etal. proposed using different super-resolution network sub-block combinations to enhance the network performance and reduce the network parameters Song etal. (2020). However, these methods are also just a mix-and-match combination of previous methods with limited optimization of the number of parameters. Depthwise separable convolution (DW Conv) has made notable achievements in the research of lightweight neural networks. However, using DW Conv fully in super-resolution neural networks leads to significant performance degradation. In this paper, we propose a framework that aims to be able to automatically insert DW Conv at appropriate locations in the network while minimizing the impact on performance. Since the search space is enormous, finding a method with high searchability is necessary to improve the search speed and quickly find a better network architecture MoralesHernández etal. (2022); Mishra and Kane (2022). Metaheuristic algorithms do not require problem-specific knowledge or information, which makes them suitable for complex problems where the problem structure and properties may not be easily understood or modeled. The meta-heuristic algorithm can perform a global search to find the approximate solution of the optimal solution Rodríguez-Molina etal. (2020); Akhand etal. (2020). During the meta-heuristic algorithm search, the exploration phase explores the search space as much as possible to find the areas where the optimal solution may exist Chu etal. (2006); Meng etal. (2019). Since the optimal solution may exist at any location throughout the search space, a detailed search of the areas near the current optimal solution is performed in the development phase. In most cases, there are some correlations between solutions. The meta-heuristic algorithm uses these correlations to adjust the solution process Chu etal. (2005); Wang et al. (2022). The mathematically based RUNge Kutta optimizer (RUN) algorithm Ahmadianfar etal. (2021) is a typical example. The RUN Optimizer balances the exploration and development phases by designing the Runge Kutta Search Mechanism (RKM) for exploration and the Enhanced Solution Quality (ESQ) mechanism for exploitation. However, the RUN optimizer is designed for continuous optimization problems and cannot be applied to combinatorial optimization problems. In this paper, we propose DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 3 of 21 23 a transfer function that maps the continuous solution space to the discrete solution space, enabling it to solve combinatorial optimization problems. In addition, in order to avoid performance degradation due to excessive use of DW Conv (or excessive number of parameters due to excessive use of DW Conv) during the optimization process, we also propose a multi-objective optimization strategy for balancing the relationship between PSNR and the number of parameters during the search process. In this paper, an efficient and straightforward method for super-resolution network optimization is proposed. The search advantage of the meta-heuristic algorithm is implemented for NAS. The main contributions of this study are as follows: • MoBRUN is employed to balance the PSNR and the number of parameters. • A grid mechanism is established for the non-dominated solution in archives. • New leader selection schemes are presented to improve the position updating method of population individuals in the binary multi-objective meta-heuristic algorithm. • A novel framework is proposed to apply the MoBRUN algorithm to NAS to optimize super-resolution neural networks. The remaining sections of this manuscript are organized as follows: Sect. 2 briefly describes the development of the meta-heuristic algorithm and its application in NAS. Section3 introduces the original RUNge algorithm in detail. Section4 presents the improved MoBRUN algorithm. Section5 provides the proposed framework. Section6 discussed the experiment and its results. Section7 is the conclusion of this paper. 2 Related works Research into meta-heuristics has a long history. In the past, most research on meta-heuristic algorithms has emphasized their use in problems such as engineering optimization. An example is the Particle Swarm Optimization (PSO) algorithm Marini and Walczak (2015), initially based on the stochastic optimization technique of populations. Simulated Annealing (SA) algorithm Delahaye etal. (2019) for simulated metal annealing design. The Ant Colony Optimization (ACO) Zhou et al. (2022) algorithm is designed to abstract ants searching for food and record their paths. Meta-heuristic algorithms have demonstrated their usefulness in several fields Wang etal. (2014); Chu etal. (2022). In recent years, some researchers have tried to apply meta-heuristic algorithms to solve NAS problems. Wang etal. combined the PSO algorithm with Convolutional Neural Network (CNN) and proposed the cPSO-CNN algorithm Wang etal. (2019), which can automatically search the CNN architecture. Lu etal. explored a multi-objective genetic algorithm for neural network search Lu etal. (2020), which is better in terms of interactivity and structural design. Together, these studies outline the critical role of meta-heuristics in NAS. However, the focus of such studies remains narrow and only deals with applying meta-heuristics to NAS. Once the meta-heuristic algorithm is converted to the binary version, the position is only selected between 0 and 1 Beheshti (2020); Akay etal. (2021). When the binary meta-heuristic optimization algorithm performs multi-objective optimization, the leader selection mechanism enables individuals to converge to the current Pareto frontier. This phenomenon reduces the diversity of individuals, and the algorithm is likely to fall into the optimal local solution Tian etal. (2021); Liu etal. (2020); Zhang etal. (2020). Therefore, S.-C.Chu et al. 1 3 23 Page 4 of 21 the MoBRUN algorithm is proposed for solving the problem and finding the best solution for the multi-objective NAS problem. 3 RUNge Kutta optimizer The RUN algorithm proposed by Iman et al. is based on the specific slope calculation of the Runge Kutta method Butcher (1987). It is an effective global optimization search strategy. RUN consists of two main parts: RKM for exploration and ESQ mechanism for exploitation. 3.1 Initialization step The meta-heuristic algorithm is a method that uses N individuals to optimize D dimensions. For the enhancement of increase the randomness and diversity of individuals in the initial stage, the initial positions of individuals in the RUN optimizer are generated using Eq.(1). where xn,d in Eq.(1) is the location of the individual and the solution of the optimization problem of dimension D. Ld and Ul are the upper and lower bounds of the d-th variable of the problem to be optimized (d=1, 2, ..., D) . The rand is a random number within [0,1]. The dominant search mechanism in RUN is an RK4 based approach. This method searches the decision space with the aid of three randomly selected solutions. The mechanism can be modeled as: in which RUN performs random global (exploration) and local ( exploitation) searches in each iteration. When rand <0.5 , perform the global search method; otherwise, perform the local search method. The search method is designed using the RK method. The new solution is determined by Eq.(4). in which where r is used to change the search direction and add diversity, and is an integer taking the value of 1 or -1. SF is an adaptive factor. Parameter 𝜇 is a random number, and randn is a normally distributed random number. Parameter h is a random number taking values in the range [0,2]. Parameters xm and xc are calculated by the following equation: (1) xn,d=Ld+rand.(Ud−Ld) (2) SM = 1 6 (xRK )Δ x (3) xRK =k1+2×k2+2×k3+k4 (4) xn+1= {(x c +SF ×X c ×h×r)+SM ×SF + 𝜇 ×randn.(x m −x c )if rand < 0.5 (x m +SF ×X m ×h×r)+SM ×SF + 𝜇 ×randn . (x r1 −x r2 ) else (5) 𝜇=0.5 +0.1 ×randn DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 5 of 21 23 where xbest is the optimal solution achieved, the xlbet is the best solution obtained for current iteration, and xr1 is the position of an individual randomly selected in the population. 3.2 Enhanced solution quality mechanism The ESQ mechanism is used to improve the quality of the solution and avoid getting trapped in a local optimum in each iteration. When rand >0.5 , the ESQ mechanism performs the following scheme to create the solution: where w is a random number that decreases as the algorithm progresses, and parameter xavg is the average of three randomly selected solutions. The xnew1 is the best solution, and xavg is the random number determined. The solution calculated in Eq.(8) may not be better than the current one. In order to obtain a better solution, When rand <w , take the following steps to generate a new solution. where v is a random number taking values in the range [0,2]. The pseudo-code of the RUN algorithm is given in Algorithm1. Algorithm1 The pseudo-code of RUN algorithm (6) xc=xn×rand +(1−rand)×xr1 (7) xm=xbest ×rand +(1−rand)×xlbest (8) xnew2= { xnew1+∣(randn −xavg +xnew1)∣×w.r.if w< 1 (xnew1−xavg)+∣randn −xavg +u.xnew1∣×w.rif w> 1 (9) xnew3=xnew2×(1−rand)+SF.(v.xb+rand.xRK −xnew2) S.-C.Chu et al. 1 3 23 Page 6 of 21 4 The proposed MoBRUN method Because the RUN algorithm was initially designed for solving problems with continuous space, we proposed MoBRUN to solve the NAS problem of super-resolution networks. 4.1 Binary conversion The original RUN algorithm is designed to solve continuous problems, so it is required to convert to a binary version to solve the NAS problem. Numerous studies show that the values in continuous space can be converted into binary space after normalization by transfer function. The common transfer functions are S-, V-, and U-shaped transfer functions Mirjalili and Lewis (2013); Mirjalili etal. (2020); He etal. (2022), and here we choose to use the V-shaped transfer function to convert the RUN algorithm. The V-shaped transfer function is shown in Eq.(10). where xn,d is the position of the n-th individual in the d-th dimension, and erf is the Gaussian error function. The V-shaped transfer function converts the individual solution space from continuous to 0-1 space by Eq.(11) after normalization. The image of the V-shaped function is shown in Fig.1. (10) V xn,d=∣ erf ( 𝜋 2 ×xn,d )∣ (11) x t+1 n,d= { ¬x t n,dif rand <Vxn, d xt n,d if rand ≥Vx n,d Fig. 1 V-shaped transfer function DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 7 of 21 23 4.2 Multi‑objective strategy This subsection applies two components to improve the RUN algorithm so that it is able to execute multi-objective optimization. One is the archive component responsible for storing the Pareto optimal solution, and another one is the leader selection component that selects leaders from the archive. The leader selection component assists the RUN algorithm in selecting the optimal solution for position updating. An archive is a storage unit with a fixed size for storing Pareto optimal solutions. In an optimization problem with m objective functions, the solution vector x=(x1,x2, ..., xn) is assumed to minimize each objective function fi(x) . In this context, a non-dominated solution is defined as follows: The non-dominated solutions fi(xnew) obtained during the iterative process are compared with all solutions in the archive. Since the archive has a fixed size, new solutions entering the archive need to run the grid mechanism to redistribute the archive when it is full: the most crowded part of the current archive is found, and one of the solutions is omitted. The new solution is then inserted into the most sparse part to multiply the diversity of the Pareto optimal frontier of the final approximation. It should be noted that in the process of using a binary meta-heuristic optimization algorithm to solve problems, a large number of identical solution spaces will be generated in the archive. Therefore, it is necessary to prioritize removing identical solutions when deciding the dominance relation. When the RUN algorithm conducts a search work, we hope to find an optimal solution to guide the next step. Therefore, a leader selection mechanism is introduced to handle this problem. The leader selection mechanism in which the best solution is selected from the archive using a roulette wheel method. The advantage of this mechanism is expressed as follows: where q is a constant greater than 1 and Ns is the best solutions number in the archive for the current iteration. However, in the RUN algorithm, using the Pareto dominance relation for location update will lead to the problem of a slow location update. Therefore, to increase location diversity and speed up searches, we use the replacement strategy shown in Algorithm2 to update the location. (12) ∀i∈1, 2, ..., m,fi(xnew)≤fi(x) (13) P t= q N s S.-C.Chu et al. 1 3 23 Page 8 of 21 Algorithm2 The pseudo-code of replacement strategy 5 DWSR Framework Super-resolution neural networks usually have three phases: feature extraction, nonlinear mapping, and reconstruction. Compared with neural networks dealing with classification problems, super-resolution networks require more computational resources and are unsuitable for mobile devices. Researchers have recently preferred to design lightweight superresolution neural networks Kim et al. (2021). However, designing new neural network architectures is time-consuming and laborious. So we can optimize based on previous neural network architectures to reduce the cost. Andrew etal. proposed using a depthwise separable filter to reduce the number of neural network parameters for operation on mobile devices in 2017 Howard etal. (2017). For example, in a standard convolution operation, the number of parameters required is proportional to the product of the input and output channels and the kernel size. Specifically, if there are CI input channels, CO output channels, and a convolution kernel of size DK×DK , then CI×CO×DK 2 parameters are needed. By contrast, using a depthwise separable convolution reduces the computational cost by the following equation. where DF is the input feature map resolution. First, as shown in Fig.2, the deep convolution operation uses only a single convolution kernel for each input channel, which reduces the number of convolution kernels to CI × DK 2 . Second, as shown in Fig.3, point-by-point convolution uses a 1×1 convolution kernel to map the result of deep convolution from CI channels to CO channels, requiring only CI×CO parameters. Deeply separable convolution reduces the computational complexity of the convolution operation by splitting it into two steps and using fewer parameters, which results in a lightweight and efficient model. In addition, it also works better with fewer data because the depth-separable convolution reduces the possibility of overfitting. However, it should be noted that the performance of the super resolution neural network will decline sharply if all convolutions are replaced by depth separable filters. Therefore, we propose the DWSR framework: the MoBRUN algorithm combined with a depthwise separable filter is used to optimize the super-resolution neural network, and the number of (14) D F ×D F ×C O ×C I ×D K ×D K +C I ×D F ×D F D F ×D F ×C I ×D K ×D K ×C O = 1 C O + 1 D K 2 DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 9 of 21 23 parameters and PSNR are evaluated to obtain the optimized super-resolution neural network architecture. Figure4 shows the architecture of the DWSR framework. The DWSR uses the MoBRUN algorithm to determine the position corresponding to the DW convolution in the neural network. MoBRUN uses 0 to indicate the corresponding position using regular convolution and 1 to indicate the corresponding position using the depthwise separable filter. This approach allows adaptive solution space Fig. 2 The depthwise convolution operation Fig. 3 The pointwise convolution operation Fig. 4 The proposed DWSR framework S.-C.Chu et al. 1 3 23 Page 16 of 21 utilized. When the archive size is set to 15, the distribution scheme is more even and has a richer PSNR distribution than when the archive size is set to 30. In order to strike a balance between performance and efficiency, we select an archive size of 15 for the remainder of the experiments. Fig. 7 Visual results of the BI degradation model using a scale factor of ×4 Table 8 Ablation study on Archive size design Archive Size Number of archives retained Max PSNR Min PSNR Average PSNR 5 5 30.11 29.92 29.98 15 15 30.49 29.85 30.23 30 18 30.50 29.91 30.25 DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 17 of 21 23 6.5.2 Design choices foriteration number Table9 presents the impact of the number of iterative searches on the final performance of the prediction model. There is a positive correlation between the number of iterations and the model’s final performance. When the number of iterations is small, the final performance of the model cannot be accurately evaluated, even though the search time is reduced accordingly. As the number of iterations increases beyond 2500, the improvement in evaluation gain gradually becomes saturated. To strike a balance between search time and evaluation accuracy, we set the number of search iterations to 2500 for the remainder of the experiments to obtain an accurate evaluation within a relatively short search time. 6.5.3 Design choices formeta‑heuristic algorithm To ensure a fair comparison, we evaluated the MoBRUN algorithm against several classical meta-heuristics over 40 iterations. The results of this comparison, in terms of PSNR and the number of archives, are presented in Table10. It is evident that MoBPSO produces only 9 archives, with a resulting network architecture that has a low PSNR. MoBDE experiences similar difficulties. While MoBMPA, MoBGWO, and MoBSMA select a larger number of archives, they suffer from a lack of diversity and tend to be densely concentrated in certain ranges. By contrast, the MoBRUN algorithm is able to deliver better solutions. 7 Conclusion Fine-tuning DW convolution has always been a challenge for obtaining satisfactory CNN network architectures. This is primarily due to the high cost of trial and error involved in the process. To overcome this obstacle, it is necessary to speed up the network search and reduce the cost required to evaluate the network. In this paper, we propose the DWSR framework, which introduces a meta-heuristic algorithm to accelerate the network architecture search. The multi-objective mechanism provides multiple network structure choices, and the network architecture search is accelerated by changing the patch size rather than reducing the number of iterations. The DWSR framework with these mechanisms minimizes the impact on the network performance while obtaining fewer parameters, and there is a substantial improvement in network search speed. Our work suggests: Developing suitable variants of meta-heuristic algorithms is a potential direction for optimizing super-resolution networks. Table 9 Ablation study on iteration number design DW Blocks Iterations 500 1000 1500 2000 2500 3000 200k 97 7.86 18.09 20.87 23.55 23.99 24.95 29.85 87 7.01 18.56 21.56 23.15 24.35 25.04 30.09 80 9.23 18.58 21.42 23.98 24.36 25.28 30.17 62 8.15 18.02 22.41 23.66 24.5 25.34 30.34 30 8.16 19.31 22.38 24.47 24.62 25.4 30.47 S.-C.Chu et al. 1 3 23 Page 18 of 21 Author contributions S-CC, Z-CD, and J-SP wrote the main manuscript text and LK, VS, and JW revised the content critically. All authors reviewed the manuscript. Funding N/A. No funding to declare. Declarations Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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Eng Appl Artif Intell 114:105139 DWSR: anarchitecture optimization framework foradaptive… 1 3 Page 21 of 21 23 Authors and Affiliations Shu‑ChuanChu1· Zhi‑ChaoDou1· Jeng‑ShyangPan1,2· LingpingKong3· VáclavSnášel3· JunzoWatada4 * Jeng-Shyang Pan jengshy[email protected] Shu-Chuan Chu [email protected] Zhi-Chao Dou [email protected] Lingping Kong lingping_k[email protected] Václav Snášel vaclav[email protected] Junzo Watada junzuo.w[email protected] 1 College ofComputer Science andEngineering, Shandong University ofScience andTechnology, Qingdao266590, Shandong, China 2 Department ofInformation Managementy, Chaoyang University ofTechnology, Taichung41349, Taiwan 3 Faculty ofElectrical Engineering andComputer Science, VšB-Technical University ofOstrava, Ostrava, CzechRepublic 4 Graduate School ofInformation, Production andSystems, Waseda University, Kitakyushu808-0135, Japan Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.