1 Introduction
Deep convolutional neural networks (CNNs) have recently achieved great success in many visual recognition tasks. Many works have been proposed to compress large CNNs or directly learn more efficient CNN models for fast inference. However, the eltwise layer’s eltwise operation largely limits the network pruning ability. It is one of the important problems that need to be solved and also the purpose of this research.
2 Related Work
Among deep neural network model compression and acceleration, parametric pruning is one of the commonly used methods. It achieves model compression by removing redundant and less informed weights from the trained network [Yu et al.(2017)Yu, Wang, Pan, and Tao]. It is robust to various settings, can achieve good performance, and support both training from scratch and pretrained models [Yu et al.(2017)Yu, Wang, Pan, and Tao]
. Srinivas and Babu had proposed a neuronoriented pruning method by studying the redundancy between neurons
[Srinivas and Babu(2015)]. Han et alhad proposed the classic trainprunefinetuning threestep pruning method, which used L2norm for neuronlevel pruning [Han et al.(2015)Han, Pool, Tran, and Dally]. Wen et alhad proposed a Structural Sparse Learning (SSL) model which enforced group lasso constraints on multiple levels CNN structures like filter and channel before performing sparse learning and pruning [Wen et al.(2016)Wen, Wu, Wang, Chen, and Li]. Li et alused L1norm for filterlevel pruning, they pointed out that the L1norm and L2norm mechanisms are different while the effects are similar [Hao et al.(2016)Hao, Kadav, Durdanovic, Samet, and Graf].CNN network with branch structure usually brings better performance, and convolutioneltwise is a commonly used module in branched CNN. Due to the specific operation of the eltwise layer, all its previous convolutional layers should vote for which filters by index should be pruned. Since only an intersection of the votes is pruned, the pruning ability is limited. One typical branched CNN is the ResNet [He et al.(2016)He, Zhang, Ren, and Sun]. Many pruning studies use the ResNet as an example to verify the effectiveness of their pruning methodology. One consensus has been reached that it is difficult to deeply prune ResNet [Luo and Wu(2017), Ullrich et al.(2017)Ullrich, Meeds, and Welling]. There are many works that have studied ResNet pruning. Wu et alproposed the SqueenzeDet [Wu et al.(2017)Wu, Iandola, Jin, and Keutzer]. Huang et alproposed a random depth network pruning method [Huang et al.(2016)Huang, Sun, Liu, Sedra, and Weinberger]. Ullrich et alproposed a simple regularization method based on soft weightsharing, which included both quantization and pruning in one simple (re)training procedure [Ullrich et al.(2017)Ullrich, Meeds, and Welling]. Wang et alused successive incremental coefficients for group lasso items in different fine tuning iterations [Wang et al.(2018)Wang, Zhang, Wang, and Hu]. The method addressed the problem that ResNet’s initial pruning cannot withstand large penalty items. Ding et alproposed the AFP method which divided different filters of the same convolutional layer into two groups named reserved/removed [Ding et al.(2018)Ding, Ding, Han, and Tang]. AFP assigned different coefficients on L1norm of filters during training and later pruned the removed group. Our method assigns different coefficients for different group lasso items, and it is similar to [Wang et al.(2018)Wang, Zhang, Wang, and Hu] and [Ding et al.(2018)Ding, Ding, Han, and Tang]. The differences lay in the problem to be solved and the design of coefficient setting. Our method assigns coefficients using an index related function for each filter of the same convolutional layer in one finetuning round, and its purpose is to increase the compression rate of CNNs having convolutioneltwise modules like ResNet.
3 Problem Definition
Following the definition of group lasso constraints on CNN in [Wen et al.(2016)Wen, Wu, Wang, Chen, and Li], set as the kth filter, as the loss on data, as nonstructured regularization (e.g. l2norm), as filterwise group lasso. and are empirical coefficients on corresponding items. The generic optimization target is shown as formula 1. is the square root of the sum of jth weight in filter k of layer l, as shown in formula 2. Group lasso can effectively zero out all weights in groups, thus filterwise group lasso would zero out a whole filter and then the filter is removed from the convolution layer. This work also uses formula 2 as the measure of filter activation.
Here we further assume that one CNN contains at least one eltwise layer (ELT) and an ELT has convolution layers () as its direct/indirect bottom layers. In a usual pruning scenario, convolutional layers are pruned separately. Each of these convolutional layers has filters, indexed from to . For , the set of the indices of the pruned filters is named as .
For pruning filters of a convolutional layer before ELT, all should vote for which filters by index to be pruned. Two voting strategies exist: 1) prune the intersection as or 2) prune the union as . The former can better preserve the precision, while it is difficult to reach a high compression rate; the latter can improve the clipping amount better, while the precision drop is large and hard to restore as many important activated filters are removed.
(a)  (b) 
As shown in Figure 1 (A), as Conv1 will pruning filter , Conv2 prefers pruning . Strategy 1 will pruning nothing while strategy 2 will loss all its precision. The disadvantage is brought by filter randomly activation. If filters of each Conv are incremental activated, the problem would be solved. As shown in Figure 1(B), if Conv1 prefers prune filter and Conv2 prefers pruning of . Combining strategy 1, are pruned, the compression rate is as while as the max precision is kept. Combining strategy 2, are pruned, the compression rate is as high as .
4 Methodology
To solve the problem brought by randomly activated filters, this work proposes a method to replace random filter activation by controlled filter activation through indexwise filter weighting, called DirectedWeighting Group Lasso (DWGL). It assigns a small coefficient on 1index filter’s item , and gradually increase the coefficient until the last. Notion that although trends in either direction is allowed, the work uses an increment coefficient as default. The coefficient setting was outputted by a coefficient function , which obeys 5 basic rules as detailed in methodology section. The modified formula is shown as formula 3. For simplify, this work named the Layer L’ sum of filterwise group lasso as in formula 4.
The design of requires the following five principles.

Directed Trend. Function should be incremental with filter index , which should lead to an decremented activation.

Exponential growth. To guarantee the decremented trend of filter activation, value of should increase exponentially with index .

Magnitude invariance. Item should keep its magnitude although the coefficient is forced.

Fix magnitude ratio. Experience shows that the most efficient value domain of is , when , the filter tends to be heavily suppressed; when , the filter tends to be loosely released. The coefficient should better adapts to this range.

Scale invariance. The range of is not affected by the range of .
Based on above 5 rules, the coefficient function is designed as formula 5. In formula 5, the softmax style formula meets principle 1, 2 and 3. The value 9.22 scales the range of , which better fits principle 4. The ratio guarantees the principle 5.
The pruning process is consistent with the way provided by [Ding et al.(2018)Ding, Ding, Han, and Tang], and here will not be described again.
5 Experiments and Result Discussion
In this section, the work first analyzes the actual impact of DWGL on one single convolutional layer, which proves that DWGL can force controlled decremental activation. Secondly, it verifies its effect by applying the Resnet56 and Resnet20 model on CIFAR10 dataset and the Resnet50 model on ImageNet dataset respectively, which proves that DWGL can release speed acceleration and good final performance.
To facilitate comparison, this work uses the same experiment configuration as those mention in [Wang et al.(2018)Wang, Zhang, Wang, and Hu, Ding et al.(2018)Ding, Ding, Han, and Tang]
. All of the following experiments use Caffe as the experimental framework. Each experiment use four Nvidia P40 GPU as the training and testing hardware.
5.1 Effect of DWGL on Convolution
This article compares the differences between the original ResNet, ResNet using group lasso, and ResNet using DWGL. Among them, the original ResNet only uses L2norm to constrain the global weights; ResNet using group lasso apply filter level group lasso. Experiments in this section were carried out on the ResNet50 network. In figure 2 and 3, the horizontal axis represents the filter index of the convolutional layer, and the vertical axis represents the filters’ activation value (formula 2). As experiment result shown in Figure 2, filter activations of the original ResNet convolutional layer is high in median 1.88, low in max activated 3.77, low in variance 0.71 and rather randomized. The group lasso has the lowest median activation (1.07E4), high in max activated 6.68, high in variance 3.06. For DWGL, the division between the activated/nonactivated filters is obvious, showing a downward trend. Split by the overall average value, the number of activated filter is 132, median 3.00, variance 2.31, while the overall average activation 1.54, median 0.43.
This paper also compares the activation pattern of conv layers with 64/512/2048 filters. As experiment result shown in Figure 3, regardless of the number of filters, the activation exhibits the same activation pattern. The low indexed filters are activated while the high indexed filters are suppressed.
The above experimental results are consistent with the design, which show DWGL produces two expected effects, 1. force the low indexed filters to be activated and the high indexed filters to be suppressed; 2. The number of filters would not affect the activation pattern.
5.2 Resnet56 / Resnet20 on CIFAR10 dataset
The work retrain the ResNet56 model accuracy to 92.99% and retrain the ResNet20 model accuracy to 90.9% on CIFAR10 dataset as the baseline, and execute DWGL on them. The pruning result show that, DWGL reach a high compression rate of 74.64% with an increase error of only 0.94%. Compared with the AFP [Ding et al.(2018)Ding, Ding, Han, and Tang] method, DWGL increase compression rate by 14.48%, while causing less increased error by 0.06%. At the same time, DWGL shows more potential, when the pruning reached an increased error of 1.05%, the compression rate could be as much as 80%.
Comparison of DWGL and SSL [Wen et al.(2016)Wen, Wu, Wang, Chen, and Li] is not intuitive enough, since which convolution layer was pruned and the compression rate were not reported in their work. As SSL prune 20 layers to 14 within an increased error of about 1.0%, DWGL reached a compression rate 52.06% with an increased error of about 0.72%.
Method  Models  Compression Rate  Increase Error% 

SSL  Resnet20  20 to 16 layers  0.6 
SSL  Resnet20  20 to 14 layers  1.0 
ours  Resnet20  52.06%  0.72 
AFP  Resnet56  60.86%  0.99 
ours  Resnet56  75.34%  0.94 
ours  Resnet56  80%  1.05 
5.3 ResNet50 on ImageNet
The work retrain the ResNet50 model accuracy to 91.3% as the baseline and execute DWGL on it. The pruning result show that, DWGL reach a compression rate of 53% with an increase error of only 0.6%, and the speed up is 2.23x. When reaching the 2x speed up, the increased error of method CP [He et al.(2017)He, Zhang, and Sun], SPP [Wang et al.(2017)Wang, Zhang, Wang, and Hu] and SPIR [Wang et al.(2018)Wang, Zhang, Wang, and Hu] is correspondingly 1.4%, 0.8% and 0.1%. A further experiment show that the proposed DWGL model can reach a compression rate as much as 75% and the corresponding speed up is 4x at a cost of increased error of only 1.2%.
Method  Models  Compression Rate  Speedup Rates  Increase Error% 

CP  Resnet50    2x  1.4 
SPP  Resnet50    2x  0.8 
SPIR  Resnet50    2x  0.1 
ours  Resnet50  53%  2.23x  0.6 
ours  Resnet50  75%  4x  1.2 
6 Conclusions and Future Work
This paper proposes directedweighting group lasso method for pruning of eltwise blocked CNNs, called DWGL. DWGL solves the pruning problem that brought by the random filter activation and eltwise operation in convolutioneltwise modules through controlled filter activation. Although reported only on ResNet, this method is widely applicable to deep CNN networks with eltwise layers. This method can raise the compression rate with little/no increased error. The method never refuses to combine other accelerating method like quantization, so the pruning performance may be further boosted. The weakness of DWGL is that, like all other parameter pruning methods, DWGL requires iteratively pruning and finetuning, which usually take a longer time. In the future, this issue needs continuous improvement. Since DWGL is effective and robust, how to apply its basic idea to nonCNN networks could also be further researched.
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