MobilePrune: Neural Network Compression via ℓ_{0} Sparse Group Lasso on the Mobile System
Abstract
:1. Introduction
2. Related Work
2.1. Sparsity for Deep Learning Models
2.2. Learning Algorithms for ${\ell}_{0}$ Norm
2.3. Software & Hardware Compatibility
3. Overview
4. Methods
4.1. ${\ell}_{0}$ Sparse Group Lasso
4.2. Exact Optimization by PALM
Algorithm 1 The framework of MobilePrune Algorithm. 

4.3. Efficient Computation of Proximal Operators
4.3.1. Proximal Operator ${\pi}_{\lambda}^{\eta}(\xb7)$
Algorithm 2 Efficient calculation of ${\pi}_{\lambda}^{\eta}\left(y\right)$ 

4.3.2. Proximal Operator ${\theta}_{\beta ,\gamma}^{\alpha}\left(y\right)$
Algorithm 3 Efficient calculation of ${\theta}_{\beta ,\gamma}^{\alpha}\left(y\right)$ 

5. Experimental Setup and Results
5.1. Performance on Image Benchmarks
5.1.1. MNIST Dataset
5.1.2. CIFAR10 Dataset
5.1.3. TinyImageNet Dataset
5.2. Performance on Human Activity Recognition Benchmarks
5.2.1. Performance on the Desktop
5.2.2. Performance of Mobile Phones
5.3. Ablation Studies
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix A.1. The Convergence Analysis of Applying PALM Algorithm to Deep Learning Models
 1.
 The regularization functions ${r}_{i},i=1,\dots ,N$ are lower semicontinuous.
 2.
 The derivatives of the loss function ℓ and all activation functions ${\sigma}_{i},i=1,\dots ,N$ are bounded and Lipschitz continuous.
 3.
 The loss function ℓ, activation function ${\sigma}_{i},\forall i$, and the regularization function ${r}_{i},\forall i$ are either real analytic or semialgebraic [37], and continuous on their domains.
 1.
 $\mathsf{\Psi}\left(\mathbf{W}\right)$ is a KurdykaLojasiewicz (KL) function.
 2.
 The partial gradient ${\nabla}_{{W}_{i}}\mathbf{H},\forall i$ is Lipschitz continuous and there exist positive constants $\underline{{l}_{i}},\overline{{l}_{i}}$ such that ${c}_{i}^{k}\in (\underline{{l}_{i}},\overline{{l}_{i}}),k=1,2,\dots $.
 3.
 ${\nabla}_{\mathbf{W}}\mathbf{H}({W}_{1},\dots ,{W}_{N})$ has Lipschitz constant on any bounded set.
Algorithm A1 PALM Algorithm for Deep Learning Models 

Appendix A.2. The Lemma Used in the Proof of Theorem 1
Appendix A.3. The Algorithm for ℓ 1 Sparse Group Lasso
Algorithm A2 DNN_PALM Algorithm for ${\ell}_{1}$ norm Group Lasso 

Appendix B
Appendix B.1. Iterative Method
Appendix B.2. HyperParameter Settings
HyperParameter  LeNet300  LeNet5  VGGLike  ResNet32  VGG19  Description 

learning rate  1 × 10^{−3}  1 × 10^{−3}  1 × 10^{−3}  1 × 10^{−3}  1 × 10^{−3}  The learning rate used in retraining process 
gradient momentum  0.9  0.9  0.9  0.9  0.9  The gradient momentum used in retraining process 
weight decay  1 × 10^{−4}  1 × 10^{−5}  5 × 10^{−4}  1 × 10^{−4}  1 × 10^{−4}  The weight decay factor used in retraining process 
minibatch size  1 × 10^{2}  6 × 10^{2}  1 × 10^{3}  3 × 10^{2}  4 × 10^{2}  The number of training samples over which each SGD update is computed during the retraining process 
${\ell}_{0}$ norm factor  4 × 10^{−4}  2 × 10^{−4}  1 × 10^{−6}  1 × 10^{−8}  1 × 10^{−10}  The shrinkage coefficient for ${\ell}_{0}$ norm regularization 
channel factor    1 × 10^{−3}  1 × 10^{−3}–1 × 10^{−2} ^{1}  5 × 10^{−2}  5 × 10^{−2}  The shrinkage coefficient of channels for group Lasso 
neuron factor  2 × 10^{−4}  2 × 10^{−4}  1 × 10^{−4}  0  1 × 10^{−2}  The shrinkage coefficient of neurons for group Lasso 
filter size factor    1 × 10^{−3}  1 × 10^{−4}  1 × 10^{−4}  1 × 10^{−4}  The shrinkage coefficient of filter shapes for group Lasso 
pruning frequency (epochs/minibatches)  10  10  1  2  1  No. of epochs(LeNet)/minibatches(VGGNet/ResNet) for pruning before retraining 
retraining epochs  30  30  20  30  15  The number of retraining epochs after pruning 
iterations  74  102  63  2  66  The number of iterations for obtaining the final results 
Appendix C
Appendix C.1. Computational Efficiency
Appendix C.2. Results about the SSL
Method  Base/Pruned Accuracy (%)  Filter Size  Remaining Filters  Remaining Parameters  FLOPs (K) 

Baseline    25–500  20–50  500–25,000  2464 
SSL [24]  99.10/99.00  7–14  1–50    63.82 
MobilePrune  99.12/99.03  14–9  4–16  46–26  51.21 
Appendix C.3. Additional Ablation Studies
Penalty  Base/Pruned Accuracy (%)  Original/Remaining Parameters (K)  Pruned Architecture  Filter Size  FLOPs (K)  Sparsity (%) 

${\ell}_{0}$ norm  99.12/99.20  431/321.00  2050800500  25–500  2293.0  74.48 
Group Lasso  99.12/99.11  431/8.81  41930129  25–99  187.00  2.04 
${\ell}_{1}$ Group Lasso  99.12/99.03  431/9.98  41727182  23–99  183.83  2.32 
${\ell}_{0}$ sparse group lasso  99.12/99.11  431/2.31  51415157  16–65  113.50  1.97 
Penalty  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  Pruned Architecture  FLOPs (Mil) 

${\ell}_{0}$ norm  92.96/93.40  15/3.39  18439299229240246507504486241114428168  210.94 
Group Lasso  92.96/92.47  15/0.84  174389992131629342322885429168  78.07 
${\ell}_{1}$ Group Lasso  92.96/92.90  15/0.61  174392992292402463231481114139159161  134.35 
${\ell}_{0}$ sparse group lasso  92.96/92.94  15/0.60  174387992011858037272594368167  77.83 
Penalty  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  FLOPs (Mil)  Sparsity (%) 

${\ell}_{0}$ norm  95.29/95.68  7.42/6.74  993.11  90.84 
Group Lasso  95.29/95.30  7.42/3.43  393.09  45.95 
${\ell}_{1}$ Group Lasso  95.29/95.04  7.42/5.66  735.12  76.28 
${\ell}_{0}$ sparse group lasso  95.29/95.47  7.42/2.93  371.30  39.49 
Penalty  Test Accuracy (%)  Remaining Parameters (Mil)  Pruned Architecture  FLOPs (Mil) 

Baseline  61.56  20.12  6464128128256256256256512512512512512512512512  1592.53 
${\ell}_{0}$ norm  61.99  19.29  4564114128256256256256512511512509512512512512  1519.23 
Group Lasso  53.25  5.93  2361801281221141642532553224124622393129512  683.99 
${\ell}_{1}$ Group Lasso  53.97  0.21  2964109128254246254256510509509509512512484512  1282.82 
${\ell}_{0}$ sparse group lasso  56.27  4.05  1948571027983100179219273317341256158116512  407.37 
Appendix C.4. Additional Comparison between ℓ 0 Sparse Group Lasso and ℓ 1 Norm Sparse Group Lasso
Appendix C.5. The Effect of the Coefficient of ℓ 0 Norm Regularizer
${\mathit{\ell}}_{0}$ Penalty Coefficient  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  Pruned Architecture  FLOPs (Mil) 

1 × 10^{−4}  92.96/89.77  15/0.06  1743839916110557282415114104157  56.43 
1 × 10^{−5}  92.96/92.19  15/0.30  1643859917115575332318103264167  66.82 
1 × 10^{−6}  92.96/92.94  15/0.60  174387992011858037272594368167  77.83 
1 × 10^{−7}  92.96/92.54  15/0.74  174387992131889140262794400168  81.64 
${\mathit{\ell}}_{0}$ Penalty Coefficient  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  FLOPs (Mil)  Sparsity 

1 × 10^{−6}  95.29/95.11  7.42/2.06  330.90  27.76 
1 × 10^{−7}  95.29/95.33  7.42/2.72  369.36  36.66 
1 × 10^{−8}  95.29/95.47  7.42/2.93  77.83  39.49 
1 × 10^{−9}  95.29/95.44  7.42/3.02  372.98  40.70 
Appendix D
Appendix D.1. Har Dataset Description
Appendix D.1.1. Wisdm Dataset
Appendix D.1.2. UCIHAR Dataset
Appendix D.1.3. PAMAP2 Dataset
Appendix D.2. 1D CNN Model
Appendix D.3. Data PreProcessing
Appendix D.3.1. ReScaling and Standardization
Appendix D.3.2. Segmentation
Appendix D.3.3. KFold CrossValidation
Appendix D.4. HyperParameters Tuning
Appendix D.4.1. CrossValidation Tuning
Appendix D.4.2. Learning Rate Tuning
Dataset  Type  Value  Base/Pruned Accuracy (%)  Parameter Nonzero (%)  Parameter Remaining (%)  Node Remaining (%) 

WISDM  Fold Number  1  93.52/92.68  11.64  32.49  57.42 
2  94.88/93.70  10.03  30.35  55.08  
3  94.45/93.48  9.45  27.97  52.13  
4  94.97/94.65  9.52  28.03  53.52  
5  93.52/92.68  11.64  32.49  57.42  
Learning Rate  1.0 $\times {10}^{5}$  89.55/86.72  27.09  93.50  96.68  
5.0 $\times {10}^{5}$  92.93/84.36  9.41  40.44  64.06  
1.0 $\times {10}^{4}$  94.97/94.65  27.09  28.03  53.52  
1.5 $\times {10}^{4}$  94.96/94.88  10.38  27.26  52.54  
1.0 $\times {10}^{4}$  94.65/94.57  10.54  32.38  56.84  
UCIHAR  Fold Number  1  78.42/78.08  15.53  31.99  56.64 
2  89.89/89.28  32.49  64.29  80.27  
3  79.13/79.37  16.02  32.25  56.84  
4  78.22/78.22  18.69  40.02  63.48  
5  90.06/89.96  23.00  46.83  68.75  
Learning Rate  1.0 $\times {10}^{5}$  85.27/85.51  77.98  94.66  97.27  
5.0 $\times {10}^{5}$  89.38/89.24  16.69  85.77  92.58  
1.0 $\times {10}^{4}$  90.06/89.96  23.00  46.83  68.75  
1.5 $\times {10}^{4}$  90.94/90.91  16.69  31.04  56.45  
2.0 $\times {10}^{4}$  90.40/90.43  13.24  29.10  54.10  
PAMAP2  Fold Number  1  96.89/96.95  1.26  3.72  10.74 
2  92.29/92.28  1.27  3.15  10.35  
3  96.49/96.28  1.81  4.74  14.84  
4  95.08/94.99  1.20  3.42  10.55  
5  94.81/94.81  1.46  3.71  11.52  
Learning Rate  1.0 $\times {10}^{5}$  93.63/85.80  7.93  28.61  49.22  
5.0 $\times {10}^{5}$  94.25/93.89  3.90  11.81  28.32  
1.0 $\times {10}^{4}$  96.89/96.95  1.26  3.72  10.74  
1.5 $\times {10}^{4}$  96.57/96.62  1.12  2.36  7.62  
2.0 $\times {10}^{4}$  94.89/94.99  0.68  2.02  7.62 
Appendix D.4.3. Number of Epochs Tuning
Appendix D.4.4. Prune Threshold Tuning
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Dataset  Model  Methods  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  FLPOs (Mil) 

MNIST  BCGNJ [9]  98.40/98.20  267.00/28.73  28.64  
BCGHS [9]  98.40/98.20  267.00/28.17  28.09  
LeNet300100  L0 [8]  /98.60    69.27  
L0sep [8]  /98.20    26.64  
MobilePrune  98.24/98.23  267.00/5.25  25.79  
SBP [7]  /99.14    212.80  
BCGNJ [9]  99.10/99.00  431.00/3.88  282.87  
BCGHS [9]  99.10/99.00  431.00/2.59  153.38  
LeNet5  L0 [8]  /99.10    1113.40  
L0sep [8]  /99.00    390.68  
MobilePrune  99.12/99.11  431.00/2.31  113.50  
CIFAR10  Original [44]  /92.45  15.00/  313.5  
PF [32]  /93.40  15.00/5.4  206.3  
VGGlike  SBP [7]  92.80/92.50  15.00/  136.0  
SBPa [7]  92.80/91.00  15.00/  99.20  
VIBNet [39]  /93.50  15.00/0.87  86.82  
MobilePrune  92.96/92.94  15.00/0.60  77.83  
COBD [40]  95.30/95.27  7.42/2.92  488.85  
COBS [2]  95.30/95.30  7.42/3.04  378.22  
ResNet32  KronOBD [40,41]  95.30/95.30  7.42/3.26  526.17  
KronOBS [2,41]  95.30/95.46  7.42/3.23  524.52  
EigenDamage [42]  95.30/95.28  7.42/2.99  457.46  
MobilePrune  95.29/95.47  7.42/2.93  371.30  
NN slimming [33]  61.56/40.05  20.12/5.83  158.62  
COBD [40]  61.56/47.36  20.12/4.21  481.90  
COBS [2]  61.56/39.80  20.12/6.55  210.05  
TinyImageNet  VGG19  KronOBD [40,41]  61.56/44.41  20.12/4.72  298.28 
KronOBS [2,41]  61.56/44.54  20.12/5.26  266.43  
EigenDamage [42]  61.56/56.92  20.12/5.21  408.17  
MobilePrune  61.56/56.27  20.12/4.05  407.37 
Dataset  Penalty  Base/Pruned Accuracy (%)  Parameter Nonzero (%)  Parameter Remaining (%)  Node Remaining (%)  Base/Pruned Response Delay (s)  Time Saving Percentage (%) 

WISDM  ${l}_{0}$ norm  94.72/94.79  63.36  100.00  100.00  0.38/0.39  0.00 
${l}_{1}$ norm  94.30/93.84  13.58  46.26  68.16  0.38/0.24  36.84  
${l}_{2}$ norm  94.61/94.54  56.28  90.46  95.12  0.38/0.35  7.89  
Group lasso  94.68/94.32  48.23  89.73  94.73  0.38/0.35  7.89  
${l}_{1}$ sparse Group lasso  94.81/94.79  17.91  53.41  73.83  0.41/0.26  36.59  
MobilePrune  94.97/94.65  9.52  28.03  52.52  0.50/0.17  66.00  
UCIHAR  ${l}_{0}$ norm  91.52/91.48  88.49  100.00  100.00  0.84/0.80  4.76 
${l}_{1}$ norm  90.46/90.33  81.58  98.47  99.22  0.81/0.82  0.00  
${l}_{2}$ norm  91.01/90.94  88.35  100.00  100.00  0.79/0.80  0.00  
Group lasso  90.80/90.84  82.91  100.00  100.00  0.83/0.78  6.02  
${l}_{1}$ sparse Group lasso  91.11/91.04  81.21  97.70  98.83  0.84/0.80  4.76  
MobilePrune  90.06/89.96  23.00  46.83  68.75  1.01/0.43  57.43  
PAMAP2  ${l}_{0}$ norm  93.15/93.07  69.27  100.00  100.00  0.41/0.41  0.00 
${l}_{1}$ norm  95.22/95.29  1.46  7.28  19.73  0.40/0.08  80.00  
${l}_{2}$ norm  92.08/92.09  65.32  94.93  97.27  0.41/0.39  4.88  
Group lasso  93.30/93.28  61.78  100.00  100.00  0.41/0.41  0.00  
${l}_{1}$ sparse Group lasso  96.87/97.20  2.67  9.72  26.17  0.40/0.10  75.00  
MobilePrune  96.89/96.95  1.26  3.72  10.74  0.51/0.05  90.20 
Dataset  Device  Penalty  Base/Pruned Response Delay (s)  Time Saving Percentage (%)  Based/Pruned Device Estimated Battery Use (%/h)  Battery Saving Percentage (%) 

WISDM  Huawei P20  ${l}_{0}$ norm  1.40/1.27  9.29  0.71/0.70  1.41 
${l}_{1}$ norm  1.33/0.71  46.62  0.74/0.65  12.16  
${l}_{2}$ norm  1.28/1.21  5.47  0.74/0.77  0.00  
Group lasso  1.27/1.27  0.00  0.74/0.77  0.00  
${l}_{1}$ sparse Group lasso  1.25/0.81  35.20  0.74/0.68  8.11  
MobilePrune  1.34/0.51  61.94  0.72/0.45  37.50  
OnePlus 8 Pro  ${l}_{0}$ norm  0.57/0.49  14.04  0.34/0.32  5.88  
${l}_{1}$ norm  0.48/0.34  29.17  0.35/0.30  14.29  
${l}_{2}$ norm  0.48/0.40  16.67  0.34/0.34  0.00  
Group lasso  0.49/0.45  8.16  0.34/0.35  0.00  
${l}_{1}$ sparse Group lasso  0.48/0.33  31.25  0.35/0.30  14.29  
MobilePrune  0.48/0.23  52.08  0.34/0.23  32.35  
HCIHAR  Huawei P20  ${l}_{0}$ norm  1.43/1.43  0.00  0.84/0.84  0.00 
${l}_{1}$ norm  1.42/1.42  0.00  0.85/0.84  1.18  
${l}_{2}$ norm  1.43/1.43  0.00  0.84/0.84  0.00  
Group lasso  1.43/1.43  0.00  0.84/0.82  2.38  
${l}_{1}$ sparse Group lasso  1.42/1.41  0.70  0.85/0.82  3.53  
MobilePrune  1.42/0.85  40.14  0.84/0.55  34.52  
OnePlus 8 Pro  ${l}_{0}$ norm  0.53/0.53  0.00  0.35/0.35  0.00  
${l}_{1}$ norm  0.54/0.51  5.56  0.37/0.36  2.70  
${l}_{2}$ norm  0.54/0.53  1.85  0.37/0.37  0.00  
Group lasso  0.53/0.52  1.89  0.36/0.36  0.00  
${l}_{1}$ sparse Group lasso  0.53/0.52  1.89  0.36/0.36  0.00  
MobilePrune  0.54/0.42  22.22  0.36/0.29  19.44  
PAMAP2  Huawei P20  ${l}_{0}$ norm  2.64/2.72  0.00  0.76/0.79  0.00 
${l}_{1}$ norm  2.74/0.45  83.58  0.79/0.53  32.91  
${l}_{2}$ norm  2.67/2.56  4.12  0.78/0.78  0.00  
Group lasso  2.67/2.68  0.00  0.78/0.78  0.00  
${l}_{1}$ sparse Group lasso  2.69/0.55  79.55  0.79/0.57  27.85  
MobilePrune  2.70/0.32  88.15  0.79/0.50  36.71  
OnePlus 8 Pro  ${l}_{0}$ norm  0.94/0.93  1.06  0.88/0.88  0.00  
${l}_{1}$ norm  0.93/0.25  73.12  0.87/0.55  36.78  
${l}_{2}$ norm  0.93/0.91  2.15  0.88/0.87  1.14  
Group lasso  0.94/0.95  0.00  0.89/0.89  0.00  
${l}_{1}$ sparse Group lasso  0.95/0.29  69.47  0.88/0.59  32.95  
MobilePrune  0.94/0.21  77.66  0.87/0.54  37.93 
Network Model  Penalty  Base/Pruned Accuracy (%)  Original/Remaining Parameters (Mil)  FLOPs  Sparsity (%) 

LetNet300  ${\ell}_{0}$ norm  98.24/98.46  267 K/57.45 K  143.20  21.55 
Group lasso  98.24/98.17  267 K/32.06 K  39.70  12.01  
${\ell}_{1}$ sparse group lasso  98.24/98.00  267 K/15.80 K  25.88  5.93  
${\ell}_{0}$ sparse group lasso  98.24/98.23  267 K/5.25 K  25.79  1.97  
LetNet5  ${\ell}_{0}$ norm  99.12/99.20  431 K/321.0 K  2293.0  74.48 
Group lasso  99.12/99.11  431 K/8.81 K  187.00  2.04  
${\ell}_{1}$ sparse group lasso  99.12/99.03  431 K/9.98 K  183.83  2.32  
${\ell}_{0}$ sparse group lasso  99.12/99.11  431 K/2.31 K  113.50  0.54  
VGGlike  ${\ell}_{0}$ norm  92.96/93.40  15 M/3.39 M  210.94  22.6 
Group lasso  92.96/92.47  15 M/0.84 M  78.07  5.60  
${\ell}_{1}$ sparse group lasso  92.96/92.90  15 M/0.61 M  134.35  4.06  
${\ell}_{0}$ sparse group lasso  92.96/92.94  15 M/0.60 M  77.83  4.00  
ResNet32  ${\ell}_{0}$ norm  95.29/95.68  7.42 M/6.74 M  993.11  90.84 
Group lasso  95.29/95.30  7.42 M/3.03 M  373.09  40.84  
${\ell}_{1}$ sparse group lasso  95.29/95.04  7.42 M/5.66 M  735.12  76.28  
${\ell}_{0}$ sparse group lasso  95.29/95.47  7.42 M/2.93 M  371.30  39.49  
VGG19  ${\ell}_{0}$ norm  61.56/61.99  138 M/19.29 M  1519.23  13.98 
Group lasso  61.56/53.25  138 M/5.93 M  683.99  4.30  
${\ell}_{1}$ sparse group lasso  61.56/53.97  138 M/0.21 M  1282.82  0.15  
${\ell}_{0}$ sparse group lasso  61.56/56.27  138 M/4.05 M  407.37  2.93 
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Shao, Y.; Zhao, K.; Cao, Z.; Peng, Z.; Peng, X.; Li, P.; Wang, Y.; Ma, J. MobilePrune: Neural Network Compression via ℓ_{0} Sparse Group Lasso on the Mobile System. Sensors 2022, 22, 4081. https://doi.org/10.3390/s22114081
Shao Y, Zhao K, Cao Z, Peng Z, Peng X, Li P, Wang Y, Ma J. MobilePrune: Neural Network Compression via ℓ_{0} Sparse Group Lasso on the Mobile System. Sensors. 2022; 22(11):4081. https://doi.org/10.3390/s22114081
Chicago/Turabian StyleShao, Yubo, Kaikai Zhao, Zhiwen Cao, Zhehao Peng, Xingang Peng, Pan Li, Yijie Wang, and Jianzhu Ma. 2022. "MobilePrune: Neural Network Compression via ℓ_{0} Sparse Group Lasso on the Mobile System" Sensors 22, no. 11: 4081. https://doi.org/10.3390/s22114081