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A novel interpolation consistency for bad generative adversarial networks (IC-BGAN)

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Abstract

Semi-supervised learning techniques utilize both labeled and unlabeled images to enhance classification performance in scenarios where labeled images are limited. However, challenges such as integrating unlabeled images with incorrect pseudo-labels, determining appropriate thresholds for the pseudo-labels, and label prediction fluctuations on low-confidence unlabeled images, hinder the effectiveness of existing methods. This research introduces a novel framework named Interpolation Consistency for Bad Generative Adversarial Networks (IC-BGAN) that utilizes a new loss function. The proposed model combines bad adversarial training, fusion techniques, and regularization to address the limitations of semi-supervised learning. IC-BGAN creates three types of image augmentations and label consistency regularization in interpolation of bad fake images, real and bad fake images, and unlabeled images. It demonstrates linear interpolation behavior, reducing fluctuations in predictions, improving stability, and facilitating the identification of decision boundaries in low-density areas. The regularization techniques boost the discriminative capability of the classifier and discriminator, and send a better signal to the bad generator. This improves the generalization and the generation of diverse inter-class fake images as support vectors with information near the true decision boundary, which helps to correct the pseudo-labeling of unlabeled images. The proposed approach achieves notable improvements in error rate from 2.87 to 1.47 on the Modified National Institute of Standards and Technology (MNIST) dataset, 3.59 to 3.13 on the Street View House Numbers (SVHN) dataset, and 12.13 to 9.59 on the Canadian Institute for Advanced Research, 10 classes (CIFAR-10) dataset using 1000 labeled training images. Additionally, it reduces the error rate from 22.11 to 18.40 on the CINIC-10 dataset when using 700 labeled images per class. The experiments demonstrate the IC-BGAN framework outperforms existing semi-supervised methods, providing a more accurate classification solution with smoother class label estimates, especially for low-confidence unlabeled images.

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Authors and Affiliations

  1. Department of Computer Engineering, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran

    Mohammad Saber Iraji, Jafar Tanha, Mohammad-Ali Balafar & Mohammad-Reza Feizi-Derakhshi

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  1. Mohammad Saber Iraji
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  2. Jafar Tanha
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  3. Mohammad-Ali Balafar
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  4. Mohammad-Reza Feizi-Derakhshi
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Contributions

Mohammad Saber Iraji: Writing – original draft- review & editing, Methodology, Investigation, Validation, Conceptualization. Jafar Tanha: Supervision, Conceptualization, Methodology, Writing – review & editing, Validation Mohammad-Ali Balafar: Supervision, Validation. Mohammad-Reza Feizi-Derakhshi: Supervision, Validation.

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Correspondence to Jafar Tanha.

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Appendix

Appendix

Open view of the method equations

$${\ddot{x}}_{{\text{mix}}_{\lambda }\left(ij\right)}{=\text{mix}}_{\lambda \left(ij\right)}({x}_{i},{x}_{j})=\lambda {x}_{i}+\left(1-\lambda \right) {x}_{j}$$
(22)
$${\ddot{y}}_{{\text{mix}}_{\lambda }\left(ij\right)}{=\text{mix}}_{\lambda \left(ij\right)}\left({y}_{i},{y}_{j}\right)=\lambda {y}_{i}+\left(1-\lambda \right) {y}_{j}$$
(23)

Where\({(x}_i,y_i)\sim p_{(x,y)},{(x}_j,y_j)\sim p_{(x,y)}\)are two images, corresponding labels

$${\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{real}-\text{fake}}={\text{mix}}_{\lambda (ij)}\left({x}_{i}^{l,u},{x}_{j}^{g}\right)=\lambda {x}_{i}^{l,u}+(1-\lambda ){x}_{j}^{g}=\lambda {x}_{i}^{l,u}+(1-\lambda )G\left({z}_{j}\right)$$
(24)
$${\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{real}-\text{fake}}={\text{mix}}_{\lambda (ij)}\left({y}_{i}^{l,u},{y}_{j}^{g}\right)=\lambda {y}_{i}^{l,u}+(1-\lambda ){y}_{j}^{g}=\lambda$$
(25)

Where \({x}_{i}^{l}\sim {p}_{{x}^{l}}^{\text{real}},{x}_{i}^{u}\sim {p}_{{x}^{u}}^{\text{real}} ,{x}_{j}^{g}=G\left({z}_{j}\right)\sim {p}_{{x}^{g}}^{\text{fake}}\) are input images, \({y}_{i}^{l,u}=1,{y}_{j}^{g}=0\) are corresponding labels

$${L}_{\text{discriminator}-\text{adversarial}-\text{mixup}}={L}_{\text{CE}}\left( D( {\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{real}-\text{fake}}),\lambda \right)={L}_{\text{CE}}\left( D( \lambda {x}_{i}^{l,u}+(1-\lambda )G({z}_{j})),\lambda \right)$$
(26)

where

$${\text{Cross Entropy Loss}=L}_{\text{CE}}\left(y,C\left(x\right)\right)=-\sum_{i=1}^{nc}{y}_{i }\text{log}\left({C\left(x\right)}_{i}\right) ,$$
(27)
$$\text{nc}=\text{number of class}$$
$${{L}_{1-\text{classifier}-\text{supervised}}=L}_{\text{CE}}\left(C\left({x}^{l}\right),y\right)={L}_{\text{CE}}\left({\widehat{y}}_{{x}^{l}},y\right)$$
(28)
$${L}_{2-\text{classifier}-\text{adversarial}-\text{fake}}={L}_{\text{ICE}}{(\widehat{y}}_{{x}^{g}},{\widetilde{y}}_{{x}^{g}})={L}_{\text{ICE}}\left(C\left(G\left(z\right)\right),\text{ arg max}\left(C\left(G\left(z\right)\right)\right)\right)$$
(29)
$${\text{Inverse Cross Entropy Loss}=L}_{\text{ICE}}\left(y,c\left(x\right)\right)=-\sum_{i=1}^{nc}{y}_{i }\text{log}\left(1-{c\left(x\right)}_{i}\right)$$
(30)
$$,\text{nc}=\text{number of class}$$
$${\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{fake}-\text{fake}}={\text{mix}}_{\lambda (ij)}\left({x}_{i}^{g},{x}_{j}^{g}\right)=\lambda {x}_{i}^{g}+(1-\lambda ){x}_{j}^{g}=\lambda G\left({z}_{i}\right)+(1-\lambda )G\left({z}_{j}\right)$$
(31)
$${\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{fake}-\text{fake}}={\text{mix}}_{\lambda (ij)}\left({y}_{i}^{g},{y}_{j}^{g}\right)=\lambda {y}_{i}^{g}+\left(1-\lambda \right){y}_{j}^{g}=\lambda {\widetilde{y}}_{{x}_{i}^{g}}+\left(1-\lambda \right){\widetilde{y}}_{{x}_{j}^{g}}$$
(32)

Where \({x}_{i}^{g}=G\left({z}_{i}\right)\sim {p}_{{x}^{g}}^{\text{fake}},{x}_{j}^{g}=G\left({z}_{j}\right)\sim {p}_{{x}^{g}}^{\text{fake}}\) are two fake images, \({y}_{i}^{g}={\widetilde{y}}_{{x}_{i}^{g}}=\text{arg max}\left(C\left(G\left({z}_{i}\right)\right)\right),{y}_{j}^{g}={\widetilde{y}}_{{x}_{j}^{g}}=\text{arg max}\left(C\left(G\left({z}_{j}\right)\right)\right)\) are corresponding labels by the classifier

$${L}_{3-\text{classifier}-\text{adversarial}-\text{fake}-\text{mixup}}{=L}_{\text{ICE}}\left( C( {\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{fake}-\text{fake}}),{\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{fake}-\text{fake}}\right)={L}_{\text{ICE}}\left({\widehat{y}}_{{\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{fake}-\text{fake}}} ,{\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{fake}-\text{fake}}\right)={L}_{\text{ICE}}\left( C\left( \lambda G\left({z}_{i}\right)+\left(1-\lambda \right)G\left({z}_{j}\right)\right),\left(\lambda {\widetilde{y}}_{{x}_{i}^{g}}+\left(1-\lambda \right){\widetilde{y}}_{{x}_{j}^{g}}\right)\right)={L}_{\text{ICE}}\left( C\left(input \right),\left(\lambda \text{arg max}\left(C\left(G\left({z}_{i}\right)\right)\right)+\left(1-\lambda \right)\text{ arg max}\left(C\left(G\left({z}_{j}\right)\right)\right)\right)\right)$$
(33)
$${L}_{4-\text{classifier}-\text{unsupervised}}={{L}_{\text{CE}}(\widehat{y}}_{{x}^{u}},{\widetilde{y}}_{{x}^{u}}){=L}_{\text{CE}}\left(C\left({x}^{u}\right),\text{ arg max}\left(C\left({x}^{u}\right)\right)\right)$$
(34)
$${\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{unlabeled}-\text{unlabeled}}={\text{mix}}_{\lambda (ij)}\left({x}_{i}^{u},{x}_{j}^{u}\right)=\lambda {x}_{i}^{u}+(1-\lambda ){x}_{j}^{u}$$
(35)
$${\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{unlabeled}-\text{unlabeled}}={\text{mix}}_{\lambda (ij)}\left({y}_{i}^{u},{y}_{j}^{u}\right)=\lambda {y}_{i}^{u}+\left(1-\lambda \right){y}_{j}^{u}=\lambda {\widetilde{y}}_{{x}_{i}^{u}}+\left(1-\lambda \right){\widetilde{y}}_{{x}_{j}^{u}}$$
(36)

Where \({x}_{i}^{u}\sim {p}_{{x}^{u}}^{\text{real}} ,{x}_{j}^{u}\sim {p}_{{x}^{u}}^{\text{real}}\) are two input unlabeled images, \({y}_{i}^{u}={\widetilde{y}}_{{x}_{i}^{u}}=\text{arg max}\left(C\left({x}_{i}^{u}\right)\right),{y}_{j}^{u}={\widetilde{y}}_{{x}_{j}^{u}}=\text{arg max}\left(C\left({x}_{j}^{u}\right)\right)\) are corresponding labels by the classifier

$${L}_{5-\text{classifier}-\text{unsupervised}-\text{mixup}}{=L}_{\text{CE}}\left( C( {\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{unlabeled}-\text{unlabeled}}),{\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{unlabeled}-\text{unlabeled}}\right)={L}_{\text{CE}}\left({\widehat{y}}_{{\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{unlabeled}-\text{unlabeled}}} ,{\ddot{y}}_{{\text{mix}}_{\uplambda }(ij)}^{\text{unlabeled}-\text{unlabeled}}\right)={L}_{\text{CE}}\left( C\left( \lambda {x}_{i}^{u}+(1-\lambda ){x}_{j}^{u}\right),\left(\lambda {\widetilde{y}}_{{x}_{i}^{u}}+\left(1-\lambda \right){\widetilde{y}}_{{x}_{j}^{u}}\right)\right)={L}_{\text{CE}}\left( C\left( \lambda {x}_{i}^{u}+(1-\lambda ){x}_{j}^{u}\right),\left(\lambda \text{arg max}\left(C\left({x}_{i}^{u}\right)\right)+\left(1-\lambda \right)\text{arg max}\left(C\left({x}_{j}^{u}\right)\right)\right)\right)$$
(37)
$${L}_{\text{classifier}}={L}_{\text{CE}}\left(C\left({x}^{l}\right),y\right){+L}_{\text{ICE}}\left(C\left(G\left(z\right)\right),\text{ arg max}\left(C\left(G\left(z\right)\right)\right)\right)+{L}_{\text{ICE}}\left( C\left(input\right),\left(\lambda \text{arg max}\left(C\left(G\left({z}_{i}\right)\right)\right)+\left(1-\lambda \right)\text{arg max}\left(C\left(G\left({z}_{j}\right)\right)\right)\right)\right)+{L}_{\text{CE}}\left(C\left({x}^{u}\right),\text{ arg max}\left(C\left({x}^{u}\right)\right)\right)+{L}_{\text{CE}}\left( C\left( \lambda {x}_{i}^{u}+(1-\lambda ){x}_{j}^{u}\right),\left(\lambda \text{arg max}\left(C\left({x}_{i}^{u}\right)\right)\left(1-\lambda \right)\text{arg max}\left(C\left({x}_{j}^{u}\right)\right)\right)\right)$$
(38)
$${L}_{1-\text{generator}-\text{adversarial}-\text{mixup}}=-{L}_{\text{CE}}\left( D\left( {\ddot{x}}_{{\text{mix}}_{\lambda \left(ij\right)}}^{\text{real}-\text{fake}}\right),\lambda \right)=-{L}_{\text{CE}}\left( D( \lambda {x}_{i}^{l,u}+(1-\lambda )G({z}_{j})),\lambda \right)$$
(39)
$${L}_{2-\text{generator}-\text{adversarial}-\text{fake}}={L}_{\text{CE}}{(\widehat{y}}_{{x}^{g}},{\widetilde{y}}_{{x}^{g}})={L}_{\text{CE}}\left(C\left(G\left(z\right)\right),\text{ arg max}\left(C\left(G\left(z\right)\right)\right)\right)$$
(40)
$$\begin{array}{c}{{L}_{3-\text{generator}-\text{adversarial}-\text{fake}-\text{mixup}}=L}_{\text{CE}}\left( C\left( {\ddot{x}}_{{\text{mix}}_{\lambda \left(ij\right)}}^{\text{fake}-\text{fake}}\right),{\ddot{y}}_{{\text{mix}}_{\lambda }\left(ij\right)}^{\text{fake}-\text{fake}}\right)=\\ \begin{array}{c}{L}_{\text{CE}}\left({\widehat{y}}_{{\ddot{x}}_{{\text{mix}}_{\lambda (ij)}}^{\text{fake}-\text{fake}}} ,{\ddot{y}}_{{\text{mix}}_{\lambda }(ij)}^{\text{fake}-\text{fake}}\right)={L}_{\text{CE}}\left( C\left( \lambda G\left({z}_{i}\right)+\left(1-\lambda \right)G\left({z}_{j}\right)\right),\left(\lambda {\widetilde{y}}_{{x}_{i}^{g}}+\left(1-\lambda \right){\widetilde{y}}_{{x}_{j}^{g}}\right)\right)\\ \begin{array}{c}={L}_{\text{CE}}\left( C\left(input\right),\left(\lambda \text{arg max}\left(C\left(G\left({z}_{i}\right)\right)\right)+\left(1-\lambda \right)\text{arg max}\left(C\left(G\left({z}_{j}\right)\right)\right)\right)\right)\\ \end{array}\end{array}\end{array}$$
(41)
$${L}_{\text{generator}}={-L}_{\text{CE}}\left( D( \lambda {x}_{i}^{l,u}+(1-\lambda )G({z}_{j})),\lambda \right)+{L}_{\text{CE}}\left(C\left(G\left(z\right)\right),\text{ arg max}\left(C\left(G\left(z\right)\right)\right)\right)+{L}_{\text{CE}}\left( C\left(input\right),\left(\lambda \text{arg max}\left(C\left(G\left({z}_{i}\right)\right)\right)+\left(1-\lambda \right)\text{arg max}\left(C\left(G\left({z}_{j}\right)\right)\right)\right)\right)$$
(42)

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Iraji, M.S., Tanha, J., Balafar, MA. et al. A novel interpolation consistency for bad generative adversarial networks (IC-BGAN). Multimed Tools Appl 83, 86161–86205 (2024). https://doi.org/10.1007/s11042-024-20333-5

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