# Semantic and Generalized Entropy Loss Functions for Semi-Supervised Deep Learning

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## Abstract

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## 1. Introduction

- If the two analyzed regularization terms prove to be effective in semi-supervised classification tasks, which loss function provides the best results?
- What is the relation between semantic loss function and generalized entropy loss function?
- What is the impact of the input and tuning parameter values on both proposed approaches on the final results?

## 2. Preliminaries

#### 2.1. Semi-Supervised Learning

- Manifold assumption, the data lie approximately on a manifold of much lower dimension than the input space. This assumption allows the use of distances and densities which are defined on a manifold;
- Continuity assumption, the algorithm assumes that (after transformed to a lower dimension) the points which are closer to each other are more likely to have the same output label;
- Cluster assumption, (after transformed to a lower dimension) the data is divided into discrete clusters and points in the same cluster are more likely to share an output label.

#### 2.2. Deep Neural Networks

_{1}or L

_{2}penalty to add costs proportional to the size of the node weights. Regularizing the weights forces small signals (noise) to have weights almost equal to zero and allows only consistently strong signals to have relatively higher weights. More specifically, for some hyper-parameter $w$, the new overall loss becomes:

#### 2.3. Propositional Logic

## 3. Theoretical Framework of the Semantic and the Generalized Entropy Loss Functions

#### 3.1. Semantic Loss Function

#### 3.2. Generalized Entropy Loss function

#### 3.3. Relation between Generalized Entropy and Semantic Loss Functions

## 4. Research Framework and Settings

#### 4.1. Datasets Characteristics

#### 4.2. Performance Measure

#### 4.3. Numerical Implementation

#### 4.4. Tuning of the Parameters

- $Q$-value $\in \left\{1\times {10}^{-6},\text{}0.25,\text{}0.5,\text{}0.75,\text{}1+1\times {10}^{-6},\text{}1.25,\text{}1.5,\text{}1.75,\text{}2\right\}$—from the Equation (6);
- Weights $\in \left\{0.001,\text{}0.005,\text{}0.01,\text{}0.05,\text{}0.1\right\}$, which is the hyper-parameter associated with the Rényi or semantic regularization term in Equation (2);
- Batch size $\in \left\{20,\text{}50,\text{}100\right\}$,which is the mini-batch size needed for adaptive stochastic gradient descent optimization algorithm;
- Number of labeled examples $\in \left\{100,\text{}1000,\text{}50,000\right\}$, which is the number of randomly chosen labeled examples from the training set with the assumption that the final set is balanced, i.e., no particular class is overrepresented.

#### 4.5. Benchmarking Models

## 5. Empirical Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gajowniczek, K.; Orłowski, A.; Ząbkowski, T. Simulation Study on the Application of the Generalized Entropy Concept in Artificial Neural Networks. Entropy
**2018**, 20, 249. [Google Scholar] [CrossRef][Green Version] - Gajowniczek, K.; Ząbkowski, T.; Sodenkamp, M. Revealing Household Characteristics from Electricity Meter Data with Grade Analysis and Machine Learning Algorithms. Appl. Sci.
**2018**, 8, 1654. [Google Scholar] [CrossRef][Green Version] - Sadarangani, A.; Jivani, A. A survey of semi-Supervised learning. Int. J. Eng. Sci. Res. Technol.
**2016**, 5. [Google Scholar] [CrossRef] - Nafkha, R.; Gajowniczek, K.; Ząbkowski, T. Do Customers Choose Proper Tariff? Empirical Analysis Based on Polish Data Using Unsupervised Techniques. Energies
**2018**, 11, 514. [Google Scholar] [CrossRef][Green Version] - Prakash, V.J.; Nithya, D.L. A survey on semi-supervised learning techniques. arXiv
**2014**, arXiv:1402.4645. [Google Scholar] - Xu, J.; Zhang, Z.; Friedman, T.; Liang, Y.; Van den Broeck, G. A semantic loss function for deep learning with symbolic knowledge. In Proceedings of the 35th International Conference on Machine Learning (ICML), Stockholm, Sweden, 10–15 July 2018. [Google Scholar]
- Xu, J.; Zhang, Z.; Friedman, T.; Liang, Y.; Van den Broeck, G. A Semantic Loss Function for Deep Learning Under Weak Supervision. In Proceedings of the NIPS 2017 Workshop on Learning with Limited Labeled Data: Weak Supervision and Beyond, Long Beach, CA, USA, 4–9 December 2017. [Google Scholar]
- LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] - Wang, J.; Chen, Y.; Hao, S.; Peng, X.; Hu, L. Deep learning for sensor-based activity recognition: A survey. Pattern Recognit. Lett.
**2019**, 119, 3–11. [Google Scholar] [CrossRef][Green Version] - Miyato, T.; Maeda, S.-I.; Koyama, M.; Ishii, S. Virtual Adversarial Training: A Regularization Method for Supervised and Semi-Supervised Learning. IEEE Trans. Pattern Anal. Mach. Intell.
**2019**, 41, 1979–1993. [Google Scholar] [CrossRef][Green Version] - Alaya, M.Z.; Bussy, S.; Gaıffas, S.; Guilloux, A. Binarsity: A penalization for one-Hot encoded features in linear supervised learning. J. Mach. Learn. Res.
**2019**, 20, 1–34. [Google Scholar] - Krizhevsky, A.; Sutskever, I.; Hinton, G.E. ImageNet classification with deep convolutional neural networks. Commun. ACM
**2017**, 60, 84–90. [Google Scholar] [CrossRef] - Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, MA, USA, 2016. [Google Scholar]
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef][Green Version] - Rényi, A. On measures of information and entropy. In the fourth Berkeley Symposium on Mathematics, Statistics and Probability; University of California Press: Berkeley, CA, USA, 1961; pp. 547–561. [Google Scholar]
- Amigó, J.; Balogh, S.; Hernández, S. A Brief Review of Generalized Entropies. Entropy
**2018**, 20, 813. [Google Scholar] [CrossRef][Green Version] - Gajowniczek, K.; Karpio, K.; Łukasiewicz, P.; Orłowski, A.; Ząbkowski, T. Q-Entropy Approach to Selecting High Income Households. Acta Phys. Pol. A
**2015**, 127, A:38–A:44. [Google Scholar] [CrossRef] - Gajowniczek, K.; Orłowski, A.; Ząbkowski, T. Entropy Based Trees to Support Decision Making for Customer Churn Management. Acta Phys. Pol. A
**2016**, 129, 971–979. [Google Scholar] [CrossRef] - Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-Based learning applied to document recognition. Proc. IEEE
**1998**, 86, 2278–2324. [Google Scholar] [CrossRef][Green Version] - Xiao, H.; Rasul, K.; Vollgraf, R. Fashion-Mnist: A novel image dataset for benchmarking machine learning algorithms. arXiv
**2017**, arXiv:1708.07747. [Google Scholar] - Schwenker, F.; Trentin, E. Pattern classification and clustering: A review of partially supervised learning approaches. Pattern Recognit. Lett.
**2014**, 37, 4–14. [Google Scholar] [CrossRef] - van Engelen, J.E.; Hoos, H.H. A survey on semi-Supervised learning. Mach. Learn.
**2019**, 1–68. [Google Scholar] [CrossRef][Green Version] - Bengio, Y.; Lee, D.H.; Bornschein, J.; Mesnard, T.; Lin, Z. Towards biologically plausible deep learning. arXiv
**2015**, arXiv:1502.04156. [Google Scholar] - Marblestone, A.H.; Wayne, G.; Kording, K.P. Toward an Integration of Deep Learning and Neuroscience. Front. Comput. Neurosci.
**2016**, 10, 94. [Google Scholar] [CrossRef] - Najafabadi, M.M.; Villanustre, F.; Khoshgoftaar, T.M.; Seliya, N.; Wald, R.; Muharemagic, E. Deep learning applications and challenges in big data analytics. J. Big Data
**2015**, 2. [Google Scholar] [CrossRef][Green Version] - Liu, W.; Wang, Z.; Liu, X.; Zeng, N.; Liu, Y.; Alsaadi, F.E. A survey of deep neural network architectures and their applications. Neurocomputing
**2017**, 234, 11–26. [Google Scholar] [CrossRef] - Nair, V.; Hinton, G.E. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML), Haifa, Israel, 21–25 June 2010; pp. 807–814. [Google Scholar]
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Ioffe, S.; Szegedy, C. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In Proceedings of the 32nd International Conference on Machine Learning (ICML), Lille, France, 6–11 July 2015. [Google Scholar]
- Darwiche, A. SDD: A new canonical representation of propositional knowledge bases. In Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), Barcelona, Spain, 16–22 July 2011. [Google Scholar]
- Tomamichel, M.; Berta, M.; Hayashi, M. Relating different quantum generalizations of the conditional Rényi entropy. J. Math. Phys.
**2014**, 55, 082206. [Google Scholar] [CrossRef][Green Version] - Fehr, S.; Berens, S. On the Conditional Rényi Entropy. IEEE Trans. Inf. Theory
**2014**, 60, 6801–6810. [Google Scholar] [CrossRef] - The R Development Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2014. [Google Scholar]
- Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Ghemawat, S. Tensorflow: Large-Scale machine learning on heterogeneous distributed systems. arXiv
**2016**, arXiv:1603.04467. [Google Scholar] - Rasmus, A.; Berglund, M.; Honkala, M.; Valpola, H.; Raiko, T. Semi-Supervised learning with ladder networks. In Proceedings of the Neural Information Processing Systems 2015 (NIPS 2015), Montreal, QC, Canada, 7–12 December 2015; pp. 3546–3554. [Google Scholar]
- Pitelis, N.; Russell, C.; Agapito, L. Semi-Supervised learning using an unsupervised atlas. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases; Springer: Berlin/Heidelberg, Germany, 2014; pp. 565–580. [Google Scholar]
- Kingma, D.P.; Mohamed, S.; Rezende, D.J.; Welling, M. Semi-Supervised learning with deep generative models. In Proceedings of the Neural Information Processing Systems 2014 (NIPS 2014), Montreal, QC, Canada, 8–11 December 2014; pp. 3581–3589. [Google Scholar]
- He, K.; Zhang, X.; Ren, S.; Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, Las Vegas, NV, USA, 26 June–1 July 2016; pp. 770–778. [Google Scholar]

**Figure 6.**Accuracy on the test set of the MNIST dataset in terms of different combinations of the tuned parameters.

**Figure 7.**Accuracy on the test set of the Fashion-MNIST dataset in terms of different combinations of the tuned parameters.

Predicted Value | |||||
---|---|---|---|---|---|

Class 1 | Class 2 | $\cdots $ | Class k | ||

Real value | Class 1 | True_{1} | False_{1} | $\cdots $ | False_{1} |

Class 2 | False_{2} | True_{2} | $\cdots $ | False_{2} | |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | |

Class k | False_{k} | False_{k} | $\cdots $ | True_{k} |

**Table 2.**Comparison of the best results for Rényi and semantic losses, along with benchmark models on the MNIST dataset.

Sample Size | Loss | $\mathit{Q}$-Value | Weight | Batch Size | Mean Validation Accuracy | Mean Test Accuracy |
---|---|---|---|---|---|---|

100 | Semantic | 0.005 | 100 | 98.02 (∓0.04) | 97.98 (∓0.04) | |

Rényi | 0.75 | 0.100 | 100 | 98.20 (∓0.02) | 98.21 (∓0.03) | |

MLP | 100 | 78.46 (∓1.94) | ||||

AtlasRBF | 91.9 (∓0.95) | |||||

Deep Generative | 96.67 (∓0.14) | |||||

Virtual Adversarial | 97.67 | |||||

Ladder Net | 98.94 (∓0.37) | |||||

1000 | Semantic | 0.100 | 100 | 97.95 (∓0.05) | 98.02 (∓0.03) | |

Rényi | 0.50 | 0.100 | 100 | 98.27 (∓0.03) | 98.25 (∓0.03) | |

MLP | 100 | 94.26 (∓0.31) | ||||

AtlasRBF | 96.32 (∓0.12) | |||||

Deep Generative | 97.60 (∓0.02) | |||||

Virtual Adversarial | 98.64 | |||||

Ladder Net | 99.16 (∓0.08) | |||||

50,000 | Semantic | 0.100 | 100 | 98.13 (∓0.03) | 98.15 (∓0.04) | |

Rényi | 0.50 | 0.100 | 100 | 98.29 (∓0.02) | 98.29 (∓0.03) | |

MLP | 100 | 98.13 (∓0.04) | ||||

AtlasRBF | 98.69 | |||||

Deep Generative | 99.04 | |||||

Virtual Adversarial | 99.36 | |||||

Ladder Net | 99.43 (∓0.02) | |||||

ResNet | 99.40 |

**Table 3.**Comparison of the best results for Rényi and semantic losses, along with benchmark models on the Fashion-MNIST dataset.

Sample Size | Loss | $\mathit{Q}$-Value | Weight | Batch Size | Mean Validation Accuracy | Mean Test Accuracy |
---|---|---|---|---|---|---|

100 | Semantic | 0.005 | 100 | 88.65 (∓0.11) | 87.65 (∓0.07) | |

Rényi | 0.50 | 0.100 | 100 | 89.62 (∓0.07) | 88.89 (∓0.10) | |

MLP | 100 | 69.45 (∓2.03) | ||||

Ladder Net | 81.46 (∓0.64) | |||||

1000 | Semantic | 0.100 | 100 | 88.71 (∓0.06) | 87.83 (∓0.07) | |

Rényi | 0.75 | 0.100 | 100 | 89.54 (∓0.35) | 88.80 (∓0.08) | |

MLP | 100 | 78.12 (∓1.41) | ||||

Ladder Net | 86.48 (∓0.15) | |||||

50,000 | Semantic | 0.100 | 100 | 89.26 (∓0.08) | 88.49 (∓0.10) | |

Rényi | 0.50 | 0.100 | 100 | 89.90 (∓0.06) | 89.03 (∓0.06) | |

MLP | 100 | 88.26 (∓0.18) | ||||

Ladder Net | 90.46 | |||||

ResNet | 92.00 |

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**MDPI and ACS Style**

Gajowniczek, K.; Liang, Y.; Friedman, T.; Ząbkowski, T.; Van den Broeck, G. Semantic and Generalized Entropy Loss Functions for Semi-Supervised Deep Learning. *Entropy* **2020**, *22*, 334.
https://doi.org/10.3390/e22030334

**AMA Style**

Gajowniczek K, Liang Y, Friedman T, Ząbkowski T, Van den Broeck G. Semantic and Generalized Entropy Loss Functions for Semi-Supervised Deep Learning. *Entropy*. 2020; 22(3):334.
https://doi.org/10.3390/e22030334

**Chicago/Turabian Style**

Gajowniczek, Krzysztof, Yitao Liang, Tal Friedman, Tomasz Ząbkowski, and Guy Van den Broeck. 2020. "Semantic and Generalized Entropy Loss Functions for Semi-Supervised Deep Learning" *Entropy* 22, no. 3: 334.
https://doi.org/10.3390/e22030334