Dropout training for SVMs with data augmentation

Abstract

Dropout and other feature noising schemes have shown promise in controlling over-fitting by artificially corrupting the training data. Though extensive studies have been performed for generalized linear models, little has been done for support vector machines (SVMs), one of the most successful approaches for supervised learning. This paper presents dropout training for both linear SVMs and the nonlinear extension with latent representation learning. For linear SVMs, to deal with the intractable expectation of the non-smooth hinge loss under corrupting distributions, we develop an iteratively re-weighted least square (IRLS) algorithm by exploring data augmentation techniques. Our algorithm iteratively minimizes the expectation of a reweighted least square problem, where the re-weights are analytically updated. For nonlinear latent SVMs, we consider learning one layer of latent representations in SVMs and extend the data augmentation technique in conjunction with first-order Taylor-expansion to deal with the intractable expected hinge loss and the nonlinearity of latent representations. Finally, we apply the similar data augmentation ideas to develop a new IRLS algorithm for the expected logistic loss under corrupting distributions, and we further develop a non-linear extension of logistic regression by incorporating one layer of latent representations. Our algorithms offer insights on the connection and difference between the hinge loss and logistic loss in dropout training. Empirical results on several real datasets demonstrate the effectiveness of dropout training on significantly boosting the classification accuracy of both linear and nonlinear SVMs.

Keywords

dropout / SVMs / logistic regression / data augmentation / iteratively reweighted least square

References

[1] Srivastava N, Hinton G, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 2014, 15: 1929–1958

[2] Wager S, Wang S, Liang P. Dropout training as adaptive regularization. In: Proceedings of the 26th International Conference on Neural Information Processing Systems. 2013

[3] Maaten L V, Chen M, Tyree S, Weinberger K Q. Learning with marginalized corrupted features. In: Proceedings of the 30th International Conference on Machine Learning. 2013, 410–418

[4] Wang S, Wang M Q, Wager S, Liang P, Manning C D. Feature noising for log-linear structured prediction. In: Proceedings of Conference on Empirical Methods on Natural Language Processing. 2013, 1170–1179

[5] Wang S, Manning C. Fast dropout training. In: Proceedings of the 30th International Conference on Machine Learning. 2013, 777–785

[6] Wang H, Shi X J, Yeung D Y. Relational stacked denoising autoencoder for tag recommendation. In: Proceedings of the 29th AAAI Conference on Artificial Intelligence. 2015, 3052–3058

[7] Vapnik V. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995

[8] Burges C J C, Scholkopf B. Improving the accuracy and speed of support vector machines. In: Proceedings of Advances in Neural Information Processing Systems. 1997, 375–381

[9] Globerson A, Roweis S. Nightmare at test time: robust learning by feature deletion. In: Proceedings of the 23rd International Conference on Machine Learning. 2006, 353–360

[10] Dekel O, Shamir O. Learning to classify with missing and corrupted features. In: Proceedings of the 25th International Conference on Machine Learning. 2008, 149–178

[11] Teo C H, Globerson A, Roweis S T, Smola A K. Convex learning with invariances. In: Proceedings of Advances in Neural Information Processing Systems. 2008, 1489–1496

[12] Polson N G, Scott S L. Data augmentation for support vector machines. Bayesian Analysis, 2011, 6(1): 1–24

[13] Polson N G, Scott J G, Windle J. Bayesian inference for logistic models using Polya-Gamma latent variables. Journal of the American Statistical Association, 2013, 108(504): 1339–1349

[14] Rosasco L, De Vito E, Caponnetto A, Piana M, Verri A. Are loss functions all the same? Neural Computation, 2004, 16(5): 1063–1076

[15] Globerson A, Koo T Y, Carreras X, Collins M. Exponentiated gradient algorithms for log-linear structured prediction. In: Proceedings of the 24th International Conference on Machine Learning. 2007, 305–312

[16] Baldi P, Sadowski P. The dropout learning algorithm. Artificial Intelligence, 2014, 210(5): 78–122

[17] Srivastava N, Hinton G E, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 2014, 15: 1929–1958

[18] Srivastava N. Improving neural networks with dropout. Dissertation for the Master Degree. Toronto: University of Toronto, 2013

[19] Huang G, Song S J, Gupta J N D, Wu C. Semi-supervised and unsupervised extreme learning machines. IEEE Transactions on Cybernetics, 2014, 44(12): 2405–2417

[20] Van Erven T, Kotlowski W, Warmuth M K. Follow the leader with dropout perturbations. Proceedings of Machine Learning Research, 2014, 35: 949–974

[21] Xu P Y, Sarikaya R. Targeted feature dropout for robust slot filling in natural language understanding. In: Proceedings of the 15th Annual Conference of the International Speech Communication Association. 2014, 258–262

[22] Rashmi R K, Gilad-Bachrach R. Dart: dropouts meet multiple additive regression trees. In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics. 2015, 489–497

[23] Chen M M, Xu Z X, Weinberger K, Sha F. Marginalized denoising autoencoders for domain adaptation. In: Proceedings of International Conference on Machine Learning. 2012, 767–774

[24] Chen M M, Weinberger K, Sha F, Bengio Y. Marginalized denoising autoencoders for nonlinear representation. In: Proceedings of the 31st International Conference on Machine Learning. 2014, 3342–3350

[25] Chen Z, Chen M M, Weinberger K Q, Zhang W X. Marginalized denoising for link prediction and multi-label learning. In: Proceedings of the 29th AAAI Conference on Artificial Intelligence. 2015, 1707–1713

[26] Chen Z, Zhang W X. A marginalized denoising method for link prediction in relational data. In: Proceedings of the SIAM International Conference on Data Mining. 2014, 298–306

[27] Chen M M, Zheng A, Weinberger K. Fast image tagging. In: Proceedings of International Conference on Machine Learning. 2013, 2311–2319

[28] Qian Q, Hu J H, Jin R, Pei J, Zhu S H. Distance metric learning using dropout: a structured regularization approach. In: Proceedings of the 20th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2014, 323–332

[29] Wager S, Fithian W, Wang S, Liang P S. Altitude training: strong bounds for single-layer dropout. In: Proceedings of Advances in Neural Information Processing Systems. 2014, 100–108

[30] Bachman P, Alsharif O, Precup D. Learning with pseudo-ensembles. In: Proceedings of Advances in Neural Information Processing Systems. 2014, 3365–3373

[31] Helmbold D P, Long P M. On the inductive bias of dropout. Journal of Machine Learning Research, 2015, 16: 3403–3454

[32] Maeda S. A Bayesian encourages dropout. 2014, arXiv:1412.7003v3

[33] Gal Y, Ghahramani Z. Dropout as a Bayesian approximation: representing model uncertainty in deep learning. In: Proceedings of International Conference on Machine Learning. 2016, 1651–1660

[34] Chen N, Zhu J, Chen J F, Zhang B. Dropout training for support vector machines. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence. 2014, 1752–1759

[35] Vincent P, Larochelle H, Bengio Y, Manzagol P A. Extracting and composing robust features with denoising autoencoders. In: Proceedings of the 25th International Conference on Machine Learning. 2008, 1096–1103

[36] Saul L K, Jaakkola T, Jordan M I. Mean field theory for sigmoid belief networks. Journal of Artificial Intelligence Research, 1996, 4: 61–76

[37] Zhu J, Chen N, Perkins H, Zhang B. Gibbs max-margin topic models with data augmentation. Journal of Machine Learning Research, 2014, 15: 1073–1110

[38] Devroye L. Non-Uniform Random Variate Generation. New York: Springer-Verlag, 1986

[39] Liu D C, Nocedal J. On the limited memory BFGS method for large scale optimization. Mathematical Programming, 1989, 45(3): 503–528

[40] Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. New York: Springer, 2009

[41] Bengio Y, Courville A, Vincent P. Representation learning: a review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013, 35(8): 1798–1828

[42] Guo J, Che W X, Yarowsky D, Wang H F, Liu T. A distributed representation-based framework for cross-lingual transfer parsing. Journal of Artificial Intelligence Research, 2016, 55: 995–1023

[43] Smola A J, Scholkopf B. A tutorial on support vector regression. Statistics and Computing, 2003, 14(3): 199–222

[44] Chen N, Zhu J, Xia F, Zhang B. Generalized relational topic models with data augmentation. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence. 2013, 1273–1279

[45] Blitzer J, Dredze M, Pereira F. Biographies, bollywood, boom-boxes and blenders: domain adaptation for sentiment classification. In: Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics. 2007, 440–447

[46] Torralba A, Fergus R, Freeman W. A large dataset for non-parametric object and scene recognition. IEEE Transaction on Pattern Analysis and Machine Intelligence, 2008, 30(11): 1958–1970

[47] Krizhevsky A. Learning multiple layers of features from tiny images. Technical Report. 2009

[48] Zhu J, Xing E P. Conditional topic random fields. In: Proceedings of International Conference on Machine Learning. 2010, 1239–1246

[49] Rifkin R, Klautau A. In defense of one-vs-all classification. Journal of Machine Learning Research, 2004, (5): 101–141

[50] Blei D, McAuliffe J D. Supervised topic models. In: Proceedings of Advances in Neural Information Processing Systems. 2007

[51] Tang Y. Deep learning with linear support vector machines. In: Proceedings of ICML workshop on Representational Learning. 2013

[52] Kingma D P, Welling M. Efficient gradient-based inference through transformations between bayes nets and neural nets. In: Proceedings of International Conference on Machine Learning. 2014, 3791–3799

[53] Bacon P L, Bengio E, Pineau J, Precup D. Conditional computation in neural networks using a decision-theoretic approach. In: Proceedings of the 2nd Multidisciplinary Conference on Reinforcement Learning and Decision Making. 2015

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature