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August  2020, 19(8): 4191-4212. doi: 10.3934/cpaa.2020188

Stochastic AUC optimization with general loss

1. 

Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY 12206, USA

2. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong, China

3. 

Department of Mathematics, The University of Hong Kong, Hong Kong, China

* Corresponding author

Received  October 2019 Revised  January 2020 Published  May 2020

Fund Project: This work was completed when Wei Shen was a visiting student at SUNY Albany. Yiming Ying is supported by the National Science Foundation (NSF, Grant IIS1816227)

Recently, there is considerable work on developing efficient stochastic optimization algorithms for AUC maximization. However, most of them focus on the least square loss which may be not the best option in practice. The main difficulty for dealing with the general convex loss is the pairwise nonlinearity w.r.t. the sampling distribution generating the data. In this paper, we use Bernstein polynomials to uniformly approximate the general losses which are able to decouple the pairwise nonlinearity. In particular, we show that this reduction for AUC maximization with a general loss is equivalent to a weakly convex (nonconvex) min-max formulation. Then, we develop a novel SGD algorithm for AUC maximization with per-iteration cost linearly w.r.t. the data dimension, making it amenable for streaming data analysis. Despite its non-convexity, we prove its global convergence by exploring the appealing convexity-preserving property of Bernstein polynomials and the intrinsic structure of the min-max formulation. Experiments are performed to validate the effectiveness of the proposed approach.

Citation: Zhenhuan Yang, Wei Shen, Yiming Ying, Xiaoming Yuan. Stochastic AUC optimization with general loss. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4191-4212. doi: 10.3934/cpaa.2020188
References:
[1]

F. Bach and E. Moulines, Non-strongly-convex smooth stochastic approximation with convergence rate O (1/n), in Advances in Neural Information Processing Systems, (2013), 773–781. Google Scholar

[2]

A. P. Bradley, The use of the area under the ROC curve in the evaluation of machine learning algorithms, Pattern Recognit., 30 (1997), 1145-1159.  doi: 10.1016/S0031-3203(96)00142-2.  Google Scholar

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C. Cortes and M. Mohri, AUCoptimization vs. error rate minimization, in Advances in Neural Information Processing Systems, (2004), 313–320. Google Scholar

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D. Davis and D. Drusvyatskiy, Stochastic model-based minimization of weakly convex functions, SIAM J. Optim., 29 (2019), 207-239.  doi: 10.1137/18M1178244.  Google Scholar

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D. Davis and B. Grimmer, Proximally Guided Stochastic Subgradient Method for Nonsmooth, Nonconvex Problems, SIAM J. Optim., 29 (2019), 1908-1930.  doi: 10.1137/17M1151031.  Google Scholar

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T. Fawcett, An introduction to ROC analysis, Pattern Recognit. Lett., 27 (2006), 861-874.   Google Scholar

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W. Gao, R. Jin, S. Zhu and Z. H. Zhou, One-pass AUC optimization, in International Conference on Machine Learning, (2013), 906–914. Google Scholar

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W. Gao and Z. H. Zhou, On the Consistency of AUC Pairwise Optimization, in IJCAI, (2015), 939–945. Google Scholar

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J. A. Hanley and B. J. McNeil, The meaning and use of the area under a receiver operating characteristic (ROC) curve, Radiology, 143 (1982), 29-36.   Google Scholar

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A. Herschtal and B. Raskutti, Optimising area under the ROC curve using gradient descent, in Proceedings of the 21st International Conference on Machine Learning, ACM, (2004), 49. Google Scholar

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T. Joachims, A support vector method for multivariate performance measures, in Proceedings of the 22nd International Conference on Machine Learning, ACM, (2005), 377–384. Google Scholar

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P. Kar, B. Sriperumbudur, P. Jain and H. Karnick, On the generalization ability of online learning algorithms for pairwise loss functions, in International Conference on Machine Learning, (2013), 441–449. Google Scholar

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S. Lacoste-Julien, M. Schmidt and F. Bach, A simpler approach to obtaining an O (1/t) convergence rate for the projected stochastic subgradient method, preprint, arXiv: 1212.2002. doi: 10.1137/1.9781611974331.ch127.  Google Scholar

[19]

J. Lin and L. Rosasco, Optimal learning for multi-pass stochastic gradient methods, in Advances in Neural Information Processing Systems, (2016), 4556–4564.  Google Scholar

[20]

M. Liu, Z. Yuan, Y. Ying and T. Yang, Stochastic AUC Maximization with Deep Neural Networks, in International Conference on Learning Representations (ICLR), 2020. Google Scholar

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M. Liu, X. Zhang, Z. Chen, X. Wang and T. Yang, Fast stochastic AUC maximization with O (1/n)-convergence rate, in International Conference on Machine Learning, (2018), 3195–3203. Google Scholar

[22]

M. Natole, Y. Ying and S. Lyu, Stochastic proximal algorithms for AUC maximization, in International Conference on Machine Learning, (2018), 3707–3716. Google Scholar

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A. NemirovskiA. JuditskyG. Lan and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM J. Optim., 19 (2009), 1574-1609.  doi: 10.1137/070704277.  Google Scholar

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E. A. Nurminskii, The quasigradient method for the solving of the nonlinear programming problems, Cybernetics, 9 (1973), 145-150.   Google Scholar

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G. M. Phillips, Interpolation and Approximation by Polynomials, Vol. 14, Springer Science & Business Media, 2003. doi: 10.1007/b97417.  Google Scholar

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H. Rafique, M. Liu, Q. Lin and T. Yang, Non-Convex Min-Max Optimization: Provable Algorithms and Applications in Machine Learning, preprint, arXiv: 1810.02060. Google Scholar

[28]

A. Rakhlin, O. Shamir and K. Sridharan, Making gradient descent optimal for strongly convex stochastic optimization, in Proceedings of the 29th International Conference on Machine Learning, (2012), 449–456. Google Scholar

[29]

O. Shamir and T. Zhang, Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes, in International Conference on Machine Learning, (2013), 71–79. Google Scholar

[30]

N. Srebro, A. Tewari, Stochastic optimization for machine learning, CML Tutorial, (2010). Google Scholar

[31]

Y. Wang, R. Khardon, D. Pechyony and R. Jones, Generalization bounds for online learning algorithms with pairwise loss functions, in Conference on Learning Theory, (2012), 13. Google Scholar

[32]

Y. Ying, L. Wen and S. Lyu, Stochastic online AUC maximization, in Advances in Neural Information Processing Systems, 2016. Google Scholar

[33]

Y. Ying and D. X. Zhou, Online regularized classification algorithms, IEEE Trans. Inform. Theory, 52 (2006), 4775-4788.  doi: 10.1109/TIT.2006.883632.  Google Scholar

[34]

Y. Ying and D. X. Zhou, Online pairwise learning algorithms, Neural Comput., 28 (2016), 743-777.  doi: 10.1162/neco_a_00817.  Google Scholar

[35]

X. ZhangA. Saha and S. V. N. Vishwanathan, Smoothing multivariate performance measures, J. Mach. Learn. Res., 13 (2012), 3623-3680.   Google Scholar

[36]

P. Zhao, R. Jin, T. Yang and S. C. Hoi, Online AUC maximization, in Proceedings of the 28th International Conference on Machine Learning (ICML-11), 2011. Google Scholar

show all references

References:
[1]

F. Bach and E. Moulines, Non-strongly-convex smooth stochastic approximation with convergence rate O (1/n), in Advances in Neural Information Processing Systems, (2013), 773–781. Google Scholar

[2]

A. P. Bradley, The use of the area under the ROC curve in the evaluation of machine learning algorithms, Pattern Recognit., 30 (1997), 1145-1159.  doi: 10.1016/S0031-3203(96)00142-2.  Google Scholar

[3]

T. Calders and S. Jaroszewicz, Efficient AUC optimization for classification, in PKDD, Vol. 4702, Springer, (2007), 42–53. Google Scholar

[4]

C. C. Chang and C. J. Lin, LIBSVM: a library for support vector machines, ACM Trans. Intell. Syst. Technol., 2 (2011), 21 pp. doi: 10.1145/1961189.1961199.  Google Scholar

[5]

S. ClémençonG. Lugosi and N. Vayatis, Ranking and empirical minimization of U-statistics, Ann. Statist., 36 (2008), 844-874.  doi: 10.1214/009052607000000910.  Google Scholar

[6]

C. Cortes and M. Mohri, AUCoptimization vs. error rate minimization, in Advances in Neural Information Processing Systems, (2004), 313–320. Google Scholar

[7]

D. Davis and D. Drusvyatskiy, Stochastic model-based minimization of weakly convex functions, SIAM J. Optim., 29 (2019), 207-239.  doi: 10.1137/18M1178244.  Google Scholar

[8]

D. Davis and B. Grimmer, Proximally Guided Stochastic Subgradient Method for Nonsmooth, Nonconvex Problems, SIAM J. Optim., 29 (2019), 1908-1930.  doi: 10.1137/17M1151031.  Google Scholar

[9]

Dheeru, Dua and Karra Taniskidou, Efi, UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, 2017. Available from: http://archive.ics.uci.edu/ml. Google Scholar

[10]

D. Drusvyatskiy, The proximal point method revisited, preprint, arXiv: 1712.06038. Google Scholar

[11]

T. Fawcett, An introduction to ROC analysis, Pattern Recognit. Lett., 27 (2006), 861-874.   Google Scholar

[12]

W. Gao, R. Jin, S. Zhu and Z. H. Zhou, One-pass AUC optimization, in International Conference on Machine Learning, (2013), 906–914. Google Scholar

[13]

W. Gao and Z. H. Zhou, On the Consistency of AUC Pairwise Optimization, in IJCAI, (2015), 939–945. Google Scholar

[14]

J. A. Hanley and B. J. McNeil, The meaning and use of the area under a receiver operating characteristic (ROC) curve, Radiology, 143 (1982), 29-36.   Google Scholar

[15]

A. Herschtal and B. Raskutti, Optimising area under the ROC curve using gradient descent, in Proceedings of the 21st International Conference on Machine Learning, ACM, (2004), 49. Google Scholar

[16]

T. Joachims, A support vector method for multivariate performance measures, in Proceedings of the 22nd International Conference on Machine Learning, ACM, (2005), 377–384. Google Scholar

[17]

P. Kar, B. Sriperumbudur, P. Jain and H. Karnick, On the generalization ability of online learning algorithms for pairwise loss functions, in International Conference on Machine Learning, (2013), 441–449. Google Scholar

[18]

S. Lacoste-Julien, M. Schmidt and F. Bach, A simpler approach to obtaining an O (1/t) convergence rate for the projected stochastic subgradient method, preprint, arXiv: 1212.2002. doi: 10.1137/1.9781611974331.ch127.  Google Scholar

[19]

J. Lin and L. Rosasco, Optimal learning for multi-pass stochastic gradient methods, in Advances in Neural Information Processing Systems, (2016), 4556–4564.  Google Scholar

[20]

M. Liu, Z. Yuan, Y. Ying and T. Yang, Stochastic AUC Maximization with Deep Neural Networks, in International Conference on Learning Representations (ICLR), 2020. Google Scholar

[21]

M. Liu, X. Zhang, Z. Chen, X. Wang and T. Yang, Fast stochastic AUC maximization with O (1/n)-convergence rate, in International Conference on Machine Learning, (2018), 3195–3203. Google Scholar

[22]

M. Natole, Y. Ying and S. Lyu, Stochastic proximal algorithms for AUC maximization, in International Conference on Machine Learning, (2018), 3707–3716. Google Scholar

[23]

A. NemirovskiA. JuditskyG. Lan and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM J. Optim., 19 (2009), 1574-1609.  doi: 10.1137/070704277.  Google Scholar

[24]

E. A. Nurminskii, The quasigradient method for the solving of the nonlinear programming problems, Cybernetics, 9 (1973), 145-150.   Google Scholar

[25]

G. M. Phillips, Interpolation and Approximation by Polynomials, Vol. 14, Springer Science & Business Media, 2003. doi: 10.1007/b97417.  Google Scholar

[26] M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.   Google Scholar
[27]

H. Rafique, M. Liu, Q. Lin and T. Yang, Non-Convex Min-Max Optimization: Provable Algorithms and Applications in Machine Learning, preprint, arXiv: 1810.02060. Google Scholar

[28]

A. Rakhlin, O. Shamir and K. Sridharan, Making gradient descent optimal for strongly convex stochastic optimization, in Proceedings of the 29th International Conference on Machine Learning, (2012), 449–456. Google Scholar

[29]

O. Shamir and T. Zhang, Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes, in International Conference on Machine Learning, (2013), 71–79. Google Scholar

[30]

N. Srebro, A. Tewari, Stochastic optimization for machine learning, CML Tutorial, (2010). Google Scholar

[31]

Y. Wang, R. Khardon, D. Pechyony and R. Jones, Generalization bounds for online learning algorithms with pairwise loss functions, in Conference on Learning Theory, (2012), 13. Google Scholar

[32]

Y. Ying, L. Wen and S. Lyu, Stochastic online AUC maximization, in Advances in Neural Information Processing Systems, 2016. Google Scholar

[33]

Y. Ying and D. X. Zhou, Online regularized classification algorithms, IEEE Trans. Inform. Theory, 52 (2006), 4775-4788.  doi: 10.1109/TIT.2006.883632.  Google Scholar

[34]

Y. Ying and D. X. Zhou, Online pairwise learning algorithms, Neural Comput., 28 (2016), 743-777.  doi: 10.1162/neco_a_00817.  Google Scholar

[35]

X. ZhangA. Saha and S. V. N. Vishwanathan, Smoothing multivariate performance measures, J. Mach. Learn. Res., 13 (2012), 3623-3680.   Google Scholar

[36]

P. Zhao, R. Jin, T. Yang and S. C. Hoi, Online AUC maximization, in Proceedings of the 28th International Conference on Machine Learning (ICML-11), 2011. Google Scholar

Figure 1.  Comparison of convergence speed between SAUC-H and $ \text{OAM}_{gra} $
Figure 2.  Evaluation of AUC scores vesus the degree of the Bernstein polynomial
Algorithm 1: Stochastic AUC Optimization (SAUC)

1: Input: $ R>0 $, $ \gamma\geq\gamma_0 $ and $ \beta>0 $.
2: Initialize $ \bar{{\mathbf{v}}}_0 = 0 $ and $ \bar{{\mathit{\boldsymbol{\alpha}}}}_0 = 0 $.
3: for $ t=1 $ to $ T-1 $ do
4: Set $ {\mathbf{v}}_0^t = \bar{{\mathbf{v}}}_{t-1}, {\mathit{\boldsymbol{\alpha}}}_0^t = \bar{{\mathit{\boldsymbol{\alpha}}}}_{t-1} $ and $ \eta_t = \frac{\beta}{\sqrt{t}}. $
5: for $ j=1 $ to $ t $ do
6: Randomly sample $ z_j^t = (x_j^t,y_j^t) $ and compute
$ \begin{align*} &{\mathbf{v}}_{j}^t = {{\bf Proj}}_{{\Omega}_1} \bigl({\mathbf{v}}_{j-1}^t - \eta_t \nabla_{{\mathbf{v}}} \varPhi_{\gamma}^t({\mathbf{v}}_{j-1}^t,{\mathit{\boldsymbol{\alpha}}}_{j-1}^t;z_j^t)\bigr), &{\mathit{\boldsymbol{\alpha}}}_{j}^t = {{\bf Proj}}_{{\Omega}_2} \bigl({\mathit{\boldsymbol{\alpha}}}_{j-1}^t + \eta_t \nabla_{{\mathit{\boldsymbol{\alpha}}}} \varPhi_{\gamma}^t({\mathbf{v}}_{j-1}^t,{\mathit{\boldsymbol{\alpha}}}_{j-1}^t;z_j^t)\bigr) \end{align*} $
7: end for
8: Compute $ \bar{{\mathbf{v}}}_{t} = \frac{1}{t}\sum_{j=0}^{t-1} {\mathbf{v}}_j^t $ and $ \bar{{\mathit{\boldsymbol{\alpha}}}}_{t} = \frac{1}{t}\sum_{j=0}^{t-1} {\mathit{\boldsymbol{\alpha}}}_j^t. $
9: end for
10: Output: $ \widetilde{{\mathbf{v}}}_T:=\frac{1}{T}\sum_{t=0}^{T-1}\bar{{\mathbf{v}}}_{t} $ and $ \widetilde{{\mathit{\boldsymbol{\alpha}}}}_T:=\frac{1}{T}\sum_{t=0}^{T-1}\bar{{\mathit{\boldsymbol{\alpha}}}}_{t}. $
Algorithm 1: Stochastic AUC Optimization (SAUC)

1: Input: $ R>0 $, $ \gamma\geq\gamma_0 $ and $ \beta>0 $.
2: Initialize $ \bar{{\mathbf{v}}}_0 = 0 $ and $ \bar{{\mathit{\boldsymbol{\alpha}}}}_0 = 0 $.
3: for $ t=1 $ to $ T-1 $ do
4: Set $ {\mathbf{v}}_0^t = \bar{{\mathbf{v}}}_{t-1}, {\mathit{\boldsymbol{\alpha}}}_0^t = \bar{{\mathit{\boldsymbol{\alpha}}}}_{t-1} $ and $ \eta_t = \frac{\beta}{\sqrt{t}}. $
5: for $ j=1 $ to $ t $ do
6: Randomly sample $ z_j^t = (x_j^t,y_j^t) $ and compute
$ \begin{align*} &{\mathbf{v}}_{j}^t = {{\bf Proj}}_{{\Omega}_1} \bigl({\mathbf{v}}_{j-1}^t - \eta_t \nabla_{{\mathbf{v}}} \varPhi_{\gamma}^t({\mathbf{v}}_{j-1}^t,{\mathit{\boldsymbol{\alpha}}}_{j-1}^t;z_j^t)\bigr), &{\mathit{\boldsymbol{\alpha}}}_{j}^t = {{\bf Proj}}_{{\Omega}_2} \bigl({\mathit{\boldsymbol{\alpha}}}_{j-1}^t + \eta_t \nabla_{{\mathit{\boldsymbol{\alpha}}}} \varPhi_{\gamma}^t({\mathbf{v}}_{j-1}^t,{\mathit{\boldsymbol{\alpha}}}_{j-1}^t;z_j^t)\bigr) \end{align*} $
7: end for
8: Compute $ \bar{{\mathbf{v}}}_{t} = \frac{1}{t}\sum_{j=0}^{t-1} {\mathbf{v}}_j^t $ and $ \bar{{\mathit{\boldsymbol{\alpha}}}}_{t} = \frac{1}{t}\sum_{j=0}^{t-1} {\mathit{\boldsymbol{\alpha}}}_j^t. $
9: end for
10: Output: $ \widetilde{{\mathbf{v}}}_T:=\frac{1}{T}\sum_{t=0}^{T-1}\bar{{\mathbf{v}}}_{t} $ and $ \widetilde{{\mathit{\boldsymbol{\alpha}}}}_T:=\frac{1}{T}\sum_{t=0}^{T-1}\bar{{\mathit{\boldsymbol{\alpha}}}}_{t}. $
Table 1.  Statistics of datasets
Table 2.  Comparison of AUC score (mean$ \pm $std) on test data; OPAUC on news20 and sector does not converge in a reasonable time limit. Best AUC value on each dataset is in bold and second is underlined
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