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Distributionally robust chance constrained svm model with $\ell_2$-Wasserstein distance
School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China |
In this paper, we propose a distributionally robust chance-constrained SVM model with $ \ell_2 $-Wasserstein ambiguity. We present equivalent formulations of distributionally robust chance constraints based on $ \ell_2 $-Wasserstein ambiguity. In terms of this method, the distributionally robust chance-constrained SVM model can be transformed into a solvable linear 0-1 mixed integer programming problem when the $ \ell_2 $-Wasserstein distance is discrete form. The DRCC-SVM model could be transformed into a tractable 0-1 mixed-integer SOCP programming problem for the continuous case. Finally, numerical experiments are given to illustrate the effectiveness and feasibility of our model.
References:
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G. Aurora and M. C. Victoria,
Towards energy efficiency smart buildings models based on in telligent data analytics, Procedia Computer Science, 83 (2016), 994-999.
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Y. Q. Bai and K. J. Shen,
Alternating direction method of multipliers for $\ell_1$-$\ell_2$ regularized logistic regression model, J. Oper. Res. Soc. China, 4 (2016), 243-253.
doi: 10.1007/s40305-015-0090-2. |
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Y. Q. Bai, Y. J. Shen and K. J. Shen,
Consensus proximal support vector machine for classficication problems with sparse solutions, J. Oper. Res. Soc. China, 2 (2014), 57-79.
doi: 10.1007/s40305-014-0037-z. |
[4] |
C. Bhattacharyya, Robust classification of noisy data using second order cone programming approach, In Intelligent Sensing and Information Processing, IEEE, (2004), 433–438. |
[5] |
J. Blanchet, L. Chen and X. Y. Zhou, Distributionally robust mean-variance portfolio selection with wasserstein distances, preprint, arXiv: 1802.04885. |
[6] |
G. C. Calafiore and L. Ghaoui,
On distributionally robust chance-constrained linear programs, J. Optim. Theory Appl., 130 (2006), 1-22.
doi: 10.1007/s10957-006-9084-x. |
[7] |
S. Chao, X. Huang and F. You,
Data-driven robust optimization based on kernel learning, Computers and Chemical Engineering, 106 (2017), 464-479.
|
[8] |
A. Charnes, W. W. Cooper and G. H. Symonds,
Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil, Management Science, 4 (1958), 235-263.
doi: 10.1287/mnsc.4.3.235. |
[9] |
C. Cortes and V. Vapnik,
Support-vector networks, Machine Learning, 20 (1995), 273-297.
doi: 10.1007/BF00994018. |
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M. D. Dias and A. R. Neto,
Training soft margin support vector machines by simulated annealing: A dual approach, Expert Systems with Applications, 87 (2017), 157-169.
|
[11] |
E. Erdogan and G. Iyengar,
Ambiguous chance constrained problems and robust optimization, Math. Program., 107 (2006), 37-61.
doi: 10.1007/s10107-005-0678-0. |
[12] |
P. M. Esfahani and D. Kuhn,
Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Program., 171 (2018), 115-166.
doi: 10.1007/s10107-017-1172-1. |
[13] |
R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with wasserstein distance, preprint, arXiv: 1604.02199. |
[14] |
Y. Guo, K. Baker, E. Dall'Anese, Z. C. Hu and T. H. Summers,
Data-based distributionally robust stochastic optimal power flow-Part I: Methodologies, IEEE Transactions on Power Systems, 34 (2019), 1483-1492.
doi: 10.1109/TPWRS.2018.2878385. |
[15] |
Y. Guo, K. Baker, E. Dall'Anese, Z. C. Hu and T. H. Summers,
Data-based distributionally robust stochastic optimal power flow-Part II: Case studies, IEEE Transactions on Power Systems, 34 (2018), 1493-1503.
|
[16] |
P. Georg and W. David,
Ambiguity in portfolio selection, Quant. Finance, 7 (2007), 435-442.
doi: 10.1080/14697680701455410. |
[17] |
B. Han, C. Shang and D. Huang,
Multiple kernel learning-aided robust optimization: Learning algorithm, computational tractability, and usage in multi-stage decision-making, European J. Oper. Res., 292 (2021), 1004-1018.
doi: 10.1016/j.ejor.2020.11.027. |
[18] |
R. Jagannathan,
Chance-constrained programming with joint constraints, Operations Res., 22 (1974), 358-372.
doi: 10.1287/opre.22.2.358. |
[19] |
R. Ji and M. A. Lejeune,
Data-driven optimization of reward-risk ratio measures, INFORMS J. Comput., 33 (2021), 1120-1137.
doi: 10.1287/ijoc.2020.1002. |
[20] |
S. Justin and G. Leonidas,
Convolutional wasserstein distances: Efficient optimal transportation on geometric domains, ACM Transactions on Graphics, 34 (2015), 1-11.
|
[21] |
G. R. Lanckriet, L. Ghaoui, C. Bhattacharyya and M. I. Jordan,
A robust minimax approach to classification, J. Mach. Learn. Res., 3 (2003), 555-582.
|
[22] |
B. Piccoli and F. Rossi,
On properties of the generalized Wasserstein distance, Arch. Ration. Mech. Anal., 222 (2016), 1339-1365.
doi: 10.1007/s00205-016-1026-7. |
[23] |
P. K. Shivaswamy, C. Bhattacharyya and A. J. Smola,
Second order cone programming approaches for handling missing and uncertain data, J. Mach. Learn. Res., 7 (2006), 1283-1314.
|
[24] |
H. Xu, C. Caramanis and S. Mannor,
Robustness and regularization of support vector machines, J. Mach. Learn. Res., 10 (2009), 1485-1510.
|
[25] |
S. Zymler, D. Kuhn and B. Rustem,
Distributionally robust joint chance constraints with second-order moment information, Math. Program., 137 (2013), 167-198.
doi: 10.1007/s10107-011-0494-7. |
[26] |
S. Zymler, D. Kuhn and B. Rustem,
Worst-case value at risk of nonlinear portfolios, Management Science, 59 (2013), 172-188.
doi: 10.1287/mnsc.1120.1615. |
show all references
References:
[1] |
G. Aurora and M. C. Victoria,
Towards energy efficiency smart buildings models based on in telligent data analytics, Procedia Computer Science, 83 (2016), 994-999.
|
[2] |
Y. Q. Bai and K. J. Shen,
Alternating direction method of multipliers for $\ell_1$-$\ell_2$ regularized logistic regression model, J. Oper. Res. Soc. China, 4 (2016), 243-253.
doi: 10.1007/s40305-015-0090-2. |
[3] |
Y. Q. Bai, Y. J. Shen and K. J. Shen,
Consensus proximal support vector machine for classficication problems with sparse solutions, J. Oper. Res. Soc. China, 2 (2014), 57-79.
doi: 10.1007/s40305-014-0037-z. |
[4] |
C. Bhattacharyya, Robust classification of noisy data using second order cone programming approach, In Intelligent Sensing and Information Processing, IEEE, (2004), 433–438. |
[5] |
J. Blanchet, L. Chen and X. Y. Zhou, Distributionally robust mean-variance portfolio selection with wasserstein distances, preprint, arXiv: 1802.04885. |
[6] |
G. C. Calafiore and L. Ghaoui,
On distributionally robust chance-constrained linear programs, J. Optim. Theory Appl., 130 (2006), 1-22.
doi: 10.1007/s10957-006-9084-x. |
[7] |
S. Chao, X. Huang and F. You,
Data-driven robust optimization based on kernel learning, Computers and Chemical Engineering, 106 (2017), 464-479.
|
[8] |
A. Charnes, W. W. Cooper and G. H. Symonds,
Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil, Management Science, 4 (1958), 235-263.
doi: 10.1287/mnsc.4.3.235. |
[9] |
C. Cortes and V. Vapnik,
Support-vector networks, Machine Learning, 20 (1995), 273-297.
doi: 10.1007/BF00994018. |
[10] |
M. D. Dias and A. R. Neto,
Training soft margin support vector machines by simulated annealing: A dual approach, Expert Systems with Applications, 87 (2017), 157-169.
|
[11] |
E. Erdogan and G. Iyengar,
Ambiguous chance constrained problems and robust optimization, Math. Program., 107 (2006), 37-61.
doi: 10.1007/s10107-005-0678-0. |
[12] |
P. M. Esfahani and D. Kuhn,
Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Program., 171 (2018), 115-166.
doi: 10.1007/s10107-017-1172-1. |
[13] |
R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with wasserstein distance, preprint, arXiv: 1604.02199. |
[14] |
Y. Guo, K. Baker, E. Dall'Anese, Z. C. Hu and T. H. Summers,
Data-based distributionally robust stochastic optimal power flow-Part I: Methodologies, IEEE Transactions on Power Systems, 34 (2019), 1483-1492.
doi: 10.1109/TPWRS.2018.2878385. |
[15] |
Y. Guo, K. Baker, E. Dall'Anese, Z. C. Hu and T. H. Summers,
Data-based distributionally robust stochastic optimal power flow-Part II: Case studies, IEEE Transactions on Power Systems, 34 (2018), 1493-1503.
|
[16] |
P. Georg and W. David,
Ambiguity in portfolio selection, Quant. Finance, 7 (2007), 435-442.
doi: 10.1080/14697680701455410. |
[17] |
B. Han, C. Shang and D. Huang,
Multiple kernel learning-aided robust optimization: Learning algorithm, computational tractability, and usage in multi-stage decision-making, European J. Oper. Res., 292 (2021), 1004-1018.
doi: 10.1016/j.ejor.2020.11.027. |
[18] |
R. Jagannathan,
Chance-constrained programming with joint constraints, Operations Res., 22 (1974), 358-372.
doi: 10.1287/opre.22.2.358. |
[19] |
R. Ji and M. A. Lejeune,
Data-driven optimization of reward-risk ratio measures, INFORMS J. Comput., 33 (2021), 1120-1137.
doi: 10.1287/ijoc.2020.1002. |
[20] |
S. Justin and G. Leonidas,
Convolutional wasserstein distances: Efficient optimal transportation on geometric domains, ACM Transactions on Graphics, 34 (2015), 1-11.
|
[21] |
G. R. Lanckriet, L. Ghaoui, C. Bhattacharyya and M. I. Jordan,
A robust minimax approach to classification, J. Mach. Learn. Res., 3 (2003), 555-582.
|
[22] |
B. Piccoli and F. Rossi,
On properties of the generalized Wasserstein distance, Arch. Ration. Mech. Anal., 222 (2016), 1339-1365.
doi: 10.1007/s00205-016-1026-7. |
[23] |
P. K. Shivaswamy, C. Bhattacharyya and A. J. Smola,
Second order cone programming approaches for handling missing and uncertain data, J. Mach. Learn. Res., 7 (2006), 1283-1314.
|
[24] |
H. Xu, C. Caramanis and S. Mannor,
Robustness and regularization of support vector machines, J. Mach. Learn. Res., 10 (2009), 1485-1510.
|
[25] |
S. Zymler, D. Kuhn and B. Rustem,
Distributionally robust joint chance constraints with second-order moment information, Math. Program., 137 (2013), 167-198.
doi: 10.1007/s10107-011-0494-7. |
[26] |
S. Zymler, D. Kuhn and B. Rustem,
Worst-case value at risk of nonlinear portfolios, Management Science, 59 (2013), 172-188.
doi: 10.1287/mnsc.1120.1615. |
AUC | (S.E.) | |
Classical SVM | 0.9039 | 0.0046 |
0.9105 | 0.0020 |
AUC | (S.E.) | |
Classical SVM | 0.9039 | 0.0046 |
0.9105 | 0.0020 |
Date set | classification | numbers | feature |
Sonar | 2 | 208 | 60 |
Pima | 2 | 267 | 22 |
Heart | 2 | 270 | 12 |
BUPA Liver | 2 | 345 | 6 |
Ionosphere | 2 | 351 | 34 |
Australian | 2 | 690 | 14 |
Breast Cancer | 2 | 699 | 13 |
Bank | 2 | 4521 | 16 |
Date set | classification | numbers | feature |
Sonar | 2 | 208 | 60 |
Pima | 2 | 267 | 22 |
Heart | 2 | 270 | 12 |
BUPA Liver | 2 | 345 | 6 |
Ionosphere | 2 | 351 | 34 |
Australian | 2 | 690 | 14 |
Breast Cancer | 2 | 699 | 13 |
Bank | 2 | 4521 | 16 |
Classic SVM | Soft margin SVM | ||||||
Data set | n | Accuracy | (S.E.) | Accuracy | (S.E.) | Accuracy | (S.E.) |
Sonar | 50 | 0.814 | 0.0064 | 0.801 | 0.0050 | 0.799 | 0.0049 |
Sonar | 70 | 0.856 | 0.0045 | 0.851 | 0.0041 | 0.850 | 0.0040 |
Sonar | 100 | 0.872 | 0.0038 | 0.862 | 0.0033 | 0.860 | 0.0032 |
Pima | 50 | 0.732 | 0.0055 | 0.723 | 0.0048 | 0.722 | 0.0047 |
Pima | 70 | 0.745 | 0.0051 | 0.741 | 0.0043 | 0.741 | 0.0043 |
Pima | 100 | 0.751 | 0.0046 | 0.743 | 0.0034 | 0.743 | 0.0034 |
Heart | 50 | 0.791 | 0.0027 | 0.788 | 0.0025 | 0.787 | 0.0024 |
Heart | 70 | 0.821 | 0.0024 | 0.815 | 0.0020 | 0.814 | 0.0018 |
Heart | 100 | 0.833 | 0.0016 | 0.831 | 0.0011 | 0.827 | 0.0009 |
BUPA Liver | 50 | 0.773 | 0.0035 | 0.768 | 0.0033 | 0.765 | 0.0028 |
BUPA Liver | 70 | 0.789 | 0.0032 | 0.788 | 0.0031 | 0.788 | 0.0031 |
BUPA Liver | 100 | 0.801 | 0.0025 | 0.801 | 0.0024 | 0.799 | 0.0023 |
Ionosphere | 50 | 0.812 | 0.0054 | 0.810 | 0.0051 | 0.808 | 0.0050 |
Ionosphere | 70 | 0.834 | 0.0037 | 0.842 | 0.0035 | 0.832 | 0.0035 |
Ionosphere | 100 | 0.852 | 0.0033 | 0.848 | 0.0022 | 0.845 | 0.0020 |
Australian | 50 | 0.812 | 0.0022 | 0.810 | 0.0020 | 0.807 | 0.0015 |
Australian | 70 | 0.823 | 0.0019 | 0.832 | 0.0016 | 0.822 | 0.0015 |
Australian | 100 | 0.833 | 0.0015 | 0.830 | 0.0013 | 0.830 | 0.0013 |
Breast Cancer | 50 | 0.952 | 0.0023 | 0.951 | 0.0022 | 0.951 | 0.0022 |
Breast Cancer | 70 | 0.955 | 0.0020 | 0.955 | 0.0018 | 0.952 | 0.0017 |
Breast Cancer | 100 | 0.964 | 0.0016 | 0.969 | 0.0011 | 0.952 | 0.0009 |
Classic SVM | Soft margin SVM | ||||||
Data set | n | Accuracy | (S.E.) | Accuracy | (S.E.) | Accuracy | (S.E.) |
Sonar | 50 | 0.814 | 0.0064 | 0.801 | 0.0050 | 0.799 | 0.0049 |
Sonar | 70 | 0.856 | 0.0045 | 0.851 | 0.0041 | 0.850 | 0.0040 |
Sonar | 100 | 0.872 | 0.0038 | 0.862 | 0.0033 | 0.860 | 0.0032 |
Pima | 50 | 0.732 | 0.0055 | 0.723 | 0.0048 | 0.722 | 0.0047 |
Pima | 70 | 0.745 | 0.0051 | 0.741 | 0.0043 | 0.741 | 0.0043 |
Pima | 100 | 0.751 | 0.0046 | 0.743 | 0.0034 | 0.743 | 0.0034 |
Heart | 50 | 0.791 | 0.0027 | 0.788 | 0.0025 | 0.787 | 0.0024 |
Heart | 70 | 0.821 | 0.0024 | 0.815 | 0.0020 | 0.814 | 0.0018 |
Heart | 100 | 0.833 | 0.0016 | 0.831 | 0.0011 | 0.827 | 0.0009 |
BUPA Liver | 50 | 0.773 | 0.0035 | 0.768 | 0.0033 | 0.765 | 0.0028 |
BUPA Liver | 70 | 0.789 | 0.0032 | 0.788 | 0.0031 | 0.788 | 0.0031 |
BUPA Liver | 100 | 0.801 | 0.0025 | 0.801 | 0.0024 | 0.799 | 0.0023 |
Ionosphere | 50 | 0.812 | 0.0054 | 0.810 | 0.0051 | 0.808 | 0.0050 |
Ionosphere | 70 | 0.834 | 0.0037 | 0.842 | 0.0035 | 0.832 | 0.0035 |
Ionosphere | 100 | 0.852 | 0.0033 | 0.848 | 0.0022 | 0.845 | 0.0020 |
Australian | 50 | 0.812 | 0.0022 | 0.810 | 0.0020 | 0.807 | 0.0015 |
Australian | 70 | 0.823 | 0.0019 | 0.832 | 0.0016 | 0.822 | 0.0015 |
Australian | 100 | 0.833 | 0.0015 | 0.830 | 0.0013 | 0.830 | 0.0013 |
Breast Cancer | 50 | 0.952 | 0.0023 | 0.951 | 0.0022 | 0.951 | 0.0022 |
Breast Cancer | 70 | 0.955 | 0.0020 | 0.955 | 0.0018 | 0.952 | 0.0017 |
Breast Cancer | 100 | 0.964 | 0.0016 | 0.969 | 0.0011 | 0.952 | 0.0009 |
Data set | n | Classic SVM | L2-Wasserstein SVM | Soft-margin SVM | |||
Accuracy | Accuracy (10% noise) | Accuracy | Accuracy (10% noise) | Accuracy | Accuracy (10% noise) | ||
Bank | 100 | 0.8421 | 0.7870 | 0.8841 | 0.8842 | 0.8823 | 0.8743 |
200 | 0.8377 | 0.7798 | 0.8844 | 0.8846 | 0.8823 | 0.8773 | |
300 | 0.8379 | 0.7785 | 0.8846 | 0.8845 | 0.8824 | 0.8768 | |
400 | 0.8377 | 0.7757 | 0.8850 | 0.8847 | 0.8821 | 0.8763 | |
500 | 0.8407 | 0.7735 | 0.8853 | 0.8854 | 0.8833 | 0.8765 | |
600 | 0.8389 | 0.7759 | 0.8855 | 0.8852 | 0.8831 | 0.8775 |
Data set | n | Classic SVM | L2-Wasserstein SVM | Soft-margin SVM | |||
Accuracy | Accuracy (10% noise) | Accuracy | Accuracy (10% noise) | Accuracy | Accuracy (10% noise) | ||
Bank | 100 | 0.8421 | 0.7870 | 0.8841 | 0.8842 | 0.8823 | 0.8743 |
200 | 0.8377 | 0.7798 | 0.8844 | 0.8846 | 0.8823 | 0.8773 | |
300 | 0.8379 | 0.7785 | 0.8846 | 0.8845 | 0.8824 | 0.8768 | |
400 | 0.8377 | 0.7757 | 0.8850 | 0.8847 | 0.8821 | 0.8763 | |
500 | 0.8407 | 0.7735 | 0.8853 | 0.8854 | 0.8833 | 0.8765 | |
600 | 0.8389 | 0.7759 | 0.8855 | 0.8852 | 0.8831 | 0.8775 |
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