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A difference of convex optimization algorithm for piecewise linear regression
1. | Faculty of Science and Technology, Federation University Australia, Victoria, Australia |
2. | Department of Mathematics, Shahrekord University, Iran |
The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.
References:
[1] |
L. T. H. An and P. D. Tao,
The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[2] |
K. Bache and M. Lichman, UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, http://archive.ics.uci.edu/ml, 2013. |
[3] |
A. M. Bagirov, C. Clausen and M. Kohler,
Estimation of a regression function by maxima of minima of linear functions, IEEE Transactions on Information Theory, 55 (2009), 833-845.
doi: 10.1109/TIT.2008.2009835. |
[4] |
A. M. Bagirov, C. Clausen and M. Kohler,
An algorithm for the estimation of a regression function by continuous piecewise linear functions, Computational Optimization and Applications, 45 (2010), 159-179.
doi: 10.1007/s10589-008-9174-9. |
[5] |
A. M. Bagirov, C. Clausen and M. Kohler,
An l2-boosting algorithm for estimation of a regression function, IEEE Transactions on Information Theory, 56 (2010), 1417-1429.
doi: 10.1109/TIT.2009.2039161. |
[6] |
A. M. Bagirov, B. Karasözen and M. Sezer,
Discrete gradient method: A derivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications, 137 (2008), 317-334.
doi: 10.1007/s10957-007-9335-5. |
[7] |
A. M. Bagirov, N. Karmitsa and M. Mäkelä,
Introduction to Nonsmooth Optimization, Cham, Springer, 2014.
doi: 10.1007/978-3-319-08114-4. |
[8] |
A. M. Bagirov, S. Taheri and J. Ugon,
Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems, Pattern Recognition, 53 (2016), 12-24.
doi: 10.1016/j.patcog.2015.11.011. |
[9] |
S. G. Bartels, L. Kuntz and S. Scholtes,
Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 385-407.
doi: 10.1016/0362-546X(95)91645-6. |
[10] |
L. Breiman, J. H. Friedman, C. J. Stone and R. A. Olshen,
Classification and Regression Trees, CRC press, 1984. |
[11] |
R. Collobert and S. Bengio,
SVMTorch: Support vector machines for large-scale regression problems, Journal of Machine Learning Research, 1 (2001), 143-160.
doi: 10.1162/15324430152733142. |
[12] |
V. F. Demyanov and A. M. Rubinov,
Constructive Nonsmooth Analysis, Springer, 1995. |
[13] |
J. H. Fridedman,
Multivariate adaptive regression splines (with discussion), Annals of Statistics, 19 (1991), 79-141.
doi: 10.1214/aos/1176347963. |
[14] |
J. Friedman, T. Hastie and R. Tibshirani,
The Elements of Statistical Learning, Springer, Berlin, 2001.
doi: 10.1007/978-0-387-21606-5. |
[15] |
V. V. Gorokhovik, O. I. Zorko and G. Birkhoff,
Piecewise affine functions and polyhedral sets, Optimization, 31 (1994), 209-221.
doi: 10.1080/02331939408844018. |
[16] |
L. Györfi, M. Kohler, A. Krzyźak and H. Walk,
A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics. Springer, Heldelberg, 2002.
doi: 10.1007/b97848. |
[17] |
R. Horst and N. V. Thoai,
DC programming: Overview, Journal of Optimization Theory and Applications, 103 (1999), 1-43.
doi: 10.1023/A:1021765131316. |
[18] |
K. Joki, A. M. Bagirov, N. Karmitsa and M. M. Mäkelä,
A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes, Journal of Global Optimization, 68 (2017), 501-535.
doi: 10.1007/s10898-016-0488-3. |
[19] |
D. Meyer, E. Dimitriadou, K. Hornik, A. Weingessel, F. Leisch, C. C. Chang and C. C. Lin, Misc functions of the department of statistics, probability theory group, TU Wien, Package "e1071",
http://cran.r-project.org/web/packages/e1071/e1071.pdf, 2017. |
[20] |
S. Milborrow, Multivariate adaptive regression splines, Package "earth",
http://cran.r-project.org/web/packages/earth/earth.pdf, 2017. |
[21] |
J. Nash and J. Sutcliffe,
River flow forecasting through conceptual models part I-A discussion of principles, Journal of Hydrology, 10 (1970), 282-290.
doi: 10.1016/0022-1694(70)90255-6. |
[22] |
B. Ripley and W. Venables, Feed-forward neural networks and multinomial log-linear models, Package "nnet",
http://cran.r-project.org/web/packages/nnet/nnet.pdf, 2016. |
[23] |
A. J. Smola and B. Schölkopf,
A tutorial on support vector regression, Statistics and Computing, 14 (2004), 199-222.
doi: 10.1023/B:STCO.0000035301.49549.88. |
[24] |
P. D. Tao and L. T. H. An,
Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.
|
[25] |
H. Tuy,
Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, Vol. 22. Kluwer, Dordrecht, 1998.
doi: 10.1007/978-1-4757-2809-5. |
[26] |
P. H. Wolfe,
Finding the nearest point in a polytope, Mathematical Programming, 11 (1976), 128-149.
doi: 10.1007/BF01580381. |
show all references
References:
[1] |
L. T. H. An and P. D. Tao,
The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[2] |
K. Bache and M. Lichman, UCI Machine Learning Repository, University of California, Irvine, School of Information and Computer Sciences, http://archive.ics.uci.edu/ml, 2013. |
[3] |
A. M. Bagirov, C. Clausen and M. Kohler,
Estimation of a regression function by maxima of minima of linear functions, IEEE Transactions on Information Theory, 55 (2009), 833-845.
doi: 10.1109/TIT.2008.2009835. |
[4] |
A. M. Bagirov, C. Clausen and M. Kohler,
An algorithm for the estimation of a regression function by continuous piecewise linear functions, Computational Optimization and Applications, 45 (2010), 159-179.
doi: 10.1007/s10589-008-9174-9. |
[5] |
A. M. Bagirov, C. Clausen and M. Kohler,
An l2-boosting algorithm for estimation of a regression function, IEEE Transactions on Information Theory, 56 (2010), 1417-1429.
doi: 10.1109/TIT.2009.2039161. |
[6] |
A. M. Bagirov, B. Karasözen and M. Sezer,
Discrete gradient method: A derivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications, 137 (2008), 317-334.
doi: 10.1007/s10957-007-9335-5. |
[7] |
A. M. Bagirov, N. Karmitsa and M. Mäkelä,
Introduction to Nonsmooth Optimization, Cham, Springer, 2014.
doi: 10.1007/978-3-319-08114-4. |
[8] |
A. M. Bagirov, S. Taheri and J. Ugon,
Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems, Pattern Recognition, 53 (2016), 12-24.
doi: 10.1016/j.patcog.2015.11.011. |
[9] |
S. G. Bartels, L. Kuntz and S. Scholtes,
Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 385-407.
doi: 10.1016/0362-546X(95)91645-6. |
[10] |
L. Breiman, J. H. Friedman, C. J. Stone and R. A. Olshen,
Classification and Regression Trees, CRC press, 1984. |
[11] |
R. Collobert and S. Bengio,
SVMTorch: Support vector machines for large-scale regression problems, Journal of Machine Learning Research, 1 (2001), 143-160.
doi: 10.1162/15324430152733142. |
[12] |
V. F. Demyanov and A. M. Rubinov,
Constructive Nonsmooth Analysis, Springer, 1995. |
[13] |
J. H. Fridedman,
Multivariate adaptive regression splines (with discussion), Annals of Statistics, 19 (1991), 79-141.
doi: 10.1214/aos/1176347963. |
[14] |
J. Friedman, T. Hastie and R. Tibshirani,
The Elements of Statistical Learning, Springer, Berlin, 2001.
doi: 10.1007/978-0-387-21606-5. |
[15] |
V. V. Gorokhovik, O. I. Zorko and G. Birkhoff,
Piecewise affine functions and polyhedral sets, Optimization, 31 (1994), 209-221.
doi: 10.1080/02331939408844018. |
[16] |
L. Györfi, M. Kohler, A. Krzyźak and H. Walk,
A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics. Springer, Heldelberg, 2002.
doi: 10.1007/b97848. |
[17] |
R. Horst and N. V. Thoai,
DC programming: Overview, Journal of Optimization Theory and Applications, 103 (1999), 1-43.
doi: 10.1023/A:1021765131316. |
[18] |
K. Joki, A. M. Bagirov, N. Karmitsa and M. M. Mäkelä,
A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes, Journal of Global Optimization, 68 (2017), 501-535.
doi: 10.1007/s10898-016-0488-3. |
[19] |
D. Meyer, E. Dimitriadou, K. Hornik, A. Weingessel, F. Leisch, C. C. Chang and C. C. Lin, Misc functions of the department of statistics, probability theory group, TU Wien, Package "e1071",
http://cran.r-project.org/web/packages/e1071/e1071.pdf, 2017. |
[20] |
S. Milborrow, Multivariate adaptive regression splines, Package "earth",
http://cran.r-project.org/web/packages/earth/earth.pdf, 2017. |
[21] |
J. Nash and J. Sutcliffe,
River flow forecasting through conceptual models part I-A discussion of principles, Journal of Hydrology, 10 (1970), 282-290.
doi: 10.1016/0022-1694(70)90255-6. |
[22] |
B. Ripley and W. Venables, Feed-forward neural networks and multinomial log-linear models, Package "nnet",
http://cran.r-project.org/web/packages/nnet/nnet.pdf, 2016. |
[23] |
A. J. Smola and B. Schölkopf,
A tutorial on support vector regression, Statistics and Computing, 14 (2004), 199-222.
doi: 10.1023/B:STCO.0000035301.49549.88. |
[24] |
P. D. Tao and L. T. H. An,
Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.
|
[25] |
H. Tuy,
Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, Vol. 22. Kluwer, Dordrecht, 1998.
doi: 10.1007/978-1-4757-2809-5. |
[26] |
P. H. Wolfe,
Finding the nearest point in a polytope, Mathematical Programming, 11 (1976), 128-149.
doi: 10.1007/BF01580381. |






Train | Test | CPU | Train | Test | CPU | |||||
1 | 1.089 | 1.084 | 0.09 | 1 | 1.138 | 1.113 | 0.37 | |||
2 | 0.890 | 0.856 | 0.14 | 2 | 0.596 | 0.587 | 5.90 | |||
3 | 0.982 | 1.146 | 0.87 | 3 | 1.063 | 1.101 | 3.37 | |||
4 | 1.143 | 1.395 | 0.59 | 4 | 1.168 | 1.289 | 4.32 | |||
7 | 1.307 | 1.409 | 0.39 | 7 | 1.310 | 1.353 | 2.12 | |||
10 | 1.311 | 1.406 | 0.28 | 10 | 1.314 | 1.355 | 1.64 | |||
1 | 1.061 | 1.101 | 0.14 | 1 | 1.120 | 1.118 | 0.62 | |||
2 | 0.665 | 0.725 | 1.70 | 2 | 0.706 | 0.692 | 9.10 | |||
3 | 1.003 | 1.077 | 1.22 | 3 | 1.067 | 1.038 | 6.96 | |||
4 | 1.312 | 1.225 | 0.59 | 4 | 1.263 | 1.271 | 7.07 | |||
7 | 1.368 | 1.257 | 0.59 | 7 | 1.337 | 1.323 | 2.56 | |||
10 | 1.363 | 1.263 | 0.92 | 10 | 1.337 | 1.322 | 4.10 |
Train | Test | CPU | Train | Test | CPU | |||||
1 | 1.089 | 1.084 | 0.09 | 1 | 1.138 | 1.113 | 0.37 | |||
2 | 0.890 | 0.856 | 0.14 | 2 | 0.596 | 0.587 | 5.90 | |||
3 | 0.982 | 1.146 | 0.87 | 3 | 1.063 | 1.101 | 3.37 | |||
4 | 1.143 | 1.395 | 0.59 | 4 | 1.168 | 1.289 | 4.32 | |||
7 | 1.307 | 1.409 | 0.39 | 7 | 1.310 | 1.353 | 2.12 | |||
10 | 1.311 | 1.406 | 0.28 | 10 | 1.314 | 1.355 | 1.64 | |||
1 | 1.061 | 1.101 | 0.14 | 1 | 1.120 | 1.118 | 0.62 | |||
2 | 0.665 | 0.725 | 1.70 | 2 | 0.706 | 0.692 | 9.10 | |||
3 | 1.003 | 1.077 | 1.22 | 3 | 1.067 | 1.038 | 6.96 | |||
4 | 1.312 | 1.225 | 0.59 | 4 | 1.263 | 1.271 | 7.07 | |||
7 | 1.368 | 1.257 | 0.59 | 7 | 1.337 | 1.323 | 2.56 | |||
10 | 1.363 | 1.263 | 0.92 | 10 | 1.337 | 1.322 | 4.10 |
(1, 1) | (2, 1) | (2, 2) | (3, 2) | (4, 2) | (5, 2) | |
RMSE | 1.094 | 1.094 | 0.063 | 0.058 | 0.055 | 0.055 |
MAE | 0.842 | 0.842 | 0.016 | 0.016 | 0.015 | 0.015 |
MSE | 1.198 | 1.198 | 0.004 | 0.003 | 0.003 | 0.003 |
CE | 0.013 | 0.013 | 0.997 | 0.997 | 0.998 | 0.998 |
r | 0.114 | 0.114 | 0.998 | 0.999 | 0.999 | 0.999 |
RMSE | 1.221 | 1.221 | 0.503 | 0.503 | 0.503 | 0.503 |
MAE | 0.937 | 0.937 | 0.404 | 0.404 | 0.404 | 0.404 |
MSE | 1.491 | 1.492 | 0.254 | 0.254 | 0.254 | 0.254 |
CE | 0.011 | 0.011 | 0.832 | 0.832 | 0.832 | 0.832 |
r | 0.105 | 0.105 | 0.912 | 0.912 | 0.912 | 0.912 |
RMSE | 1.498 | 1.498 | 1.242 | 1.000 | 1.000 | 1.000 |
MAE | 1.169 | 1.169 | 0.993 | 0.801 | 0.801 | 0.801 |
MSE | 2.246 | 2.247 | 1.546 | 1.003 | 1.004 | 1.005 |
CE | 0.008 | 0.008 | 0.318 | 0.558 | 0.558 | 0.558 |
r | 0.087 | 0.087 | 0.564 | 0.747 | 0.747 | 0.747 |
(1, 1) | (2, 1) | (2, 2) | (3, 2) | (4, 2) | (5, 2) | |
RMSE | 1.094 | 1.094 | 0.063 | 0.058 | 0.055 | 0.055 |
MAE | 0.842 | 0.842 | 0.016 | 0.016 | 0.015 | 0.015 |
MSE | 1.198 | 1.198 | 0.004 | 0.003 | 0.003 | 0.003 |
CE | 0.013 | 0.013 | 0.997 | 0.997 | 0.998 | 0.998 |
r | 0.114 | 0.114 | 0.998 | 0.999 | 0.999 | 0.999 |
RMSE | 1.221 | 1.221 | 0.503 | 0.503 | 0.503 | 0.503 |
MAE | 0.937 | 0.937 | 0.404 | 0.404 | 0.404 | 0.404 |
MSE | 1.491 | 1.492 | 0.254 | 0.254 | 0.254 | 0.254 |
CE | 0.011 | 0.011 | 0.832 | 0.832 | 0.832 | 0.832 |
r | 0.105 | 0.105 | 0.912 | 0.912 | 0.912 | 0.912 |
RMSE | 1.498 | 1.498 | 1.242 | 1.000 | 1.000 | 1.000 |
MAE | 1.169 | 1.169 | 0.993 | 0.801 | 0.801 | 0.801 |
MSE | 2.246 | 2.247 | 1.546 | 1.003 | 1.004 | 1.005 |
CE | 0.008 | 0.008 | 0.318 | 0.558 | 0.558 | 0.558 |
r | 0.087 | 0.087 | 0.564 | 0.747 | 0.747 | 0.747 |
Data set | ||
1. Airfoil Self-noise | 1503 | 5 |
2. Red Wine Quality | 1599 | 11 |
3. White Wine Quality | 4898 | 11 |
4. Combined Cycle Power Plant | 9568 | 4 |
5. Physicochemical Properties | 45730 | 9 |
of Protein Tertiary Structure |
Data set | ||
1. Airfoil Self-noise | 1503 | 5 |
2. Red Wine Quality | 1599 | 11 |
3. White Wine Quality | 4898 | 11 |
4. Combined Cycle Power Plant | 9568 | 4 |
5. Physicochemical Properties | 45730 | 9 |
of Protein Tertiary Structure |
PWLREG | MLR | SVM(Lin) | SVM(RBF) | MARS | ANN | |
Airfoil Self-noise | ||||||
RMSE | 4.445 | 5.024 | 5.037 | 4.710 | 5.672 | 5.499 |
MAE | 3.404 | 3.905 | 3.877 | 3.327 | 4.614 | 4.342 |
MSE | 24.641 | 25.750 | 25.885 | 22.941 | 32.822 | 31.059 |
CE | 0.653 | 0.558 | 0.556 | 0.611 | 0.437 | 0.471 |
r | 0.875 | 0.761 | 0.760 | 0.825 | 0.704 | 0.709 |
Red Wine Quality | ||||||
RMSE | 0.639 | 0.643 | 0.657 | 0.735 | 0.658 | 0.636 |
MAE | 0.489 | 0.490 | 0.489 | 0.566 | 0.503 | 0.482 |
MSE | 0.480 | 0.430 | 0.450 | 0.569 | 0.450 | 0.423 |
CE | 0.319 | 0.312 | 0.282 | 0.101 | 0.280 | 0.326 |
r | 0.582 | 0.570 | 0.537 | 0.394 | 0.555 | 0.585 |
White Wine Quality | ||||||
RMSE | 0.637 | 0.690 | 0.717 | 0.675 | 0.713 | 0.693 |
MAE | 0.508 | 0.532 | 0.551 | 0.522 | 0.527 | 0.545 |
MSE | 0.449 | 0.482 | 0.521 | 0.463 | 0.515 | 0.487 |
CE | 0.330 | 0.216 | 0.153 | 0.249 | 0.164 | 0.209 |
r | 0.616 | 0.494 | 0.467 | 0.521 | 0.537 | 0.520 |
Combined Cycle Power Plant | ||||||
RMSE | 4.163 | 4.484 | 4.503 | 3.907 | 4.187 | 4.283 |
MAE | 3.276 | 3.572 | 3.559 | 2.920 | 3.295 | 3.396 |
MSE | 17.702 | 20.159 | 20.341 | 15.377 | 17.577 | 18.411 |
CE | 0.942 | 0.933 | 0.932 | 0.949 | 0.942 | 0.939 |
r | 0.970 | 0.966 | 0.966 | 0.975 | 0.971 | 0.969 |
Physicochemical Properties Protein | ||||||
RMSE | 4.802 | 5.203 | 5.302 | 4.321 | 5.044 | 5.783 |
MAE | 3.840 | 4.374 | 4.240 | 3.018 | 4.146 | 4.987 |
MSE | 23.061 | 27.101 | 28.145 | 18.699 | 25.470 | 33.487 |
CE | 0.390 | 0.285 | 0.257 | 0.507 | 0.328 | 0.116 |
r | 0.625 | 0.534 | 0.528 | 0.719 | 0.573 | 0.341 |
PWLREG | MLR | SVM(Lin) | SVM(RBF) | MARS | ANN | |
Airfoil Self-noise | ||||||
RMSE | 4.445 | 5.024 | 5.037 | 4.710 | 5.672 | 5.499 |
MAE | 3.404 | 3.905 | 3.877 | 3.327 | 4.614 | 4.342 |
MSE | 24.641 | 25.750 | 25.885 | 22.941 | 32.822 | 31.059 |
CE | 0.653 | 0.558 | 0.556 | 0.611 | 0.437 | 0.471 |
r | 0.875 | 0.761 | 0.760 | 0.825 | 0.704 | 0.709 |
Red Wine Quality | ||||||
RMSE | 0.639 | 0.643 | 0.657 | 0.735 | 0.658 | 0.636 |
MAE | 0.489 | 0.490 | 0.489 | 0.566 | 0.503 | 0.482 |
MSE | 0.480 | 0.430 | 0.450 | 0.569 | 0.450 | 0.423 |
CE | 0.319 | 0.312 | 0.282 | 0.101 | 0.280 | 0.326 |
r | 0.582 | 0.570 | 0.537 | 0.394 | 0.555 | 0.585 |
White Wine Quality | ||||||
RMSE | 0.637 | 0.690 | 0.717 | 0.675 | 0.713 | 0.693 |
MAE | 0.508 | 0.532 | 0.551 | 0.522 | 0.527 | 0.545 |
MSE | 0.449 | 0.482 | 0.521 | 0.463 | 0.515 | 0.487 |
CE | 0.330 | 0.216 | 0.153 | 0.249 | 0.164 | 0.209 |
r | 0.616 | 0.494 | 0.467 | 0.521 | 0.537 | 0.520 |
Combined Cycle Power Plant | ||||||
RMSE | 4.163 | 4.484 | 4.503 | 3.907 | 4.187 | 4.283 |
MAE | 3.276 | 3.572 | 3.559 | 2.920 | 3.295 | 3.396 |
MSE | 17.702 | 20.159 | 20.341 | 15.377 | 17.577 | 18.411 |
CE | 0.942 | 0.933 | 0.932 | 0.949 | 0.942 | 0.939 |
r | 0.970 | 0.966 | 0.966 | 0.975 | 0.971 | 0.969 |
Physicochemical Properties Protein | ||||||
RMSE | 4.802 | 5.203 | 5.302 | 4.321 | 5.044 | 5.783 |
MAE | 3.840 | 4.374 | 4.240 | 3.018 | 4.146 | 4.987 |
MSE | 23.061 | 27.101 | 28.145 | 18.699 | 25.470 | 33.487 |
CE | 0.390 | 0.285 | 0.257 | 0.507 | 0.328 | 0.116 |
r | 0.625 | 0.534 | 0.528 | 0.719 | 0.573 | 0.341 |
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