Smooth augmented Lagrangian  method for twin bounded support vector machine

Fatemeh Bazikar; Saeed Ketabchi; Hossein Moosaei

doi:10.3934/naco.2021027

Article Contents

2022, Volume 12, Issue 4: 659-678. Doi: 10.3934/naco.2021027

This issue Previous Article Next Article A dual Bregman proximal gradient method for relatively-strongly convex optimization

Smooth augmented Lagrangian method for twin bounded support vector machine

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2.
Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran

^* Corresponding author: Saeed Ketabchi
^* Corresponding author: Saeed Ketabchi

Received: September 2020

Revised: June 2021

Early access: July 2021

Published: December 2022

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Abstract

In this paper, we propose a method for solving the twin bounded support vector machine (TBSVM) for the binary classification. To do so, we use the augmented Lagrangian (AL) optimization method and smoothing technique, to obtain new unconstrained smooth minimization problems for TBSVM classifiers. At first, the augmented Lagrangian method is recruited to convert TBSVM into unconstrained minimization programming problems called as AL-TBSVM. We attempt to solve the primal programming problems of AL-TBSVM by converting them into smooth unconstrained minimization problems. Then, the smooth reformulations of AL-TBSVM, which we called AL-STBSVM, are solved by the well-known Newton's algorithm. Finally, experimental results on artificial and several University of California Irvine (UCI) benchmark data sets are provided along with the statistical analysis to show the superior performance of our method in terms of classification accuracy and learning speed.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Illustration of TWSVM

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Figure 2. Results of linear TWSVM, TBSVM, I$ \nu $-TBSVM and AL-STBSVM on generated data set

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Table 1. Descriptions of the data sets from the UCI repository

Data set	$ \# $ Cases	$ \# $ Features	$ \# $ Classes	Source
Sonar	208	60	2	UCI
Cancer	699	9	2	UCI
Diabet	768	8	2	UCI
Wdbc	569	30	2	UCI
Ionosphere	351	34	2	UCI
Australian	690	14	2	UCI
Heart	270	14	2	UCI
Haberman	306	3	2	UCI
German	1000	24	2	UCI
House Votes	435	16	2	UCI
Spect	237	22	2	UCI
Splice	1000	60	2	UCI
Lung-cancer	32	56	2	UCI
F-diagnosis	100	9	2	UCI
Breast-cancer	116	9	2	UCI
Bupa	345	6	2	UCI
Pima	768	9	2	UCI
Housing	506	14	2	UCI

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Table 2. Comparison of linear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets

Data set	TWSVM	TBSVM	I$ \nu $-TBSVM	AL-STBSVM
size	Acc($ \% $), Time(s) $ c_{1}=c_{2} $	Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $	Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $	Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $
Sonar	76.88, 0.73	77.45, 1.53	70.29, 1.53	79.14, 0.52
208$ \times $ 60	$ 2^{6} $	$ 2^{6} $, $ 2^{-3} $	$ 2^{6} $, 0.2	1, $ 2^{-8} $
Cancer	96.13, 1.63	96.28, 2.63	90.13, 2.28	96.15, 1.99
699$ \times $9	$ 2^{-2} $	$ 2^{-5},2^{-3} $	1, 0.1	$ 2^{-5},2^{-8} $
Diabet	69.13, 1.90	73.58, 2.68	62.24, 3.39	71.74, 27.35
768$ \times $8	$ 2^{3} $	$ 2^{5},2^{-4} $	$ 2^{4} $, 0.5	$ 2^{3},2^{-7} $
Wdbc	93.66, 1.6	94.74, 2.37	90.87, 2	95.62, 6.16
569$ \times $30	$ 2^{6} $	$ 2^{3},2^{-3} $	$ 2^{5} $, 0.8	$ 2^{-5},2^{-4} $
Ionosphere	84.89, 0.86	86.30, 1.71	84.03, 1.67	85.21, 0.62
351$ \times $34	$ 2^{-3} $	$ 2^{-2},2^{-2} $	$ 2^{-2} $, 0.7	$ 2^{3},2^{-7} $
Australian	83.05, 1.62	84.05, 2.68	67.66, 3.30	83.90, 2.93
690 $ \times $ 14	$ 2^{5} $	$ 2^{5} $, 1	$ 2^{3} $, 0.6	$ 2,2^{4} $
Heart	84.44, 0.98	84.44, 1.57	83.70, 1.64	85.19, 0.63
270$ \times $14	$ 2^{2} $	$ 2,2^{3} $	$ 2^{2} $, 0.9	$ 2^{5},2^{-7} $
Haberman	74.51, 0.86	75.91, 1.64	72.91, 1.66	77.05, 0.55
306$ \times $3	$ 2^{-2} $	$ 2^{3},2^{2} $	1, 0.6	$ 2^{4},2^{4} $
German	73.9, 1.78	75.4, 3.94	61.8, 6.59	75.1, 11.42
1000$ \times $24	$ 2^{-2} $	$ 2^{-5},2^{-3} $	$ 2^{-3} $, 0.4	$ 2^{10},2^{2} $
House Votes	95.62, 0.85	95.85, 1.90	93.11, 1.67	96.08, 0.65
435$ \times $16	$ 2^{8} $	$ 2,2^{-3} $	1, 0.1	$ 2^{2},2^{2} $
Spect	68.60, 0.75	70.44, 1.59	73.57, 1.6	71.24, 0.55
237$ \times $22	$ 2^{-5} $	$ 2^{-5},2^{-6} $	$ 2^{9} $, 0.9	$ 2^{3},2^{4} $
Splice	75.40, 2.44	79.70, 3.77	78.20, 5.63	79.18, 10.19
1000$ \times $60	$ 2^{-5} $	$ 2^{-5},2^{-6} $	$ 2^{-5} $, 0.7	$ 2^{5},2^{4} $
Lung-cancer	82.5, 0.62	82.5, 1.40	85, 1.42	85.83, 0.42
32$ \times $56	$ 2^{-1} $	$ 2^{-2},2^{-3} $	$ 2^{-3} $, 0.5	$ 2^{6},2^{-8} $
F-diagnosis	76.49, 0.62	82.34, 1.40	68.35, 1.48	75.01, 0.49
100$ \times $9	$ 2^{5} $	$ 2^{8},2^{5} $	$ 2^{10} $, 0.2	$ 2^{4},2^{-7} $
Breast-cancer	72.52, 0.61	71.84, 1.38	57.04, 1.52	73.81, 0.45
116$ \times $9	$ 2^{6} $	$ 2^{3},2^{-2} $	$ 2^{6} $, 0.9	$ 2^{5},2^{-7} $
Bupa	64.35, 0.73	65.23, 1.69	69.29, 1.72	66.68, 0.58
345$ \times $6	$ 2^{-4} $	$ 2^{-3} $, 2	$ 2^{-4} $, 0.2	$ 2^{2},2^{4} $
Pima	71.36, 0.86	73.30, 1.64	64.72, 2.65	73.62, 2.04
768$ \times $9	$ 2^{-7} $	$ 2^{-5},2^{-3} $	$ 2^{-7} $, 0.2	$ 2^{2},2^{4} $
Housing	80.08, 0.96	80.24, 2.89	61.11, 2.01	93.09, 1.5
506$ \times $14	$ 2^{-8} $	$ 2^{-7},2^{-6} $	$ 2^{-7} $, 0.2	$ 2^{6},2^{-8} $
Avg.acc	79.08	80.53	74.11	83.31

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Table 3. Comparison of nonlinear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets

Dataset	TWSVM	TBSVM	I$ \nu $-TBSVM	AL-STBSVM
size	Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \gamma $	Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $	Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $, $ \gamma $	Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $
Sonar	84.53, 0.80	86.14, 1.62	83.27, 1.67	87.54, 0.84
208$ \times $ 60	$ 2^{5},2^{-6} $	$ 2^{6},2^{-3},2^{-3} $	$ 2^{5} $, 0.2, $ 2^{-3} $	$ 2^{2},2^{-8},2^{-6} $
Cancer	96.42, 4.71	96.42, 6.59	96.02, 2.90	96.85, 3.85
699$ \times $9	$ 2^{-2},2^{-6} $	$ 2^{-3},2^{-3},2^{-6} $	$ 2 $, 0.1, $ 2^{-4} $	$ 2^{-5},2^{-8},2^{-3} $
Diabet	65.49, 10.18	66.28, 8.21	65.11, 5.33	69.14, 21.28
768$ \times $8	$ 2^{5},2^{-6} $	$ 2^{5},2^{-4},2^{-6} $	$ 2^{4} $, 0.4, $ 2^{-3} $	$ 2^{5},2^{-6},2^{-3} $
Wdbc	62.74, 2.64	62.74, 3.53	91.76, 2.17	87.18, 10.69
569$ \times $30	$ 2^{5},2^{-6} $	$ 2^{4},2^{-3},2^{-6} $	$ 2^{-9} $, 0.3, $ 2^{-8} $	$ 2^{-5},2^{-6},2^{-1} $
Ionosphere	94.89, 1.42	94.54, 2.15	87.17, 1.88	94.89, 2.96
351$ \times $34	$ 2^{3},2^{-6} $	$ 2^{-4},2^{-2},2^{-3} $	$ 2^{-2} $, 0.2, $ 2^{-2} $	$ 2^{4},2^{-6},2^{-6} $
Australian	55.65, 4.89	55.22, 5.47	55.51, 6.94	65.21, 17.95
690$ \times $14	$ 2^{3},2^{-3} $	$ 2^{5},2,2^{-6} $	$ 2^{3} $, 0.5, $ 2^{-3} $	$ 2^{2},2^{4},2^{-5} $
Heart	82.59, 0.92	83.33, 1.86	81.48, 1.76	83.33, 1.58
270$ \times $14	$ 2^{2},2^{-3} $	$ 2,2,2^{-6} $	$ 2^{2} $, 0.9, $ 2^{9} $	$ 2^{5},2^{-7},2^{-6} $
Haberman	73.85, 1.28	73.52, 2.01	73.19, 1.88	73.52, 2.67
306$ \times $3	$ 2^{-1},2^{-6} $	$ 2,2,2^{-6} $	$ 1 $, 0.6, $ 2^{9} $	$ 2^{3},2^{4},2^{-6} $
German	70.1, 14.69	70.2, 14.34	70, 7.33	71.5, 35.33
1000$ \times $24	$ 2^{-2},2^{-6} $	$ 2^{-4},2^{-4},2^{-6} $	$ 2^{-3} $, 0.4, $ 2^{7} $	$ 2^{9},2^{4},2^{-6} $
House Votes	92.64, 1.22	93.55, 2.20	91.70, 1.78	94.71, 4.12
435$ \times $16	$ 2^{8},2^{-6} $	$ 2,2^{-2},2^{-6} $	$ 1 $, 0.1, $ 2^{-5} $	$ 2^{3},2^{4},2^{-3} $
Spect	71.89, 0.86	73.76, 1.66	68.90, 1.66	74.17, 1.45
237$ \times $22	$ 2^{-5},2^{-6} $	$ 2^{-3},2^{-6},2^{-4} $	$ 2^{-7} $, 0.6, $ 2^{-5} $	$ 2^{3},2^{4},2^{-1} $
Splice	76.71, 15.98	75.20, 16.58	74.41, 14.66	87.20, 42.10
1000$ \times $60	$ 2^{-6},2^{-6} $	$ 2^{-5},2^{-5},2^{-6} $	$ 2^{-4} $, 0.2, $ 2^{-3} $	$ 2^{5},2^{5},2^{-6} $
Lung-caner	80.83, 0.67	85.83, 1.41	84.16, 1.51	85, 0.35
32$ \times $56	$ 2,2^{-6} $	$ 2^{-2},2^{-3},2^{-6} $	$ 2^{-5} $, 0.1, $ 2^{-3} $	$ 2^{5},2^{-8},2^{-6} $
F-diagnosis	88.14, 0.64	88.14, 1.42	87.16, 1.59	89.25, 0.44
100$ \times $9	$ 2^{5},2^{-3} $	$ 2^{6},2^{5},2^{-3} $	$ 2^{10} $, 0.2, $ 2^{7} $	$ 2^{4},2^{-7},2^{-2} $
Breast-cancer	55.16, 0.65	55.16, 1.48	55.15, 1.56	60.31, 0.48
116$ \times $9	$ 2^{5},2^{-3} $	$ 2^{3},2^{-5},2^{-3} $	$ 2^{7} $, 0.9, $ 2^{-6} $	$ 2^{4},2^{-7},2^{-6} $
Bupa	64.37, 1.19	64.35, 2.02	65.80, 1.94	66.36, 2.62
345$ \times $6	$ 2^{-4},2^{-6} $	$ 2^{3},2,2^{-3} $	$ 2^{-5} $, 0.2, $ 2^{-3} $	$ 2^{3},2^{4},2^{-6} $
Pima	65.63, 9.66	66.15, 8.08	65.12, 4.74	69, 20.07
768$ \times $9	$ 2^{-3},2^{-6} $	$ 2^{-6},2^{-6},2^{-6} $	$ 2^{-3} $, 0.6, $ 2^{-6} $	$ 2^{2},2^{5},2^{-3} $
Housing	93.09, 2.12	93.09, 3.65	93.09, 4.30	93.09, 7.01
506$ \times $14	$ 2^{-6},2^{6} $	$ 2^{-5},2^{-3},2^{7} $	$ 2^{10} $, 0.7, $ 2^{7} $	$ 2^{7},2^{-8},2^{-6} $
Avg.acc	76.37	76.86	77.16	80.46

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Table 4. Rank of accuracy linear classifiers on UCI benchmark data sets

Data set	TWSVM	TBSVM	I$ \nu $-TBSVM	AL-STBSVM
Sonar	3	2	4	1
Cancer	3	1	4	2
Diabet	3	1	4	2
Wdbc	3	2	4	1
Ionosphere	3	1	4	2
Australian	3	1	4	2
Heart	2.5	2.5	4	1
Haberman	3	2	4	1
German	3	1	4	2
House Votes	3	2	4	1
Spect	4	3	1	2
Splice	4	1	3	2
Lung-cancer	3.5	3.5	2	1
F-diagnosis	2	1	4	3
Breast-cancer	2	3	4	1
Bupa	4	3	1	2
Pima	3	2	4	1
Housing	3	2	4	1
Average rank	3.06	1.89	3.5	1.56

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Table 5. Rank of accuracy nonlinear classifiers on UCI benchmark data sets

Data set	TWSVM	TBSVM	I$ \nu $-TBSVM	AL-STBSVM
Sonar	3	2	4	1
Cancer	2.5	2.5	4	1
Diabet	3	2	4	1
Wdbc	3.5	3.5	1	2
Ionosphere	1.5	3	4	1.5
Australian	2	4	3	1
Heart	3	1.5	4	1.5
Haberman	1	2.5	4	2.5
German	3	2	4	1
House Votes	3	2	4	1
Spect	3	2	4	1
Splice	2	3	4	1
Lung-cancer	4	1	3	2
F-diagnosis	2.5	2.5	4	1
Breast-cancer	2.5	2.5	4	1
Bupa	3	4	2	1
Pima	2	3	4	1
Housing	2.5	2.5	2.5	2.5
Average rank	2.61	2.53	3.53	1.33

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