支持向量機是一種有效的二元分類器，在各種應用中具有出色的性能。然而，在
處理大型資料集或來自分散來源的資料時，計算資源的限制和資料隱私問題可能
會阻礙支持向量機的效能。在本文中，我們研究分散共識支持向量機，它具有兩
個主要優點：它允許每位工作者在不共享資料的情況下推導出更通用的超平面，
從而保持資料的隱私；它能夠將大型問題分解為可處理的子問題，透過分散式計
算提高處理速度。儘管如此，由於分散共識支持向量機的目標函數的不可微性而
面臨挑戰，我們採用了平滑支持向量機的處理方法來解決這個問題，結合平滑函
數來增強目標函數的可微性，從而產生了所謂的分散共識平滑支持向量機，它利
用1-範數進行懲罰計算，優化效率和準確性。最後，我們透過多次數值實驗驗證
該演算法的效能。

The support vector machine (SVM) is an effective binary classifier with excellent
performance across various applications. However, computing limitations and data
privacy concerns can hinder SVM’s performance when handling large datasets or
data from distributed sources. In this thesis, we study the distributed consensus
SVM, which offers two primary advantages: it maintains data privacy by allowing
each worker to derive a more generalized hyperplane without data sharing, and it
enables the decomposition of large-scale problems into manageable sub-problems
for enhanced processing speed through distributed computing. Nonetheless, the
distributed consensus SVM faces challenges due to the non-differentiability of its
objective function. We adopt the smoothing SVM approach to address this issue,
incorporating a smoothing function to enhance function differentiability. It leads
to the so-called distributed consensus smoothing SVM, which utilizes the 1-norm
for penalty calculation, optimizing efficiency and accuracy. Finally, we validate the
performance of this algorithm through several numerical experiments.

[1] A. Beck, Introduction to Nonlinear Optimization: Theory, Algorithms, and Ap
plications with Matlab, MOS-SIAM Series on Optimization, SIAM, Philadel
phia, PA, 2014.
[2] B. E. Boser, I. M. Guyon, and V. N. Vapnik, A training algorithm for optimal
margin classifiers, COLT ’92: Proceedings of the Fifth Annual Workshop on
Computational Learning Theory, July 1992, pp. 144-152.
[3] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed opti
mization and statistical learning via the ADMM, Foundations and Trends in
Machine Learning, 3 (2010), pp. 1-122.
[4] J. Broniarek, SVM-parallel consensus and sharing optimization, Project Report,
Signal Processing@Data Science, February 2020.
[5] D. Calvetti and E. Somersalo, Mathematics of Data Science: A Computational
Approach to Clustering and Classification, SIAM, Philadelphia, PA, 2021.
[6] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear
and mixed complementarity problems, Computational Optimization and Appli
cations, 5 (1996), pp. 97-138.
[7] C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities
and linear complementarity problems, Mathematical Programming, 71 (1995),
pp. 51-69.
[8] H.-H. Chen, Distributed consensus reduced support vector machine, Master
Thesis, National Chiao Tung University, July 2019.
[9] L. J. Chien, Y. J. Lee, Z. P. Kao, and C. C. Chang, Robust 1-norm soft mar
gin smooth support vector machine, In: C. Fyfe et al. (eds.), Intelligent Data
Engineering and Automated Learning– IDEAL 2010, pp. 145-152, 2010.
37
[10] L. R. Jen and Y.-J. Lee, Clustering model selection for reduced support vec
tor machines, In: Z. R. Yang et al. (eds.), Intelligent Data Engineering and
Automated Learning– IDEAL 2004, pp. 714-719, 2004.
[11] Y.-J. Lee and S.-Y. Huang, Reduced support vector machines: A statistical
theory, IEEE Transactions on Neural Networks, 18 (2007), pp. 1-13.
[12] Y.-J. Lee and O. L. Mangasarian, SSVM: A smooth support vector machine
for classification, Computational Optimization and Applications, 20 (2001), pp.
5-22.
[13] Y.-J. Lee and O. L. Mangasarian, RSVM: reduced support vector machines,
Proceedings of the 2001 SIAM International Conference on Data Mining
(SDM), April 2001.
[14] J. Li, E. Elhamifar, I.-J. Wang, and R. Vidal, Consensus with robustness to
outliers via distributed optimization, 49th IEEE Conference on Decision and
Control (CDC), Atlanta, GA, USA, 2010, pp. 2111-2117.
[15] X. Liu, Y. Ding, and F. S. Bao, General scaled support vector machines, ICMLC
2011: Proceedings of the 3rd International Conference on Machine Learning and
Computing, Piscataway, NJ, 2011.
[16] H.-Y. Lo, Incremental reduced support vector machines, Master Thesis, Na
tional Taiwan University of Science and Technology, July 2004.
[17] O. L. Mangasarian, Generalized support vector machines, Mathematics Pro
gramming Technical Reports, Department of Computer Sciences, University of
Wisconsin-Madison, October 1998.
[18] S. W. Purnami, J. M. Zain, and A. Embong, Reduced support vector machine
based on k-mode clustering for classification large categorical dataset, In: J. M.
Zain et al. (eds.), Software Engineering and Computer Systems, ICSECS 2011,
Part II, CCIS 180, pp. 694-702, 2011.
38
[19] F. Rahimi, Support vector machines with convex combination of kernels, Master
Thesis, Concordia University, Canada, 2018.
[20] A. Rumpf, Overview of dual ascent, Short Notes, Department of Applied Math
ematics, Illinois Institute of Technology, Chicago, IL, February 19, 2019.