### Abstract

In this paper we study the problem of designing SVM classifiers when the kernel matrix,

**, is affected by uncertainty. Specifically***K***is modeled as a positive affine combination of given positive semi definite kernels, with the coefficients ranging in a norm-bounded uncertainty set. We treat the problem using the Robust Optimization methodology. This reduces the uncertain SVM problem into a deterministic conic quadratic problem which can be solved in principle by a polynomial time Interior Point (IP) algorithm. However, for large-scale classification problems, IP methods become intractable and one has to resort to first-order gradient type methods. The strategy we use here is to reformulate the robust counterpart of the uncertain SVM problem as a saddle point problem and employ a special gradient scheme which works directly on the convex-concave saddle function. The algorithm is a simplified version of a general scheme due to Juditski and Nemirovski (2011). It achieves an***K**O(1/T*reduction of the initial error after^{2})*T*iterations. A comprehensive empirical study on both synthetic data and real-world protein structure data sets show that the proposed formulations achieve the desired robustness, and the saddle point based algorithm outperforms the IP method significantly.Original language | English |
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Pages (from-to) | 2923-2954 |

Journal | Journal of Machine Learning Research |

Volume | 13 |

Publication status | Published - 2012 |

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## Cite this

Ben-Tal, A., Bhadra, S., Bhattacharyya, C., & Nemirovski, A. (2012). Efficient methods for robust classification under uncertainty in kernel matrices.

*Journal of Machine Learning Research*,*13*, 2923-2954. http://jmlr.csail.mit.edu/papers/v13/ben-tal12a.html