Wasserstein Distributionally Robust Multiclass Support Vector Machine
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arXiv
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| Format: | Preprint |
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2024
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| author | Ibrahim, Michael Rozas, Heraldo Gebraeel, Nagi |
| author_facet | Ibrahim, Michael Rozas, Heraldo Gebraeel, Nagi |
| contents | We study the problem of multiclass classification for settings where data features $\mathbf{x}$ and their labels $\mathbf{y}$ are uncertain. We identify that distributionally robust one-vs-all (OVA) classifiers often struggle in settings with imbalanced data. To address this issue, we use Wasserstein distributionally robust optimization to develop a robust version of the multiclass support vector machine (SVM) characterized by the Crammer-Singer (CS) loss. First, we prove that the CS loss is bounded from above by a Lipschitz continuous function for all $\mathbf{x} \in \mathcal{X}$ and $\mathbf{y} \in \mathcal{Y}$, then we exploit strong duality results to express the dual of the worst-case risk problem, and we show that the worst-case risk minimization problem admits a tractable convex reformulation due to the regularity of the CS loss. Moreover, we develop a kernel version of our proposed model to account for nonlinear class separation, and we show that it admits a tractable convex upper bound. We also propose a projected subgradient method algorithm for a special case of our proposed linear model to improve scalability. Our numerical experiments demonstrate that our model outperforms state-of-the art OVA models in settings where the training data is highly imbalanced. We also show through experiments on popular real-world datasets that our proposed model often outperforms its regularized counterpart as the first accounts for uncertain labels unlike the latter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_08409 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein Distributionally Robust Multiclass Support Vector Machine Ibrahim, Michael Rozas, Heraldo Gebraeel, Nagi Machine Learning We study the problem of multiclass classification for settings where data features $\mathbf{x}$ and their labels $\mathbf{y}$ are uncertain. We identify that distributionally robust one-vs-all (OVA) classifiers often struggle in settings with imbalanced data. To address this issue, we use Wasserstein distributionally robust optimization to develop a robust version of the multiclass support vector machine (SVM) characterized by the Crammer-Singer (CS) loss. First, we prove that the CS loss is bounded from above by a Lipschitz continuous function for all $\mathbf{x} \in \mathcal{X}$ and $\mathbf{y} \in \mathcal{Y}$, then we exploit strong duality results to express the dual of the worst-case risk problem, and we show that the worst-case risk minimization problem admits a tractable convex reformulation due to the regularity of the CS loss. Moreover, we develop a kernel version of our proposed model to account for nonlinear class separation, and we show that it admits a tractable convex upper bound. We also propose a projected subgradient method algorithm for a special case of our proposed linear model to improve scalability. Our numerical experiments demonstrate that our model outperforms state-of-the art OVA models in settings where the training data is highly imbalanced. We also show through experiments on popular real-world datasets that our proposed model often outperforms its regularized counterpart as the first accounts for uncertain labels unlike the latter. |
| title | Wasserstein Distributionally Robust Multiclass Support Vector Machine |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2409.08409 |