Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations
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| Format: | Preprint |
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2025
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| _version_ | 1866911298189524992 |
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| author | Bertsimas, Dimitris Iglesias, Caio de Prospero Johnson, Nicholas A. G. |
| author_facet | Bertsimas, Dimitris Iglesias, Caio de Prospero Johnson, Nicholas A. G. |
| contents | We study Sparse Multiple Kernel Learning (SMKL), which is the problem of selecting a sparse convex combination of prespecified kernels for support vector binary classification. Unlike prevailing l1 regularized approaches that approximate a sparsifying penalty, we formulate the problem by imposing an explicit cardinality constraint on the kernel weights and add an l2 penalty for robustness. We solve the resulting non-convex minimax problem via an alternating best response algorithm with two subproblems: the alpha subproblem is a standard kernel SVM dual solved via LIBSVM, while the beta subproblem admits an efficient solution via the Greedy Selector and Simplex Projector algorithm. We reformulate SMKL as a mixed integer semidefinite optimization problem and derive a hierarchy of semidefinite convex relaxations which can be used to certify near-optimality of the solutions returned by our best response algorithm and also to warm start it. On ten UCI benchmarks, our method with random initialization outperforms state-of-the-art MKL approaches in out-of-sample prediction accuracy on average by 3.34 percentage points (relative to the best performing benchmark) while selecting a small number of candidate kernels in comparable runtime. With warm starting, our method outperforms the best performing benchmark's out-of-sample prediction accuracy on average by 4.05 percentage points. Our convex relaxations provide a certificate that in several cases, the solution returned by our best response algorithm is the globally optimal solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations Bertsimas, Dimitris Iglesias, Caio de Prospero Johnson, Nicholas A. G. Machine Learning We study Sparse Multiple Kernel Learning (SMKL), which is the problem of selecting a sparse convex combination of prespecified kernels for support vector binary classification. Unlike prevailing l1 regularized approaches that approximate a sparsifying penalty, we formulate the problem by imposing an explicit cardinality constraint on the kernel weights and add an l2 penalty for robustness. We solve the resulting non-convex minimax problem via an alternating best response algorithm with two subproblems: the alpha subproblem is a standard kernel SVM dual solved via LIBSVM, while the beta subproblem admits an efficient solution via the Greedy Selector and Simplex Projector algorithm. We reformulate SMKL as a mixed integer semidefinite optimization problem and derive a hierarchy of semidefinite convex relaxations which can be used to certify near-optimality of the solutions returned by our best response algorithm and also to warm start it. On ten UCI benchmarks, our method with random initialization outperforms state-of-the-art MKL approaches in out-of-sample prediction accuracy on average by 3.34 percentage points (relative to the best performing benchmark) while selecting a small number of candidate kernels in comparable runtime. With warm starting, our method outperforms the best performing benchmark's out-of-sample prediction accuracy on average by 4.05 percentage points. Our convex relaxations provide a certificate that in several cases, the solution returned by our best response algorithm is the globally optimal solution. |
| title | Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.21890 |