Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations

• URL: http://arxiv.org/abs/2511.21890v2
• Date: Tue, 02 Dec 2025 04:05:44 GMT
• Title: Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations
• Authors: Dimitris Bertsimas, Caio de Prospero Iglesias, Nicholas A. G. Johnson,
• Abstract summary: We consider the problem of selecting a sparse combination of prespecified kernels for support vector binary classification.<n>We derive a hierarchy of predictions which can be used to certify the solutions returned by our best response and also to warm start it.
• Score: 4.86840788259762
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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