Multinoulli Extension: A Lossless Continuous Relaxation for Partition-Constrained Subset Selection
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915881495298048 |
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| author | Zhang, Qixin Huang, Wei Sun, Yan Shu, Yao Yu, Yi Tao, Dacheng |
| author_facet | Zhang, Qixin Huang, Wei Sun, Yan Shu, Yao Yu, Yi Tao, Dacheng |
| contents | Identifying the most representative subset for a close-to-submodular objective while satisfying the predefined partition constraint is a fundamental task with numerous applications in machine learning. However, the existing distorted local-search methods are often hindered by their prohibitive query complexities and the rigid requirement for prior knowledge of difficult-to-obtain structural parameters. To overcome these limitations, we introduce a novel algorithm titled Multinoulli-SCG, which not only is parameter-free, but also can achieve the same approximation guarantees as the distorted local-search methods with significantly fewer function evaluations. More specifically, when the objective function is monotone $α$-weakly DR-submodular or $(γ,β)$-weakly submodular, our Multinoulli-SCG algorithm can attain a value of $(1-e^{-α})\text{OPT}-ε$ or $(\frac{γ^{2}(1-e^{-(β(1-γ)+γ^2)})}{β(1-γ)+γ^2})\text{OPT}-ε$ with only $O(1/ε^{2})$ function evaluations, where OPT denotes the optimal value. The cornerstone of our Multinoulli-SCG algorithm is an innovative continuous-relaxation framework named Multinoulli Extension(ME), which can effectively convert the discrete subset selection problem subject to partition constraints into a solvable continuous maximization focused on learning the optimal multinoulli priors across the concerned partition. In sharp contrast with the well-established multi-linear extension for submodular subset selection, a notable advantage of our proposed ME is its intrinsic capacity to provide a lossless rounding scheme for any set function. Furthermore, based on our proposed ME, we also present two novel online algorithms, namely, Multinoulli-OSCG and Multinoulli-OSGA, for the unexplored online subset selection problems over partition constraints. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21492 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multinoulli Extension: A Lossless Continuous Relaxation for Partition-Constrained Subset Selection Zhang, Qixin Huang, Wei Sun, Yan Shu, Yao Yu, Yi Tao, Dacheng Machine Learning Optimization and Control Identifying the most representative subset for a close-to-submodular objective while satisfying the predefined partition constraint is a fundamental task with numerous applications in machine learning. However, the existing distorted local-search methods are often hindered by their prohibitive query complexities and the rigid requirement for prior knowledge of difficult-to-obtain structural parameters. To overcome these limitations, we introduce a novel algorithm titled Multinoulli-SCG, which not only is parameter-free, but also can achieve the same approximation guarantees as the distorted local-search methods with significantly fewer function evaluations. More specifically, when the objective function is monotone $α$-weakly DR-submodular or $(γ,β)$-weakly submodular, our Multinoulli-SCG algorithm can attain a value of $(1-e^{-α})\text{OPT}-ε$ or $(\frac{γ^{2}(1-e^{-(β(1-γ)+γ^2)})}{β(1-γ)+γ^2})\text{OPT}-ε$ with only $O(1/ε^{2})$ function evaluations, where OPT denotes the optimal value. The cornerstone of our Multinoulli-SCG algorithm is an innovative continuous-relaxation framework named Multinoulli Extension(ME), which can effectively convert the discrete subset selection problem subject to partition constraints into a solvable continuous maximization focused on learning the optimal multinoulli priors across the concerned partition. In sharp contrast with the well-established multi-linear extension for submodular subset selection, a notable advantage of our proposed ME is its intrinsic capacity to provide a lossless rounding scheme for any set function. Furthermore, based on our proposed ME, we also present two novel online algorithms, namely, Multinoulli-OSCG and Multinoulli-OSGA, for the unexplored online subset selection problems over partition constraints. |
| title | Multinoulli Extension: A Lossless Continuous Relaxation for Partition-Constrained Subset Selection |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2603.21492 |