Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization
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| Format: | Preprint |
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2026
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| _version_ | 1866914578853527552 |
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| author | Rostami, Mohammadreza Kia, Solmaz S. |
| author_facet | Rostami, Mohammadreza Kia, Solmaz S. |
| contents | Submodular maximization under matroid constraints is a fundamental problem in combinatorial optimization with applications in sensing, data summarization, active learning, and resource allocation. While the Sequential Greedy (SG) algorithm achieves only a $\frac{1}{2}$-approximation due to irrevocable selections, Continuous Greedy (CG) attains the optimal $\bigl(1-\frac{1}{e}\bigr)$-approximation via the multilinear relaxation, at the cost of a progressively dense decision vector that forces agents to exchange feature embeddings for nearly every ground-set element. We propose \textit{ATCG} (\underline{A}daptive \underline{T}hresholded \underline{C}ontinuous \underline{G}reedy), which gates gradient evaluations behind a per-partition progress ratio $η_i$, expanding each agent's active set only when current candidates fail to capture sufficient marginal gain, thereby directly bounding which feature embeddings are ever transmitted. Theoretical analysis establishes a curvature-aware approximation guarantee with effective factor $τ_{\mathrm{eff}}=\max\{τ,1-c\}$, interpolating between the threshold-based guarantee and the low-curvature regime where \textit{ATCG} recovers the performance of CG. This shows that the problem structure, as captured by curvature, determines the amount of coordination and communication required to approach full-CG performance. Experiments on a class-balanced prototype selection problem over a subset of the CIFAR-10 animal dataset show that \textit{ATCG} achieves objective values comparable to those of the full CG method while substantially reducing communication overhead through adaptive active-set expansion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization Rostami, Mohammadreza Kia, Solmaz S. Machine Learning Combinatorics Submodular maximization under matroid constraints is a fundamental problem in combinatorial optimization with applications in sensing, data summarization, active learning, and resource allocation. While the Sequential Greedy (SG) algorithm achieves only a $\frac{1}{2}$-approximation due to irrevocable selections, Continuous Greedy (CG) attains the optimal $\bigl(1-\frac{1}{e}\bigr)$-approximation via the multilinear relaxation, at the cost of a progressively dense decision vector that forces agents to exchange feature embeddings for nearly every ground-set element. We propose \textit{ATCG} (\underline{A}daptive \underline{T}hresholded \underline{C}ontinuous \underline{G}reedy), which gates gradient evaluations behind a per-partition progress ratio $η_i$, expanding each agent's active set only when current candidates fail to capture sufficient marginal gain, thereby directly bounding which feature embeddings are ever transmitted. Theoretical analysis establishes a curvature-aware approximation guarantee with effective factor $τ_{\mathrm{eff}}=\max\{τ,1-c\}$, interpolating between the threshold-based guarantee and the low-curvature regime where \textit{ATCG} recovers the performance of CG. This shows that the problem structure, as captured by curvature, determines the amount of coordination and communication required to approach full-CG performance. Experiments on a class-balanced prototype selection problem over a subset of the CIFAR-10 animal dataset show that \textit{ATCG} achieves objective values comparable to those of the full CG method while substantially reducing communication overhead through adaptive active-set expansion. |
| title | Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization |
| topic | Machine Learning Combinatorics |
| url | https://arxiv.org/abs/2604.03419 |