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Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization

arXiv:2604.03419v1 Announce Type: new Abstract: 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 $\eta_i$, 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. Theor

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Mohammadreza Rostami, Solmaz S. Kia · · 1 min read · 4 views
#cs.LG #math.CO

arXiv:2604.03419v1 Announce Type: new Abstract: 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 $\eta_i$, 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 $\tau_{\mathrm{eff}}=\max\{\tau,1-c\}$, interpolating between the threshold-based guarantee and the low-curvature regime where \textit{ATCG} recovers the performance of CG. 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.

Executive Summary

This article introduces Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization (ATCG), a novel algorithm that improves upon the Sequential Greedy (SG) and Continuous Greedy (CG) methods for submodular maximization under matroid constraints. ATCG achieves a curvature-aware approximation guarantee and significantly reduces communication overhead by adaptively expanding each agent's active set. Experimental results demonstrate ATCG's effectiveness in solving a prototype selection problem, outperforming CG in terms of communication efficiency while maintaining comparable objective values. The proposed algorithm paves the way for scalable submodular optimization in applications involving large-scale datasets and complex decision-making processes.

Key Points

  • ▸ ATCG achieves a curvature-aware approximation guarantee by adapting the active-set expansion strategy.
  • ▸ The algorithm reduces communication overhead by selectively transmitting feature embeddings based on a per-partition progress ratio.
  • ▸ Experiments demonstrate ATCG's effectiveness in solving a prototype selection problem, outperforming CG in terms of communication efficiency.

Merits

Strength in Adaptability

ATCG's adaptive threshold-driven approach allows for efficient active-set expansion, making it a more scalable solution for large-scale submodular optimization problems.

Improved Communication Efficiency

By selectively transmitting feature embeddings, ATCG reduces communication overhead, making it a more practical solution for distributed optimization applications.

Demerits

Complexity in Implementation

The adaptive threshold-driven approach may introduce additional complexity in the implementation of ATCG, requiring careful tuning of parameters to achieve optimal performance.

Limited Experimental Evaluation

While the experiments demonstrate ATCG's effectiveness in solving a specific prototype selection problem, further evaluation is needed to assess its performance on a broader range of applications.

Expert Commentary

The article presents a novel and innovative approach to scalable submodular optimization, addressing the limitations of existing algorithms such as SG and CG. The experimental results demonstrate ATCG's effectiveness in solving a specific prototype selection problem, and the algorithm's adaptability and communication efficiency make it a promising solution for large-scale distributed optimization applications. However, further evaluation is needed to assess its performance on a broader range of applications and to address potential implementation complexities.

Recommendations

  • ✓ Recommendation 1: Further research is needed to evaluate ATCG's performance on a broader range of applications and to assess its scalability in large-scale distributed optimization settings.
  • ✓ Recommendation 2: The development of ATCG highlights the need for further research in scalable submodular optimization, with potential policy implications for addressing complex decision-making problems in applications such as resource allocation and active learning.

Sources

Original: arXiv - cs.LG

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