Multinoulli Extension: A Lossless Continuous Relaxation for Partition-Constrained Subset Selection

2026-03-22

AI Summary

This work addresses the problem of approximate submodular subset selection under partition constraints, which existing methods often struggle with due to high query complexity or reliance on hard-to-obtain structural parameters. The authors propose the Multinoulli-SCG algorithm, built upon a novel continuous relaxation framework termed the Multinoulli Extension, which transforms the discrete optimization problem into a continuous one without requiring prior knowledge of problem-specific parameters. This framework enables lossless rounding for arbitrary set functions and further yields two online variants—Multinoulli-OSCG and Multinoulli-OSGA—suitable for dynamic settings. With only $O(1/\varepsilon^2)$ function evaluations, the approach achieves a $(1 - e^{-\alpha})\text{OPT} - \varepsilon$ approximation guarantee for monotone $\alpha$-weakly DR-submodular objectives and corresponding theoretical guarantees for $(\gamma, \beta)$-weakly submodular functions.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

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.

Problem

• subset selection
• partition constraints
• submodular optimization
• representative subset
• combinatorial optimization

Innovation

• Multinoulli Extension
• continuous relaxation
• partition-constrained subset selection
• weak submodularity
• lossless rounding