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

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 $\alpha$-weakly DR-submodular or $(\gamma,\beta)$-weakly submodular, our Multinoulli-SCG algorithm can attain a value of $(1-e^{-\alpha})\text{OPT}-\epsilon$ or $(\frac{\gamma^{2}(1-e^{-(\beta(1-\gamma)+\gamma^2)})}{\beta(1-\gamma)+\gamma^2})\text{OPT}-\epsilon$ with only $O(1/\epsilon^{2})$ 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.