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Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization

Mohammadreza Rostami, Solmaz S. Kia

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 $η_i$, 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. 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.

Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization

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 -approximation due to irrevocable selections, Continuous Greedy (CG) attains the optimal -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 , 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 , 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.

Paper Structure

This paper contains 11 sections, 2 theorems, 40 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Server-Assisted Distributed Realization
  4. Proposed Algorithm and Theoretical Guarantees
  5. Progress ratio.
  6. Ascent direction and update.
  7. Distributed realization.
  8. Communication efficiency.
  9. Performance Guarantee of ATCG
  10. Numerical Evaluation
  11. Conclusion

Key Result

Theorem III.1

Consider the continuous-time version of ATCG for a monotone submodular function $f:2^{\mathcal{P}}\to\mathbb{R}_{+}$, and let $F$ denote its multilinear extension over the matroid polytope $\mathcal{M}$. By construction of ATCG, the active-set update rule ensures that, at each iteration and for ever for some $\tau\in(0,1]$ and all $t\in[0,1]$. Let $\mathbf{x}^\star\in\mathcal{M}$ be an optimal sol

Figures (5)

  • Figure 1: Illustration of the early-commitment effect of SG (left) and its mitigation via CG (right) in a 2D sensor placement task. Deployment points (solid circles) are selected from candidates (red $\times$) to cover a clustered data distribution (blue dots) under partition-matroid constraints, where each colored region represents the candidate set of the corresponding agent.
  • Figure 2: Inter-class mean RBF similarity matrix for the CIFAR animal subset, illustrating the coupling effect between the animal images.
  • Figure 3: Objective $f$ vs. iteration for CG and ATCG ($\tau{=}0.30$) on the CIFAR animal subset.
  • Figure 4: Cumulative bytes uploaded to the server for CG and ATCG.
  • Figure 5: Total active-set size $\sum_i|\mathcal{A}_i|$ vs. iteration for ATCG.

Theorems & Definitions (7)

  • Remark 1: Exact-gradient and continuous-time idealization
  • Theorem III.1: Performance guarantee of ATCG, Algorithm \ref{['alg:atcg']}
  • Remark 2: Comparison with classical CG
  • Theorem III.2: Curvature-aware guarantee for ATCG
  • Remark 3: Communication efficiency under low curvature
  • proof : Proof of Theorem \ref{['thm:atcg_tau']}
  • proof : Proof of Theorem \ref{['cor:atcg_curvature']}