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 $\frac{1}{2}$-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 $η<em>i$, 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 $τ</em>{\mathrm{eff}}=\max{τ,1-c}$, 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.

• The paper proposes ATCG, integrating adaptive threshold gating into the continuous greedy framework to reduce communication overhead in distributed settings.
• It offers theoretical guarantees interpolating between the full (1-1/e) approximation and improved results under low curvature by dynamically expanding active sets.
• Empirical results on CIFAR-10 demonstrate that ATCG nearly matches full CG performance while significantly limiting the number of transmitted feature embeddings.

Adaptive Threshold-Driven Continuous Greedy for Scalable Submodular Optimization

Problem Context and Prior Art

Submodular maximization over matroid constraints constitutes a central combinatorial optimization problem with extensive applications in machine learning, control, and networked systems. The monotone submodular maximization framework, especially under matroid constraints, models various resource allocation, sensor placement, and data summarization objectives. The difficulty of this maximization arises from NP-hardness, which precludes tractable exact solutions for large-scale problems.

The Sequential Greedy (SG) approach selects elements with the highest marginal gain in a sequential, irrevocable manner. SG provides a $(1/2)$-approximation under general matroid constraints, but its myopic commitment results in suboptimal configurations, particularly when early selections restrict subsequent optimization opportunities.

The Continuous Greedy (CG) algorithm leverages the multilinear extension relaxation of submodular functions, optimizing over the matroid polytope. CG maintains a fractional decision variable and, at each iteration, makes an infinitesimal move along the direction of steepest ascent given by the gradient of the extension. This approach achieves the tight $(1-1/e)$-approximation. Crucially, CG does not commit early and explores a dense fractional solution space, which theoretically closes the optimality gap present in SG.

However, the communication complexity of CG, especially in distributed/partitioned-agent settings, is prohibitive. Each agent ultimately needs to have access to feature embeddings for nearly every element in the ground set, incurring communication costs scaling with $|P|$—a significant bottleneck for large $P$.

The ATCG Algorithm: Main Contributions

The paper proposes the Adaptive Thresholded Continuous Greedy (ATCG) method to address the communication bottleneck in distributed submodular maximization via CG. ATCG introduces a threshold-driven gating mechanism for gradient computation and active set expansion. Each agent restricts gradient evaluations to a dynamically expanding active set $A_i$; the set is expanded only if the ratio ($\eta_i$) of the best marginal gain within the active set to that over the entire partition falls below a user-defined threshold $T$. This explicit gating achieves two goals:

• Communication is bounded by the cumulative size of active sets (which is generally much less than $|P|$) since feature embeddings are transmitted only upon first activation of each element.
• The approximation guarantees interpolate between classical CG and a threshold-based variant, capturing the best tradeoff between communication efficiency and objective performance.

Formally, at each iteration, if the active set's coverage is at least a $T$-fraction of the maximum marginal gain, no expansion occurs. Otherwise, the most promising inactive element is added to the active set and its corresponding embedding is communicated.

Theoretical Guarantees

ATCG preserves the essential ascent structure of CG with guarantees parameterized by the threshold $T$. The main theoretical results can be summarized as:

• Threshold-Approximate Guarantee: If the per-partition active sets always maintain at least $(1-1/e)$0 coverage of the maximal marginal gain, the multilinear extension value after unit-time CG evolution satisfies $(1-1/e)$1. For $(1-1/e)$2 (i.e., full coverage as in standard CG), the classical $(1-1/e)$3 performance is recovered.
• Curvature-Aware Guarantee: For functions with low total curvature $(1-1/e)$4, ATCG's approximation factor improves further. If per-partition active sets contain the individually best singleton element, the effective rate $(1-1/e)$5 applies. As function curvature diminishes, ATCG approaches the approximation guarantee of classical CG even with relatively small active sets.

These results provide explicit, analytically tight trade-offs between communication cost (via threshold $(1-1/e)$6 and the structure of $(1-1/e)$7) and achievable objective value.

Algorithmic Details and Communication Protocol

ATCG admits an efficient server-assisted distributed realization. In this architecture, $(1-1/e)$8 agents own distinct partitions of the ground set, maintaining local sub-vectors of the global decision variable. A central server coordinates the evolution by:

• Broadcasting the current active embedding set $(1-1/e)$9 and (incrementally updated) global decision vector $|P|$0 to all agents.
• Aggregating updates and new feature embeddings from agents as active sets expand.
• Orchestrating distributed Monte Carlo gradient estimation by agents over only their local active sets, enabled by cached embeddings.
• Performing rounding (pipage or swap) to recover a feasible integral solution at the end of the trajectory.

Crucially, communication is strictly limited to the number of unique elements that ever enter the active sets. Once all agents reach the state where their active sets capture threshold coverage, no further feature transmission is required, and the overall communication overhead plateaus—contrasting strongly with standard CG, which leads to dense broadcast requirements throughout optimization.

Empirical Assessment

ATCG is benchmarked on a prototype selection task over a partitioned version of the CIFAR-10 animal subset, instantiating non-trivial cross-partition coupling and invoking a monotone submodular facility-location objective. Using a server-assisted distributed protocol, feature embeddings are exchanged only upon element activation. Major empirical findings are:

• Objective Value: ATCG matches the performance of full CG to within 1% (final objective values: 143.63 for ATCG versus 144.89 for full CG), demonstrating negligible empirical loss from active set restriction.
• Communication Efficiency: The number of feature embeddings ever transmitted to the server is substantially reduced. The upload curve flattens early, exhibiting the effect of adaptive active set stabilization. After sufficient active set expansion in early iterations, no further uploads are required, validating the communication bounds established in the theoretical analysis.
• Curvature Influence: Due to the low functional curvature arising from cross-category similarity structure in the data, small active sets quickly reach effective coverage, allowing ATCG to closely track CG with minimal overhead.

Practical and Theoretical Implications

ATCG advances the state of the art in scalable submodular optimization under communication constraints. The algorithm's use of partition-wise adaptive gating makes it particularly suitable for large-scale, distributed, or federated environments, such as sensor networks and decentralized learning. The guarantee structure—parameterized by both algorithmic thresholds and the objective's curvature—allows practitioners explicit control over the tradeoff between communication cost and solution quality.

Theoretically, the results refine the understanding of gradient truncation and sparsity mechanisms in continuous-domain submodular maximization. The curvature-aware extension bridges classical approximation bounds (curvature-dependent for discrete greedy, curvature-free for CG) within one unified framework.

Future Directions

There are several clear avenues for further research:

• Extension of ATCG to fully decentralized architectures, obviating the central coordination bottleneck.
• Adaptive scheduling or dynamic tuning of the coverage threshold $|P|$1 during optimization, potentially exploiting observed local curvature structure or measured marginal gain decay.
• Extension to non-monotone submodular maximization, where gradient ascent directions and rounding become more involved.

Conclusion

The Adaptive Thresholded Continuous Greedy (ATCG) method provides a scalable and communication-efficient algorithm for monotone submodular maximization under matroid constraints. By adaptively controlling active set expansion based on marginal gain coverage, ATCG interpolates between the approximation guarantees of classical Continuous Greedy and the communication frugality of partitioned selection. Curvature-aware analysis further tightens performance bounds, particularly in low-curvature regimes. Empirical results demonstrate that ATCG preserves solution quality while dramatically reducing communication overhead, underpinning its suitability for large-scale networked and distributed optimization applications (2604.03419).