Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains

2025-12-04

AI Summary

Existing traversal coverage planning methods exhibit numerical instability under nonlinear constraints in multi-scale scenarios, as kernel-based optimization is highly sensitive to numerical scaling. To address this, we propose a scale-invariant, adaptive optimization framework: we formulate a scale-adaptive objective using Maximum Mean Discrepancy (MMD), integrate a logarithmic-space traversability metric, and employ a hyperparameter annealing strategy to ensure stable differentiable optimization under motion constraints. Our method eliminates the need for manual scale preprocessing, preserves physical consistency, and significantly improves the numerical condition number. Experiments across diverse geometric coverage tasks demonstrate that, while maintaining high coverage rates, our approach achieves superior multi-scale robustness and convergence stability compared to state-of-the-art methods.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Recent methods in ergodic coverage planning have shown promise as tools that can adapt to a wide range of geometric coverage problems with general constraints, but are highly sensitive to the numerical scaling of the problem space. The underlying challenge is that the optimization formulation becomes brittle and numerically unstable with changing scales, especially under potentially nonlinear constraints that impose dynamic restrictions, due to the kernel-based formulation. This paper proposes to address this problem via the development of a scale-agnostic and adaptive ergodic coverage optimization method based on the maximum mean discrepancy metric (MMD). Our approach allows the optimizer to solve for the scale of differential constraints while annealing the hyperparameters to best suit the problem domain and ensure physical consistency. We also derive a variation of the ergodic metric in the log space, providing additional numerical conditioning without loss of performance. We compare our approach with existing coverage planning methods and demonstrate the utility of our approach on a wide range of coverage problems.

Problem

• Improves numerical stability in ergodic coverage optimization across scales
• Addresses sensitivity to scaling in multi-domain geometric coverage planning
• Enhances optimizer adaptability under nonlinear dynamic constraints

Innovation

• Scale-agnostic ergodic coverage using MMD metric
• Optimizes differential constraints scale with hyperparameter annealing
• Log-space ergodic metric for improved numerical conditioning