Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning

TL;DR

This paper introduces a graph value extension method, AMLE, for sparse goal-conditioned planning, demonstrating significant improvements over harmonic extension in success rates across various graph configurations.

Contribution

The paper presents a novel local action-gap certificate and applies the Absolutely Minimal Lipschitz Extension (AMLE) to improve goal-reaching success in sparse planning tasks.

Findings

AMLE achieves 0.970 success rate compared to 0.584 for harmonic extension.

High-p methods like p=8 and p=16 also show high success rates, 0.973 and 0.982 respectively.

AMLE corrects most harmonic inversions, improving local action ranking accuracy.

Abstract

Sparse goal-conditioned planning with few cost-to-go labels can be viewed as a graph-PDE Dirichlet extension problem: extend sparse labels on a goal-dependent boundary to unlabelled graph vertices so that greedy rollouts reach the goal. We study which graph value extensions are planner-admissible under the operational argmin-Q planner. Our main result is a local action-gap certificate: if the surrogate value error along the rollout stays below half the true action gap, then the greedy rollout reaches the goal. Absolutely Minimal Lipschitz Extension (AMLE), the p=infinity endpoint of the graph p-Laplacian family, instantiates this certificate through a comparison-principle fill-distance bound. Harmonic extension, by contrast, can mis-rank local actions because its values reflect boundary hitting probabilities rather than shortest-path greedy order. On 120 AntMaze layout-derived graph configurations, harmonic extension achieves 0.584 aggregate rollout success, while AMLE reaches 0.970. Finite high-p methods also enter a high-success regime, with success 0.903 for p=4, 0.973 for p=8, and 0.982 for a fixed-budget p=16 solver, though the p=16 row is not used as a converged endpoint ranking due to incomplete solver certification. Mechanism audits show that many rollout decisions occur in AMLE-compatible but harmonic-incompatible local geometry, and that AMLE corrects most harmonic inversions on the rollout-weighted decision scope.