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

Sparse goal-conditioned reinforcement learning can produce policies whose failures are hidden by aggregate success rates. We analyze trained goal-conditioned value policies through the deterministic functional graphs induced by greedy evaluation: for each goal, every state maps to a single successor, decomposing behavior into attractors and basins. This reveals a local-to-global structure in learned policies. We define local goal support (LGS), a one-step statistic measuring the fraction of valid neighboring states whose greedy successor is the goal. In deterministic sparse GridWorlds, zero LGS exactly precludes goal entry from non-goal starts. Empirically, weak LGS is a strong diagnostic of goal-level failure across update rules, curricula, larger grids, and bottleneck geometries: the fixed rule LGS <= 0.5 identifies low-success goals with precision 0.921, recall 0.929, and F1 0.925 in the main 8x8 TD setting, with similar performance across variants. However, local support is not sufficient for global success: some supported goals still fail because distant states are captured by competing attractors or fragmented basin structure. We therefore introduce a compact post-hoc taxonomy of policy-induced graphs -- goal-dominant, competitor-dominated, partial/contested, and fragmented -- to characterize residual failure modes beyond local support. These results show that sparse GCRL failures can be understood as structured policy-induced dynamics, and that local one-step policy structure provides a cheap post-training diagnostic for goal-level failure.

Automated Planning is a subfield of Artificial Intelligence (AI) where the main objective is generating a sequence of actions, known as a plan, that helps us reach a goal state from an initial state. A planning problem is defined by a set of objects, an initial state and a desired goal state. The objective is to compute a plan that'll lead us from the inital state to the goal state. Programs that generate plans are called planners. In this paper, we did a complementary study to the state-of-the-art LLM called PlanGPT which was released last year. We redid some experiments to verify whether planning with LLMs is \textbf{pertinent} and \textbf{worthwhile}. We also check whether the results obtained in the official PlanGPT paper for plan coverage were correct, and we also performed a more comprehensive study on PlanGPT's performance: in our paper PlanGPT's performance was evaluated using two metrics: Plan Cost and Plan Generation Time. The results of planGPT were compared to those produced by a traditional planner for the same plans and same metrics. We discovered that PlanGPT is no better than a Greedy search strategy.

Planning under uncertainty in continuous domains is essential for autonomous systems, yet computationally demanding. Tree-based search methods such as Monte Carlo Tree Search (MCTS) remain popular, but their branching structure can require sampling budgets that grow exponentially with lookahead depth in the worst case. From a tree perspective, continuous state or action spaces become especially challenging, since the planner must decide where to search in an infinite branching hierarchy. We propose Graph Sparse Sampling (GSS), an online planning algorithm that shares sampled futures across many candidate decisions, rather than sampling separate successors for each candidate action. This branch-free graph exposes large GPU-friendly batches, while using heuristics to focus computation. We prove finite-sample performance guarantees for GSS covering full-rank or low-rank generative simulators via smoothed backups, and discrete or sampled continuous action spaces. Under suitable overlap, regularity, and action-coverage conditions, these bounds have polynomial dependence on the planning horizon, formalizing when shared futures can avoid the exponential horizon dependence of tree-shaped sparse sampling. We demonstrate continuous-control simulations where GSS substantially outperforms tree-based planners on long horizons or achieves near-optimal performance, supporting no-branching graph planning as a complementary design principle for online control.