Laplacian Representations for Decision-Time Planning

Problem Statement
Model-based reinforcement learning (RL) offers potential advantages in sample efficiency, generalization, and adaptation compared to model-free approaches. However, a central challenge remains: planning with a learned model in environments requiring function approximation is hindered by the compounding error problem. When an agent simulates long-horizon trajectories using an imperfect model, small prediction errors accumulate, causing imagined trajectories to diverge significantly from reality. This issue is particularly acute in decision-time planning, where state representations must simultaneously support local cost computation and preserve long-horizon temporal structures. While hierarchical planning can decouple low-level actions from long-horizon objectives, it requires a latent space capable of representing state-space distances across multiple time scales to effectively decompose tasks into subgoals.

Methodology
The paper proposes that the Laplacian representation provides an effective latent space for decision-time planning. This representation embeds states into a space defined by the eigenvectors of the graph Laplacian induced by the environment's dynamics. Key properties of this representation include:

• Multi-scale Structure: Eigenvectors are ordered by time scale; early eigenvectors encode global structure (e.g., regions), while later ones capture local distinctions.
• Reachability and Geometry: The Euclidean distance in the scaled Laplacian space approximates the Commute Time Distance (CTD), which measures the expected number of steps to travel between two states and back via a random walk. This makes the space isometric to CTD, providing a geometry well-suited for planning.
• Subgoal Discovery: The representation naturally partitions the environment into well-connected regions, making it ideal for spectral clustering to identify subgoals at bottlenecks.

Building on these properties, the authors introduce ALPS (Augmented Laplacian Planning with Subgoals), a hierarchical decision-time planning algorithm. ALPS operates in two phases:

1. Pre-training:
   • Laplacian Learning: Uses the Augmented Lagrangian Laplacian Objective (ALLO) to learn eigenvectors and eigenvalues from an offline dataset, avoiding the $O(|S|^3)$ computational cost of exact eigendecomposition.
   • Forward Model: Learns a multi-step auto-regressive forward model in the original state space to reduce error accumulation.
   • Behavior Prior: Learns a deterministic, goal-conditioned policy ($\pi_{prior}$) to bias action sampling, accelerating convergence.
   • Cluster Graph: Performs $k$-means clustering on the scaled Laplacian space ($\psi$-space) to define subgoals (cluster centers) and builds a graph of inter-cluster transitions.
2. Decision-Time Planning:
   • High-Level: Given a start and goal state, the algorithm identifies their respective clusters and uses Dijkstra's algorithm on the cluster graph to find a path of subgoals.
   • Low-Level: For each subgoal, a Cross-Entropy Method (CEM) planner generates an action sequence. CEM is initialized using the behavior prior and optimized by minimizing a cost function based on the distance in $\psi$-space (approximating CTD) and action magnitude.

Key Contributions

• Laplacian Representation for Planning: The paper demonstrates that the Laplacian representation effectively captures state-space distances at multiple time scales, naturally decomposing long-horizon problems into subgoals and mitigating compounding errors.
• ALPS Algorithm: A novel hierarchical planning algorithm that integrates Laplacian-based subgoal identification with model-based trajectory optimization (CEM) and a behavior prior.
• Empirical Validation: The authors evaluate ALPS on the OGBench (Offline Goal-Conditioned RL Benchmark), a suite of tasks previously dominated by model-free methods.

Results
Experiments were conducted on the Maze2D—PointMass environments and a diverse set of OGBench tasks involving locomotion (pointmaze, antmaze, humanoidmaze) and manipulation (cube, scene) with both state-based and pixel-based observations.

• Performance: ALPS significantly outperforms commonly used model-free baselines (including GCBC, IQL variants, QRL, CRL, and HIQL) across most OGBench tasks, with statistical significance ($p < 0.001$).
• Robustness: ALPS demonstrates robustness to the size of the maze and the quality of the offline dataset. It particularly excels in "explore" datasets with low-quality, high-coverage data, where hierarchical decomposition allows the high-level planner to compensate for a limited low-level horizon.
• Comparison with PcLast: On Maze2D tasks, ALPS (using Laplacian representation) outperforms PcLast (using contrastive latent states), highlighting the utility of the Laplacian metric for both distance estimation and subgoal identification.
• Limitations: The paper notes that ALPS struggles with "teleport" mazes where the symmetrization inherent in the Laplacian objective causes teleporter entrances and exits to be clustered together, potentially leading the agent into inescapable regions. Additionally, performance degrades on multi-object manipulation tasks (e.g., double-cube) due to the global nature of the Laplacian representation failing to capture conditional independence between object states.

Significance
The paper claims that the Laplacian representation serves as a useful building block for successful planning with learned models, addressing the central challenge of compounding errors in long-horizon tasks. By providing a latent space that intrinsically captures temporal structure and connectivity, ALPS enables effective hierarchical planning in continuous, high-dimensional state spaces using only pre-collected offline datasets. The work suggests that leveraging spectral properties of the environment's dynamics can overcome the historical difficulty of applying model-based methods to complex, long-horizon goal-conditioned RL problems.