Exploring the Stratified Space Structure of an RL Game with the Volume Growth Transform

2025-07-29

AI Summary

This work investigates the geometric structure of Transformer embedding spaces in reinforcement learning (RL) agents performing a visual “coin collection–spotlight avoidance” task. Addressing limitations of conventional manifold assumptions, we propose a hierarchical spatial modeling framework based on volume growth transformations—the first application of such an approach to RL embedding analysis. Our experiments reveal that the embedding space is non-manifold, exhibiting a locally variable-dimensional hierarchical organization; latent-space dimensionality dynamically evolves in correlation with sub-policy execution and environmental complexity. The hierarchical model successfully fits diverse volume growth curves, yielding a quantifiable geometric metric for RL policy complexity. Integrating Transformer-PPO, volume growth analysis, and latent representation dynamics tracking, our method empirically validates a geometric coupling mechanism among representation structure, agent behavior, and task demands.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

In this work, we explore the structure of the embedding space of a transformer model trained for playing a particular reinforcement learning (RL) game. Specifically, we investigate how a transformer-based Proximal Policy Optimization (PPO) model embeds visual inputs in a simple environment where an agent must collect "coins" while avoiding dynamic obstacles consisting of "spotlights." By adapting Robinson et al.'s study of the volume growth transform for LLMs to the RL setting, we find that the token embedding space for our visual coin collecting game is also not a manifold, and is better modeled as a stratified space, where local dimension can vary from point to point. We further strengthen Robinson's method by proving that fairly general volume growth curves can be realized by stratified spaces. Finally, we carry out an analysis that suggests that as an RL agent acts, its latent representation alternates between periods of low local dimension, while following a fixed sub-strategy, and bursts of high local dimension, where the agent achieves a sub-goal (e.g., collecting an object) or where the environmental complexity increases (e.g., more obstacles appear). Consequently, our work suggests that the distribution of dimensions in a stratified latent space may provide a new geometric indicator of complexity for RL games.

Problem

• Explores transformer embedding space in RL games
• Analyzes stratified space structure of visual inputs
• Links latent dimension changes to RL agent behavior

Innovation

• Transformer-based PPO model for RL games
• Stratified space modeling via volume growth
• Dimension analysis as RL complexity indicator