- The paper introduces a formal framework linking goal-conditioned policies to bounded regression width for efficient planning.
- It demonstrates that RelNNs, including Graph Neural Networks and Transformers, can encode policies using finite-breadth circuits.
- Empirical results confirm that Serialized Goal Regression Search enables polynomial-time solutions in structured planning scenarios.
Introduction
Goal-conditioned policies are intended to direct an agent's actions towards achieving a specific objective within a given state space. While neural networks in the form of policy circuits have shown promise in learning such policies, the question of their effectiveness across various planning problems presents a theoretical challenge. A key aspect of this challenge lies in determining when a polynomial-sized circuit can effectively manifest such a policy and understanding its complexity.
Planning and Learning
The paper focuses on classical planning problems characterized by an object-centric representation and sparse transition models. In these problems, one seeks an action sequence bringing the world from an initial state to a state that satisfies certain goals, with machine learning approaches aimed at finding policies that correctly identify the next action from any given state and goal. The paper introduces a formal definition of planning problems under consideration, grounding the subsequent analysis in a structured framework.
Search Complexity and Regression Width
The concept of Serialized Goal Regression Search (S-GRS) is introduced, highlighting an approach aimed at enhancing efficiency by serializing the preemptive goals. The fundamental measure, regression width, indicates the number of constraints one must track during search. Under the notion of optimally serializable rules, it is shown that if a problem presents a bounded regression width, it can be effectively solved in polynomial time using S-GRS.
Policy Realization Through Neural Networks
Relational Neural Networks (RelNNs), such as Graph Neural Networks and Transformers, can generalize to handle variable-sized inputs. The paper outlines methods to construct RelNNs that represent goal-conditioned policies, revealing that for problems with bounded regression width, finite-breadth networks can encode policies. In scenarios where a problem's regression rule selector can be efficiently approximated, compilation into even more compact RelNN circuits is possible.
Practical Implications and Results
Empirical observations further confirm the theoretical findings, with RelNNs demonstrating success in generalization and efficiency when tailored to the problem's regression width. The study sheds light on why RelNNs or similar policy circuits could perform well for specific domains and stresses the potential need for unbounded depths in networks when handling certain planning tasks, like those in Sokoban or Task and Motion Planning (TAMP) problems.
Conclusion
In summary, the paper innovatively ties the complexity of goal-conditioned policies to the structure of planning problems, offering a fresh understanding of policy circuit complexity. The insights provided could guide the design of neural networks for policy learning while predicting their capabilities and limitations across different planning domains.