Analyzing the Circuit Complexity of Goal-Conditioned Policies for Classical Planning Problems

Core Concepts

The circuit complexity of goal-conditioned policies for classical planning problems can be characterized by the regression width of the planning problems, which determines the necessary size and depth of relational neural network circuits to represent such policies.

Abstract

The paper analyzes the circuit complexity of goal-conditioned policies for classical planning problems, represented as relational neural networks (RelNNs). It makes the following key contributions:

It introduces the concept of serialized goal regression search (S-GRS) and defines the notion of strong optimally-serializable (SOS) width, which characterizes the complexity of planning problems. Problems with constant SOS width can be solved efficiently using S-GRS.

It shows how to compile the BWD and S-GRS algorithms into RelNN circuits, where the circuit complexity depends on the SOS width of the planning problem. Problems with constant SOS width can be compiled into finite-breadth, possibly unbounded-depth RelNN circuits.

It introduces the concept of a regression rule selector (RRS), which can further reduce the circuit complexity by directly predicting the appropriate regression rule to use. If the RRS can be computed by a finite-depth, finite-breadth RelNN, then the overall policy can be compiled into a shallow RelNN circuit.

It analyzes the SOS width and circuit complexity for several classical planning problems, including Blocks World, Logistics, Gripper, Rover, Elevator, and Sokoban. It shows that most problems have constant SOS width except for Sokoban, which has unbounded width.

It presents experimental results on learning RelNN policies for some of these planning problems, demonstrating that the theoretical predictions about circuit complexity match the empirical performance.

The paper provides a principled understanding of the circuit complexity of goal-conditioned policies for classical planning problems, and suggests insights into why certain planning problems are harder than others.

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Key Insights Distilled From

What Planning Problems Can A Relational Neural Network Solve?

by Jiay... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2312.03682.pdf

Deeper Inquiries

What are the implications of this analysis for the design of general-purpose planning algorithms and policies that can handle a wide variety of planning domains

The analysis presented in the context has significant implications for the design of general-purpose planning algorithms and policies that can handle a wide variety of planning domains. By understanding the relationship between planning width, regression width, search complexity, and policy circuit complexity, we can tailor algorithms and policies to be more efficient and effective across different problem domains.
One key implication is the importance of considering the regression width of a problem when designing policies. By analyzing the circuit complexity in relation to the regression width, we can determine the optimal depth and breadth of the policy circuit for a given problem. This insight allows for the development of more targeted and efficient policies that can generalize well across different planning domains.
Furthermore, the concept of serialized goal regression provides a structured approach to decomposing complex planning problems into simpler subgoals. By leveraging this serialization technique, algorithms and policies can be designed to tackle individual subgoals sequentially, leading to more manageable and scalable solutions for a wide range of planning domains.
Overall, the analysis underscores the importance of understanding the underlying complexity of planning problems and leveraging this knowledge to design adaptive and robust planning algorithms and policies that can handle diverse and challenging scenarios.

How can the insights from this work be extended to planning problems with continuous state and action spaces, where the number of possible actions is infinite

Extending the insights from this work to planning problems with continuous state and action spaces, where the number of possible actions is infinite, presents a unique set of challenges and opportunities. In such continuous domains, the notion of regression width and circuit complexity may need to be redefined to account for the infinite action space and continuous state representations.
One approach to adapting these insights to continuous planning problems is to consider approximations or discretizations of the continuous spaces. By discretizing the state and action spaces, it may be possible to apply similar concepts of regression width and circuit complexity to develop policies that can handle the continuous nature of the problem.
Additionally, techniques from reinforcement learning, such as function approximation and policy gradient methods, can be integrated with the insights from this work to design continuous planning algorithms and policies. By combining the structured approach of serialized goal regression with continuous action spaces, it may be possible to develop adaptive and efficient policies for continuous planning domains.
Overall, extending these insights to continuous planning problems requires a careful consideration of how to adapt the concepts of regression width and circuit complexity to the continuous domain while leveraging techniques from reinforcement learning to design effective policies.

Are there other properties of planning problems, beyond regression width, that could be used to further characterize the circuit complexity of goal-conditioned policies

While regression width is a crucial property for characterizing the circuit complexity of goal-conditioned policies, there are other properties of planning problems that could further enhance our understanding of policy circuit complexity. Some additional properties that could be considered include:

Temporal Dependencies: Analyzing the temporal dependencies between actions and states in a planning problem can provide insights into the sequential nature of the problem. Understanding how actions influence future states and goals can help in designing policies that account for these dependencies efficiently.

Action Space Complexity: Considering the complexity of the action space, such as the branching factor or the diversity of possible actions, can impact the circuit complexity of policies. A larger or more diverse action space may require more complex policies to handle the variability in actions.

Goal Hierarchies: Examining the hierarchical structure of goals in a planning problem can reveal insights into the level of abstraction and decomposition required for solving the problem. Policies that can handle hierarchical goal structures may have different circuit complexities compared to those that do not consider such hierarchies.

By incorporating these additional properties into the analysis of policy circuit complexity, we can gain a more comprehensive understanding of the factors that influence the design and efficiency of goal-conditioned policies for a wide range of planning domains.