Policy Iteration for Factored MDPs (1301.3869v1)

Abstract: Many large MDPs can be represented compactly using a dynamic Bayesian network. Although the structure of the value function does not retain the structure of the process, recent work has shown that value functions in factored MDPs can often be approximated well using a decomposed value function: a linear combination of <I>restricted</I> basis functions, each of which refers only to a small subset of variables. An approximate value function for a particular policy can be computed using approximate dynamic programming, but this approach (and others) can only produce an approximation relative to a distance metric which is weighted by the stationary distribution of the current policy. This type of weighted projection is ill-suited to policy improvement. We present a new approach to value determination, that uses a simple closed-form computation to directly compute a least-squares decomposed approximation to the value function <I>for any weights</I>. We then use this value determination algorithm as a subroutine in a policy iteration process. We show that, under reasonable restrictions, the policies induced by a factored value function are compactly represented, and can be manipulated efficiently in a policy iteration process. We also present a method for computing error bounds for decomposed value functions using a variable-elimination algorithm for function optimization. The complexity of all of our algorithms depends on the factorization of system dynamics and of the approximate value function.

An Expert Review of "Policy Iteration for Factored MDPs"

This paper introduces an innovative approach to the policy iteration process for Markov Decision Processes (MDPs) that are structured with factored states, using a dynamic Bayesian network (DBN) representation to compactly model transition dynamics. The authors, Daphne Koller and Ronald Parr, present new methodologies aimed at improving the effectiveness and efficiency of policy iteration within large-scale, factored MDPs.

Key Contributions

The paper's primary contribution is the introduction of a novel algorithm for value determination that enables the computation of factored value functions in closed-form with arbitrary weights, rather than relying on the stationary distribution of the current policy. This approach addresses the inherent limitations observed in previous methods where stationary-weights least-squares projections often led to misleading value estimates in states outside the range of the current policy, hence hindering policy improvement.

The authors demonstrate that for factored MDPs, value functions can be efficiently approximated as linear combinations of restricted basis functions. These basis functions focus only on small subsets of the state variables, thus effectively reducing the computational burden associated with handling the exponentially large state space typical of real-world applications. Additionally, the paper provides a pragmatic solution for computing error bounds for the decomposed value functions utilizing a variable-elimination algorithm for function optimization.

Technical Approach

The authors achieve efficient policy iteration through a structured approach:

• Representation of Policies: Factored value functions are represented as decision lists, allowing for compact and efficient manipulation during policy iteration. This representation leverages the restricted variable domains in basis functions, which is critical for maintaining tractability in large state spaces.
• Closed-Form Value Determination: The paper outlines an efficient computation mechanism for weighted least squares that is independent of the stationary distribution, thereby facilitating a value determination process conducive to policy iteration.
• Computational Complexity: The algorithms introduced exhibit complexity directly tied to the factorization of system dynamics and the approximate value function, making them suitable for practical application in large-scale MDPs.

Numerical Results and Implications

The numerical paper conducted illustrates the practical viability of the algorithm. The authors provide theoretical justification that the closed-form solutions for their approximate dynamic programming equations exist for nearly all discount factors, ensuring that the value determination procedure can be applied broadly.

The paper's findings implicate significant potential improvements in AI planning efficiency in domains characterized by large, factored state spaces. By overcoming the limitations of traditional MDP approaches in handling complex systems, the algorithms proposed could lead to more robust and scalable solutions in AI-driven decision-making processes.

Future Directions

Looking ahead, the methods introduced open avenues for extending policy iteration algorithms to more intricate scenarios, such as partially observable MDPs and planning domains with parallel actions and context-sensitive dynamics. Bridging these advanced scenarios could lead to transformative improvements in AI systems dealing with real-world complexity and uncertainty.

This paper offers valuable insights into the structural nuances of factored MDPs and presents a compelling case for the adoption of factored value functions in policy iteration, paving the way for significant advancements in the modeling and planning capabilities of AI systems.