Efficient Solution Algorithms for Factored MDPs (1106.1822v1)

Abstract: This paper addresses the problem of planning under uncertainty in large Markov Decision Processes (MDPs). Factored MDPs represent a complex state space using state variables and the transition model using a dynamic Bayesian network. This representation often allows an exponential reduction in the representation size of structured MDPs, but the complexity of exact solution algorithms for such MDPs can grow exponentially in the representation size. In this paper, we present two approximate solution algorithms that exploit structure in factored MDPs. Both use an approximate value function represented as a linear combination of basis functions, where each basis function involves only a small subset of the domain variables. A key contribution of this paper is that it shows how the basic operations of both algorithms can be performed efficiently in closed form, by exploiting both additive and context-specific structure in a factored MDP. A central element of our algorithms is a novel linear program decomposition technique, analogous to variable elimination in Bayesian networks, which reduces an exponentially large LP to a provably equivalent, polynomial-sized one. One algorithm uses approximate linear programming, and the second approximate dynamic programming. Our dynamic programming algorithm is novel in that it uses an approximation based on max-norm, a technique that more directly minimizes the terms that appear in error bounds for approximate MDP algorithms. We provide experimental results on problems with over 10^40 states, demonstrating a promising indication of the scalability of our approach, and compare our algorithm to an existing state-of-the-art approach, showing, in some problems, exponential gains in computation time.

• The paper introduces two approximate solution algorithms that utilize LP decomposition and max-norm projection to efficiently compute value functions in factored MDPs.
• It transforms exponential constraint sets into polynomial-sized linear programs by applying linear combinations of basis functions to exploit additive and context-specific structures.
• Experimental results on the SysAdmin problem reveal significant efficiency gains and scalability in managing large, structured decision-making models.

Efficient Solution Algorithms for Factored MDPs

The paper "Efficient Solution Algorithms for Factored MDPs" by Carlos Guestrin, Daphne Koller, Ronald Parr, and Shobha Venkataraman addresses the computational challenges in solving large Markov Decision Processes (MDPs) through the utilization of factored representations. Factored MDPs offer a compact representation model using state variables and dynamic Bayesian networks (DBNs), which can lead to exponential reductions in representation size.

Contribution and Methodology

The paper introduces two approximate solution algorithms specifically designed to exploit the structural properties of factored MDPs: approximate linear programming and approximate dynamic programming. These methods utilize linear combinations of basis functions to approximate the value function, significantly improving computational efficiency through closed-form operations that leverage both additive and context-specific structures.

One of the central innovations is the linear program decomposition technique, analogous to variable elimination in Bayesian networks. This method reduces an exponentially large linear program (LP) to a polynomial-sized equivalent, enhancing the tractability of LP solutions in factored settings.

Approximate Solution Algorithms

1. Approximate Linear Programming: This algorithm seeks to approximate the value function through a linear program formulation restricted to the space defined by the basis functions. A significant challenge addressed is the representation of exponentially many constraints, which is compactly achieved through the factored LP technique.
2. Approximate Dynamic Programming: Here, the policy iteration method is adapted for factored MDPs using a max-norm projection technique. This approach seeks to directly optimize the terms in error bounds for MDP algorithms and offers strong theoretical guarantees. It relies on generating compact decision lists for policy representation, substantially reducing computational complexity.

Experimental Evaluation

The algorithms are tested on the SysAdmin problem, demonstrating scalability across different network topologies, with problems involving over 100 states. Notably, the approximate policy iteration algorithm with max-norm projection showed promising results in achieving efficient computation times for large state spaces.

Comparison with Existing Approaches

The presented methods show exponential gains in computation time over traditional exact algorithms. Furthermore, in comparison to tree-based approaches such as ADDs utilized in prior work, the linear value function approximations presented here can effectively manage problems where context-specific independence and additive structures are both prevalent.

Implications and Future Directions

The ability to efficiently solve large and structured MDPs has significant implications in various domains, including automated decision-making and robotics. The techniques showcased can be further extended to collaborative multiagent systems and partially observable environments (POMDPs), an area that the authors have begun to explore. Future research could also focus on automatic feature selection for basis functions, enhancing the usability and effectiveness of these methods in diverse applications.

In summary, this paper presents substantial advancements in the field of MDPs, providing scalable and efficient algorithms for handling complex decision-making models where structural properties can be strategically exploited.