Planning under uncertainty is a central problem in the study of automated sequential decision making, and has been addressed by researchers in many different fields, including AI planning, decision analysis, operations research, control theory and economics. While the assumptions and perspectives adopted in these areas often differ in substantial ways, many planning problems of interest to researchers in these fields can be modeled as Markov decision processes (MDPs) and analyzed using the techniques of decision theory. This paper presents an overview and synthesis of MDP-related methods, showing how they provide a unifying framework for modeling many classes of planning problems studied in AI. It also describes structural properties of MDPs that, when exhibited by particular classes of problems, can be exploited in the construction of optimal or approximately optimal policies or plans. Planning problems commonly possess structure in the reward and value functions used to de...

Recently, structured methods for solving factored Markov decisions processes (MDPs) with large state spaces have been proposed recently to allow dynamic programming to be applied without the need for complete state enumeration. We propose and examine a new value iteration algorithm for MDPs that uses algebraic decision diagrams (ADDs) to represent value functions and policies, assuming an ADD input representation of the MDP. Dynamic programming is implemented via ADD manipulation. We demonstrate our method on a class of large MDPs (up to 63 million states) and show that significant gains can be had when compared to tree-structured representations (with up to a thirty-fold reduction in the number of nodes required to represent optimal value functions). 1

Markov decision processes(MDPs) have proven to be popular models for decision-theoretic planning, but standard dynamic programming algorithms for solving MDPs rely on explicit, state-based specifications and computations. To alleviate the combinatorial problems associated with such methods, we propose new representational and computational techniques for MDPs that exploit certain types of problem structure. We use dynamic Bayesian networks (with decision trees representing the local families of conditional probability distributions) to represent stochastic actions in an MDP, together with a decision-tree representation of rewards. Based on this representation, we develop versions of standard dynamic programming algorithms that directly manipulate decision-tree representations of policies and value functions. This generally obviates the need for state-by-state computation, aggregating states at the leaves of these trees and requiring computations only for each aggregate state. The key to these algorithms is a decision-theoretic generalization of classic regression analysis, in which we determine the features relevant to predicting expected value. We demonstrate the method empirically on several planning problems,

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.

This dissertation investigates the use of hierarchy and problem decomposition as a means of solving large, stochastic, sequential decision problems. These problems are framed as Markov decision problems (MDPs). The new technical content of this dissertation begins with a discussion of the concept of temporal abstraction. Temporal abstraction is shown to be equivalent to the transformation of a policy defined over a region of an MDP to an action in a semi-Markov decision problem (SMDP). Several algorithms are presented for performing this transformation efficiently. This dissertation introduces the HAM method for generating hierarchical, temporally abstract actions. This method permits the partial specification of abstract actions in a way that corresponds to an abstract plan or strategy. Abstr...

Many stochastic planning problems can be represented using Markov Decision Processes (MDPs). A difficulty with using these MDP representations is that the common algorithms for solving them run in time polynomial in the size of the state space, where this size is extremely large for most real-world planning problems of interest. Recent AI research has addressed this problem by representing the MDP in a factored form. Factored MDPs, however, are not amenable to traditional solution methods that call for an explicit enumeration of the state space. One familiar way to solve MDP problems with very large state spaces is to form a reduced (or aggregated) MDP with the same properties as the original MDP by combining "equivalent" states. In this paper, we discuss applying this approach to solving factored MDP problems--- we avoid enumerating the state space by describing large blocks of "equivalent" states in factored form, with the block descriptions being inferred directly from the original factored representation. The resulting reduced MDP may have exponentially fewer states than the original factored MDP, and can then be solved using traditional methods. The reduced MDP found depends on the notion of equivalence between states used in the aggregation. The notion of equivalence chosen will be fundamental in designing and analyzing algorithms for reducing MDPs. Optimally, these algorithms will be able to find the smallest possible reduced MDP for any given input MDP and notion of equivalence (i.e. find the "minimal model" for the input MDP). Unfortunately, the classic notion of state equivalence from non-deterministic finite state machines generalized to MDPs does not prove useful. We present here a notion of equivalence that is based upon the notion of bisimulation from th...

We present a technique for computing approximately optimal solutions to stochastic resource allocation problems modeled as Markov decision processes (MDPs). We exploit two key properties to avoid explicitly enumerating the very large state and action spaces associated with these problems. First, the problems are composed of multiple tasks whose utilities are independent. Second, the actions taken with respect to (or resources allocated to) a task do not influence the status of any other task. We can therefore view each task as an MDP. However, these MDPs are weakly coupled by resource constraints: actions selected for one MDP restrict the actions available to others. We describe heuristic techniques for dealing with several classes of constraints that use the solutions for individual MDPs to construct an approximate global solution. We demonstrate this technique on problems involving thousandsof tasks, approximating the solution to problems that are far beyond the reach of standard methods. 1

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 suggested that value functions in factored MDPs can often be approximated well using a factored value function: a linear combination of restricted basis functions, each of which refers only to a small subset of variables. An approximate factored 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.

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We present a method for solving implicit (factored) Markov decision processes (MDPs) with very large state spaces. We introduce a property of state space partitions which we call ffl-homogeneity. Intuitively, an ffl-homogeneous partition groups together states that behave approximately the same under all or some subset of policies. Borrowing from recent work on model minimization in computer-aided software verification, we present an algorithm that takes a factored representation of an MDP and an 0 ffl 1 and computes a factored ffl-homogeneous partition of the state space. This partition defines a family of related MDPs---those MDP's with state space equal to the blocks of the partition, and transition probabilities "approximately" like those of any (original MDP) state in the source block. To formally study such families of MDPs, we introduce the new notion of a "bounded parameter MDP" (BMDP), which is a family of (traditional) MDPs defined by specifying upper...