Factored Value Iteration Converges

In this paper we propose a novel algorithm, factored value iteration (FVI), for the approximate solution of factored Markov decision processes (fMDPs). The traditional approximate value iteration algorithm is modified in two ways. For one, the least-squares projection operator is modified so that it does not increase max-norm, and thus preserves convergence. The other modification is that we uniformly sample polynomially many samples from the (exponentially large) state space. This way, the complexity of our algorithm becomes polynomial in the size of the fMDP description length. We prove that the algorithm is convergent. We also derive an upper bound on the difference between our approximate solution and the optimal one, and also on the error introduced by sampling. We analyze various projection operators with respect to their computation complexity and their convergence when combined with approximate value iteration.Comment: 17 pages, 1 figur

Least-Squares Covariance Matrix Adjustment

We consider the problem of finding the smallest adjustment to a given symmetric $n \times n$ matrix, as measured by the Euclidean or Frobenius norm, so that it satisfies some given linear equalities and inequalities, and in addition is positive semidefinite. This least-squares covariance adjustment problem is a convex optimization problem, and can be efficiently solved using standard methods when the number of variables (i.e., entries in the matrix) is modest, say, under $1000$. Since the number of variables is $n(n+1)/2$, this corresponds to a limit around $n=45$. Malick [{\it SIAM J. Matrix Anal.\ Appl.,} 26 (2005), pp. 272--284] studies a closely related problem and calls it the semidefinite least-squares problem. In this paper we formulate a dual problem that has no matrix inequality or matrix variables, and a number of (scalar) variables equal to the number of equality and inequality constraints in the original least-squares covariance adjustment problem. This dual problem allows us to solve far larger least-squares covariance adjustment problems than would be possible using standard methods. Assuming a modest number of constraints, problems with $n=1000$ are readily solved by the dual method. The dual method coincides with the dual method proposed by Malick when there are no inequality constraints and can be obtained as an extension of his dual method when there are inequality constraints. Using the dual problem, we show that in many cases the optimal solution is a low rank update of the original matrix. When the original matrix has structure, such as sparsity, this observation allows us to solve very large least-squares covariance adjustment problems

Solving Factored MDPs with Hybrid State and Action Variables

Efficient representations and solutions for large decision problems with continuous and discrete variables are among the most important challenges faced by the designers of automated decision support systems. In this paper, we describe a novel hybrid factored Markov decision process (MDP) model that allows for a compact representation of these problems, and a new hybrid approximate linear programming (HALP) framework that permits their efficient solutions. The central idea of HALP is to approximate the optimal value function by a linear combination of basis functions and optimize its weights by linear programming. We analyze both theoretical and computational aspects of this approach, and demonstrate its scale-up potential on several hybrid optimization problems