We consider finite and infinite horizon dynamic programming problems, where the control at each stage consists of several distinct decisions, each one made by one of several agents. We introduce an algorithm, whereby at every stage, each agent's decision is made by executing a local rollout algorithm that uses a base policy, together with some coordinating information from the other agents. The amount of local computation required at every stage by each agent is independent of the number of agents, while the amount of global computation (over all agents) grows linearly with the number of agents. By contrast, with the standard rollout algorithm, the amount of global computation grows exponentially with the number of agents. Despite the drastic reduction in required computation, we show that our algorithm has the fundamental cost improvement property of rollout: an improved performance relative to the base policy. We also explore related reinforcement learning and approximate policy iteration algorithms, and we discuss how this cost improvement property is affected when we attempt to improve further the method's computational efficiency through parallelization of the agents' computations.

We propose a new aggregation framework for approximate dynamic programming, which provides a connection with rollout algorithms, approximate policy iteration, and other single and multistep lookahead methods. The central novel characteristic is the use of a bias function $V$ of the state, which biases the values of the aggregate cost function towards their correct levels. The classical aggregation framework is obtained when $V\equiv0$, but our scheme works best when $V$ is a known reasonably good approximation to the optimal cost function $J^*$. When $V$ is equal to the cost function $J_{\mu}$ of some known policy $\mu$ and there is only one aggregate state, our scheme is equivalent to the rollout algorithm based on $\mu$ (i.e., the result of a single policy improvement starting with the policy $\mu$). When $V=J_{\mu}$ and there are multiple aggregate states, our aggregation approach can be used as a more powerful form of improvement of $\mu$. Thus, when combined with an approximate policy evaluation scheme, our approach can form the basis for a new and enhanced form of approximate policy iteration. When $V$ is a generic bias function, our scheme is equivalent to approximation in value space with lookahead function equal to $V$ plus a local correction within each aggregate state. The local correction levels are obtained by solving a low-dimensional aggregate DP problem, yielding an arbitrarily close approximation to $J^*$, when the number of aggregate states is sufficiently large. Except for the bias function, the aggregate DP problem is similar to the one of the classical aggregation framework, and its algorithmic solution by simulation or other methods is nearly identical to one for classical aggregation, assuming values of $V$ are available when needed.

In this paper, we propose a new lower approximation scheme for POMDP with discounted and average cost criterion. The approximating functions are determined by their values at a finite number of belief points, and can be computed efficiently using value iteration algorithms for finite-state MDP. While for discounted problems several lower approximation schemes have been proposed earlier, ours seems the first of its kind for average cost problems. We focus primarily on the average cost case, and we show that the corresponding approximation can be computed efficiently using multi-chain algorithms for finite-state MDP. We give a preliminary analysis showing that regardless of the existence of the optimal average cost J in the POMDP, the approximation obtained is a lower bound of the liminf optimal average cost function, and can also be used to calculate an upper bound on the limsup optimal average cost function, as well as bounds on the cost of executing the stationary policy associated with the approximation. Weshow the convergence of the cost approximation, when the optimal average cost is constant and the optimal differential cost is continuous.