  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.
