The predominant state-space planning methods in artificial intelligence are collectively known as heuristic search. Although superficially different from the planning methods we have discussed so far in this chapter, heuristic search and some of its component ideas can be combined with these methods in useful ways. Unlike these methods, heuristic search is not concerned with changing the approximate, or "heuristic," value function, but only with making improved action selections given the current value function. In other words, heuristic search is planning as part of a policy computation.
In heuristic search, for each state encountered, a large
tree of possible continuations is considered. The approximate value function is
applied to the leaf nodes and then backed up toward the current state at the root.
The backing up within the search tree is just the same as in the max-backups (those
for 
and 
) discussed throughout this book. The backing up stops at the
state-action nodes for the current state. Once the backed-up values of these
nodes are computed, the best of them is chosen as the current action, and then
all backed-up values are discarded.
In conventional heuristic search no effort is made to save the backed-up values by
changing the approximate value function. In fact, the value function is generally
designed by people and never changed as a result of search. However, it is natural
to consider allowing the value function to be improved over time, using either the
backed-up values computed during heuristic search or any of the other methods
presented throughout this book. In a sense we have taken this approach all along.
Our greedy and
-greedy action-selection methods are not unlike heuristic search, albeit on a
smaller scale. For example, to compute the greedy action given a model
and a state-value function, we must look ahead from each possible action to each
possible next state, backup the rewards and estimated values, and then pick the
best action. Just as in conventional heuristic search, this process computes
backed-up values of the possible actions, but does not attempt to
save them. Thus, heuristic search can be viewed as an extension of the
idea of a greedy policy beyond a single step.
The point of searching deeper than one step is to obtain better action selections.
If one has a
perfect model and an imperfect action-value function, then in fact deeper search
will usually yield better policies.9.1 Certainly, if the search
is all the way to the end of the episode, then the effect of the imperfect value
function is eliminated, and the action determined in this way must be optimal. If
the search is of sufficient depth 
such that 
is very small, then the
actions will be correspondingly near optimal. On the other hand, the deeper the
search, the more computation is required, usually resulting in a slower response
time. A good example is provided by Tesauro's grandmaster-level backgammon player,
TD-Gammon (Section 11.1). This system used TD(
) to learn an
afterstate value function through many games of self-play, using a form
of heuristic search to make its moves. As a model,
TD-Gammon used a priori knowledge of the probabilities of dice rolls and
the assumption that the opponent always selected the actions that TD-Gammon
rated as best for it. Tesauro found that the deeper the heuristic search, the
better the moves made by TD-Gammon, but the longer it took to make each move.
Backgammon has a large branching factor, yet moves must be made within a few
seconds. It was only feasible to search ahead selectively a few steps, but even so
the search resulted in significantly better action selections.
So far we have emphasized heuristic search as an action-selection technique, but this may not be its most important aspect. Heuristic search also suggests ways of selectively distributing backups that may lead to better and faster approximation of the optimal value function. A great deal of research on heuristic search has been devoted to making the search as efficient as possible. The search tree is grown selectively, deeper along some lines and shallower along others. For example, the search tree is often deeper for the actions that seem most likely to be best, and shallower for those that the agent will probably not want to take anyway. Can we use a similar idea to improve the distribution of backups? Perhaps it can be done by preferentially updating state-action pairs whose values appear to be close to the maximum available from the state. To our knowledge, this and other possibilities for distributing backups based on ideas borrowed from heuristic search have not yet been explored.
We should not overlook the most obvious way in which heuristic search focuses backups: on the current state. Much of the effectiveness of heuristic search is due to its search tree being tightly focused on the states and actions that might immediately follow the current state. You may spend more of your life playing chess than checkers, but when you play checkers, it pays to think about checkers and about your particular checkers position, your likely next moves, and successor positions. However you select actions, it is these states and actions that are of highest priority for backups and where you most urgently want your approximate value function to be accurate. Not only should your computation be preferentially devoted to imminent events, but so should your limited memory resources. In chess, for example, there are far too many possible positions to store distinct value estimates for each of them, but chess programs based on heuristic search can easily store distinct estimates for the millions of positions they encounter looking ahead from a single position. This great focusing of memory and computational resources on the current decision is presumably the reason why heuristic search can be so effective.
The distribution of backups can be altered in similar ways to focus on the current state and its likely successors. As a limiting case we might use exactly the methods of heuristic search to construct a search tree, and then perform the individual, one-step backups from bottom up, as suggested by Figure 9.15. If the backups are ordered in this way and a table-lookup representation is used, then exactly the same backup would be achieved as in heuristic search. Any state-space search can be viewed in this way as the piecing together of a large number of individual one-step backups. Thus, the performance improvement observed with deeper searches is not due to the use of multistep backups as such. Instead, it is due to the focus and concentration of backups on states and actions immediately downstream from the current state. By devoting a large amount of computation specifically relevant to the candidate actions, a much better decision can be made than by relying on unfocused backups.
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