  Finite-horizon lookahead policies are abundantly used in Reinforcement
Learning and demonstrate impressive empirical success. Usually, the lookahead
policies are implemented with specific planning methods such as Monte Carlo
Tree Search (e.g. in AlphaZero). Referring to the planning problem as tree
search, a reasonable practice in these implementations is to back up the value
only at the leaves while the information obtained at the root is not leveraged
other than for updating the policy. Here, we question the potency of this
approach. Namely, the latter procedure is non-contractive in general, and its
convergence is not guaranteed. Our proposed enhancement is straightforward and
simple: use the return from the optimal tree path to back up the values at the
descendants of the root. This leads to a $\gamma^h$-contracting procedure,
where $\gamma$ is the discount factor and $h$ is the tree depth. To establish
our results, we first introduce a notion called \emph{multiple-step greedy
consistency}. We then provide convergence rates for two algorithmic
instantiations of the above enhancement in the presence of noise injected to
both the tree search stage and value estimation stage.
