  Bandit based methods for tree search have recently gained popularity when
applied to huge trees, e.g. in the game of go (Gelly et al., 2006). The UCT
algorithm (Kocsis and Szepesvari, 2006), a tree search method based on Upper
Confidence Bounds (UCB) (Auer et al., 2002), is believed to adapt locally to
the effective smoothness of the tree. However, we show that UCT is too
``optimistic'' in some cases, leading to a regret O(exp(exp(D))) where D is the
depth of the tree. We propose alternative bandit algorithms for tree search.
First, a modification of UCT using a confidence sequence that scales
exponentially with the horizon depth is proven to have a regret O(2^D
\sqrt{n}), but does not adapt to possible smoothness in the tree. We then
analyze Flat-UCB performed on the leaves and provide a finite regret bound with
high probability. Then, we introduce a UCB-based Bandit Algorithm for Smooth
Trees which takes into account actual smoothness of the rewards for performing
efficient ``cuts'' of sub-optimal branches with high confidence. Finally, we
present an incremental tree search version which applies when the full tree is
too big (possibly infinite) to be entirely represented and show that with high
probability, essentially only the optimal branches is indefinitely developed.
We illustrate these methods on a global optimization problem of a Lipschitz
function, given noisy data.
