89,321 research outputs found
Bandit Algorithms for Tree Search
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
On quantum vs. classical probability
Quantum theory shares with classical probability theory many important
properties. I show that this common core regards at least the following six
areas, and I provide details on each of these: the logic of propositions,
symmetry, probabilities, composition of systems, state preparation and
reductionism. The essential distinction between classical and quantum theory,
on the other hand, is shown to be joint decidability versus smoothness; for the
latter in particular I supply ample explanation and motivation. Finally, I
argue that beyond quantum theory there are no other generalisations of
classical probability theory that are relevant to physics.Comment: Major revision: key results unchanged, but derivation and discussion
completely rewritten; 33 pages, no figure
Minimax estimation of smooth optimal transport maps
Brenier's theorem is a cornerstone of optimal transport that guarantees the
existence of an optimal transport map between two probability distributions
and over under certain regularity conditions. The main
goal of this work is to establish the minimax estimation rates for such a
transport map from data sampled from and under additional smoothness
assumptions on . To achieve this goal, we develop an estimator based on the
minimization of an empirical version of the semi-dual optimal transport
problem, restricted to truncated wavelet expansions. This estimator is shown to
achieve near minimax optimality using new stability arguments for the semi-dual
and a complementary minimax lower bound. Furthermore, we provide numerical
experiments on synthetic data supporting our theoretical findings and
highlighting the practical benefits of smoothness regularization. These are the
first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure
Random rewards, fractional Brownian local times and stable self-similar processes
We describe a new class of self-similar symmetric -stable processes
with stationary increments arising as a large time scale limit in a situation
where many users are earning random rewards or incurring random costs. The
resulting models are different from the ones studied earlier both in their
memory properties and smoothness of the sample paths.Comment: Published at http://dx.doi.org/10.1214/105051606000000277 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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