Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and variance by Cowan et al. [2015] and of uniform distributions with unknown support [Cowan and Katehakis, 2015]. The results derived in these specialised cases are generalised here to the non-parametric setup, where the learner knows only a bound on the kurtosis of the noise, which is a scale free measure of the extremity of outliers.

Neural Information Processing Systems http://nips.cc/

We introduce two novel tree search algorithms that use a policy to guide search. The first algorithm is a best-first enumeration that uses a cost function that allows us to prove an upper bound on the number of nodes to be expanded before reaching a goal state. We show that this best-first algorithm is particularly well suited for "needle-in-a-haystack" problems. The second algorithm is based on sampling and we prove an upper bound on the expected number of nodes it expands before reaching a set of goal states. We show that this algorithm is better suited for problems where many paths lead to a goal. We validate these tree search algorithms on 1,000 computer-generated levels of Sokoban, where the policy used to guide the search comes from a neural network trained using A3C. Our results show that the policy tree search algorithms we introduce are competitive with a state-of-the-art domain-independent planner that uses heuristic search.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Bandits

Bandits

We prove an upper bound on the minimax cumulative regret in this problem of (d p n), which matches known lower bounds up to logarithmic factors and improves on the best known upper bound by a factor of p d. We also show that the minimax simple regret is (d/ p n) and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling [Russo and Van Roy, 2014] are not sufficient for optimal regret.