Intuitively, learning should be easier when the data points lie on a low-dimensional submanifold of the input space. Recently there has been a growing interest in algorithms that aim to exploit such geometrical properties of the data. Oftentimes these algorithms require estimating the dimension of the manifold first. In this paper we propose an algorithm for dimension estimation and study its finite-sample behaviour. The algorithm estimates the dimension locally around the data points using nearest neighbor techniques and then combines these local estimates. We show that the rate of convergence of the resulting estimate is independent of the dimension of the input space and hence the algorithm is "manifold-adaptive". Thus, when the manifold supporting the data is low dimensional, the algorithm can be exponentially more efficient than its counterparts that are not exploiting this property. Our computer experiments confirm the obtained theoretical results.

Reinforcement learning with linear and non-linear function approximation has been studied extensively in the last decade. However, as opposed to other fields of machine learning such as supervised learning, the effect of finite sample has not been thoroughly addressed within the reinforcement learning framework. In this paper we propose to use regularization in reinforcement learning and planning. More specifically, we control the complexity of the value function approximation using L 2 regularization. We consider the fitted Q-iteration algorithm, provide generalization bounds that account for small sample sizes. A realistic visual-servoing problem is used to illustrate the benefits of using a regularized procedure.

We consider the problem of model selection in the batch (offline, non-interactive) reinforcement learning setting when the goal is to find an action-value function with the smallest Bellman error among a countable set of candidates functions. We propose a complexity regularization-based model selection algorithm, BERMIN, and prove that it enjoys an oracle-like property: the estimator's error differs from that of an oracle, who selects the candidate with the minimum Bellman error, by only a constant factor and a small remainder term that vanishes at a parametric rate as the number of samples increases. As an application, we consider a problem when the true action-value function belongs to an unknown member of a nested sequence of function spaces. We show that under some additional technical conditions BERMIN leads to a procedure whose rate of convergence, up to a constant factor, matches that of an oracle who knows which of the nested function spaces the true action-value function belongs to, i.e., the procedure achieves adaptivity.

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