Online allocation and homogeneous partitioning for piecewise constant mean-approximation

Odalric-Ambrym Maillard (1) , Alexandra Carpentier (2)

1 MUL - Montanuniversität Leoben
2 Statistical Laboratory [Cambridge]

In the setting of active learning for the multi-armed bandit, where the goal of a learner is to estimate with equal precision the mean of a finite number of arms, recent results show that it is possible to derive strategies based on finite-time confidence bounds that are competitive with the best possible strategy. We here consider an extension of this problem to the case when the arms are the cells of a finite partition P of a continuous sampling space X \subset \Real^d. Our goal is now to build a piecewise constant approximation of a noisy function (where each piece is one region of P and P is fixed beforehand) in order to maintain the local quadratic error of approximation on each cell equally low. Although this extension is not trivial, we show that a simple algorithm based on upper confidence bounds can be proved to be adaptive to the function itself in a near-optimal way, when |P| is chosen to be of minimax-optimal order on the class of \alpha-Hölder functions.

Mots clés: Multi-armed bandits; Active sampling; Histograms

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Apprentissage [cs.LG]; Machine Learning [stat.ML]

Odalric-Ambrym Maillard, Alexandra Carpentier. Online allocation and homogeneous partitioning for piecewise constant mean-approximation. 2012. ⟨hal-00742893⟩