Upper-Confidence-Bound Algorithms for Active Learning in Multi-Armed Bandits

International audienceIn this paper, we study the problem of estimating the mean values of all the arms uniformly well in the multi-armed bandit setting. If the variances of the arms were known, one could design an optimal sampling strategy by pulling the arms proportionally to their variances. However, since the distributions are not known in advance, we need to design adaptive sampling strategies to select an arm at each round based on the previous observed samples. We describe two strategies based on pulling the arms proportionally to an upper-bound on their variances and derive regret bounds for these strategies. %on the excess estimation error compared to the optimal allocation. We show that the performance of these allocation strategies depends not only on the variances of the arms but also on the full shape of their distributions

Fifty years of business confidence surveys on manufacturing sector

In this work the evolution of the Italian Business Confidence Survey on manufacturing sector is presented starting from the preliminary European project for harmonized statistics launched in the late fifties of the last century. Survey changes are described, focusing in particular on the so-called confidence indicator. The continuing increase of statistical accuracy in sampling is recalled, from the initial purposive sample and controls, up to the present state of the art. Specific attention is devoted to the role of administrative archives in the sampling plan. Emphasis is also given to the increasing use of computer simulation in assessing the validity of the estimates. The role of cyclical analysis is finally highlighted with regard to two aspects: (1) the business confidence has not a corresponding variable in the economic system\u2014the validation can only be performed in comparison with correlated variables (e.g. IP, GDP); (2) confidence shows forecasting capability for the economic system

On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators With Discontinuous Coefficients

International audienceIn this paper we review some recent results on stochastic analytical and numericalapproaches to parabolic and elliptic partial differential equationsinvolving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. In the one-dimensional case existence and uniqueness results for such PDEscan be obtained by stochastic methods. The probabilistic interpretation of the solutionsallows one to develop and analyze a low complexity Monte Carlo numerical resolutionmethod. In addition, it allows one to get accurate pointwise estimates for thederivatives of the solutions from which sharp convergence rate estimates are deducedfor the stochastic numerical method.A stochastic approach is also developed for the linearized Poisson-Boltzmannequation in Molecular Dynamics.As in the one-dimensional case, the probabilistic interpretation of the solution involves the solution of a SDE including a non standard localtime term related to the discontinuity interface. We presentan extended Feynman-Kac formula for the Poisson-Boltzmann equation. Thisformula justifies various probabilistic numerical methods to approximate thefree energy of a molecule and bases error analyzes.We finally present probabilistic interpretations of the non-linearizedPoisson-Boltzmann equation in terms of backward stochastic differentialequations