On the power of standard information for tractability for L2-approximation in the randomized setting

We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λ consisting of function values or general linear information Λ consisting of arbitrary linear functionals. We use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using Λ. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ and Λ for the normalized or absolute error criterion. We show that in the randomized setting for the normalized or absolute error criterion, the power of Λ is the same as that of Λ for all notions of exponential and algebraic tractability without any condition. Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.