Bernstein Numbers and Lower Bounds for the Monte Carlo Error

Abstract

We are interested in lower bounds for the approximation of linear operators between Banach spaces with algorithms that may use at most n arbitrary linear functionals as information. Lower error bounds for deterministic algorithms can easily be found by Bernstein widths; for mappings between Hilbert spaces it is already known how Bernstein widths (which are the singular values in that case) provide lower bounds for Monte Carlo methods. Here, a similar connection between Bernstein numbers and lower bounds is shown for the Monte Carlo approximation of operators between arbitrary Banach spaces. For non-adaptive algorithms we consider the average case setting with the uniform distribution on finite dimensional balls and in this way we obtain almost optimal prefactors. By combining known results about Gaussian measures and their connection to the Monte Carlo error we also cover adaptive algorithms, however with weaker constants. As an application, we find that for the $L_{\infty }$ approximation of smooth functions from the class $C^{\infty }([0,1]^d)$ with uniformly bounded partial derivatives, randomized algorithms suffer from the curse of dimensionality, as it is known for deterministic algorithms.