Noisy Information and Computational Complexity

In this work noisy information is studied in the context of computational complexity - in other words it deals with the computational complexity of mathematical problems for which available information is partial, noisy and priced. The author develops a general theory of computational complexity of continuous problems with noisy information and gives a number of applications; deterministic as well as stochastic noise is considered. He presents optimal algorithms, optimal information, and complexity bounds in different settings: worst case, average case, mixed worst-average and average-worst, and asymptotic. Particular topics include: existence of optimal linear (affine) algorithms, optimality properties of smoothing spline, regularization and least squares algorithms (with the optimal choice of the smoothing and regularization parameters), adaption versus nonadaption, relations between different settings. The book integrates the work of researchers since the mid-1980s in such areas as computational complexity, approximation theory and statistics, and includes many new results.