Noisy Information and Computational Complexity

This book 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; he considers deterministic as well as stochastic noise. He also presents optimal algorithms, optimal information, and complexity bounds in different settings: worst case, average case, mixed worst-average, average-worst, and asymptotic. Particular topics include: the 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, and relations between different settings. The book integrates the work of researchers over the past decade in such areas as computational complexity, approximation theory, and statistics, and includes many new results as well. The author supplies two hundred exercises to increase the reader's understanding of the subject.