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We estimate convergence rates for fixed-point iterations of a class ofnonlinear operators which are partially motivated from solving convexoptimization problems. We introduce the notion of the generalized averagednonexpansive (GAN) operator with a positive exponent, and provide a convergencerate analysis of the fixed-point iteration of the GAN operator. The proposedgeneralized averaged nonexpansiveness is weaker than the averagednonexpansiveness while stronger than nonexpansiveness. We show that thefixed-point iteration of a GAN operator with a positive exponent converges toits fixed-point and estimate the local convergence rate (the convergence ratein terms of the distance between consecutive iterates) according to the rangeof the exponent. We prove that the fixed-point iteration of a GAN operator witha positive exponent strictly smaller than 1 can achieve an exponential globalconvergence rate (the convergence rate in terms of the distance between aniterate and the solution). Furthermore, we establish the global convergencerate of the fixed-point iteration of a GAN operator, depending on both theexponent of generalized averaged nonexpansiveness and the exponent of theH$\ddot{\text{o}}$lder regularity, if the GAN operator is alsoH$\ddot{\text{o}}$lder regular. We then apply the established theory to threetypes of convex optimization problems that appear often in data science todesign fixed-point iterative algorithms for solving these optimization problemsand to analyze their convergence properties.

Convergence Rate Analysis for Fixed-Point Iterations of Generalized Averaged Nonexpansive Operators