Gradient algorithms for quadratic optimization with fast convergence rates

We propose a family of gradient algorithms for minimizing a quadratic function f(x) = (Ax, x)/2− (x, y) in R or a Hilbert space, with simple rules for choosing the step-size at each iteration. We show that when the step-sizes are generated by a dynamical system with ergodic distribution having the arcsine density on a subinterval of the spectrum of A, the asymptotic rate of convergence of the algorithm can approach the (tight) bound on the rate of convergence of a conjugate gradient algorithm stopped before d iterations, with d ≤ ∞ the space dimension.