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 ^d 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.