Convergence analysis of the Riemannian proximal gradient method with inexact oracle

Abstract: We extend the notion of an inexact first-order oracle from the Euclidean setting to Riemannian optimization, and conduct a convergence analysis for the Riemannian proximal gradient method equipped with this oracle, which we refer to as RPG-IO. Under mild conditions on the oracle errors, we establish the global convergence of RPG-IO. Specifically, we prove that (i) the norm of the search direction converges to zero; (ii) the sequence of function values converges to the function value at any accumulation point of the iterates; and (iii) every accumulation point of the iterates is a stationary point. Under the additional assumption of the Riemannian Kurdyka--Łojasiewicz (KL) property, we prove that the full sequence of iterates generated by RPG-IO converges to a single stationary point. Moreover, we derive explicit convergence rates when the KL exponent is specified. Finally, when a strong inexact oracle is employed, we establish the convergence rate of the sequence of function values to the optimal value.