Explicit iteration bound for unit step acceptance by line search in BFGS/quasi-Newton methods

Determine an explicit upper bound on the number of iterations required before the line search subroutine used with the BFGS (or more broadly, quasi-Newton) method accepts the unit step size η_k = 1, starting from an arbitrary initial point and symmetric positive definite initial Hessian approximation, under the assumptions that the objective f is strongly convex with Lipschitz continuous gradient and Hessian. This bound is needed to bridge line-search-based global convergence guarantees with local non-asymptotic analyses that rely on η_k = 1.

Background

Local non-asymptotic analyses of quasi-Newton methods such as BFGS often assume that the step size is fixed to one (η_k = 1) once the iterates are sufficiently close to the optimum, enabling superlinear rate proofs. To extend these local results to global guarantees, a common strategy is to use a line search to ensure global progress and then transition to the unit-step regime.

However, the authors point out a key obstacle to this approach: there is no explicit characterization of how many iterations are needed before the line search procedure begins to accept the unit step size. Without such a bound, it is difficult to connect the global phase with the local superlinear convergence analyses that assume η_k = 1.