Accelerated Concatenation of Stepsize Schedules for Smooth Strongly Convex Gradient Descent

Abstract

This work considers stepsize schedules for gradient descent on smooth strongly convex objectives. We develop a curvature-dependent concatenation framework for constructing stepsize schedules of arbitrary length with sharp worst-case guarantees. Within this framework, we introduce two families of schedules for the function-value gap and gradient norm of the last iterate, respectively. For each fixed condition number $κ=L/μ>1$, both families attain iteration complexity $Θ(κ^ρ\log(1/ε))$ for sufficiently small $ε$, where $ρ=\log_{\sqrt2+1}2\approx0.7864$. This complexity is optimal up to a subpolynomial factor in $κ$ among predetermined nonnegative stepsize schedules. The framework also recovers the strongly convex silver schedule at power-of-two iteration budgets and applies to smooth convex and smooth weakly convex functions.