Non-Acceleration of the Heavy-Ball Method in 1D

Abstract

Despite having an asymptotically optimal rate of convergence on convex quadratics, the acceleration of Polyak's heavy-ball method and a characterization of its dynamics within more general settings have remained long standing open problems. Recently, the work of Goujaud et al. [Math. Program. (2025)] made progress regarding acceleration in the context of smooth, strongly convex minimization: for a small enough inverse condition number, either the method does not accelerate on quadratics, or there exists a smooth strongly convex function such that heavy-ball cycles over the roots of unity. The constructed functions have two variables, thus leaving the question of acceleration over univariate functions open. In this work, we use a Fourier-based analysis to i) provide a proof for the non-acceleration of the heavy-ball method over the class of univariate smooth strongly convex functions. We also ii) provide a strict enlargement of the previous root-of-unity cycling region, with insights regarding the "fractional" sets of Goujaud et al. [arXiv:2502.19916], and iii) give a higher dimensional cycling instance which is not covered by root-of-unity cycling.