Year: 2026
Type: preprint
Abstract: Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \[ f(x_T)-f^\star=Ω\!\left(\frac{1}{T^α\log T}\right), \qquad α=\frac{1+\sqrt5}{2}. \] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.
Source: arXiv (Cornell University)
Authors: Ma, Jianhao, Zhang, Jingzhao
Institution: Tsinghua University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212029601