Cycling regions of the heavy-ball method

For the heavy-ball optimization method on smooth strongly convex functions, which parameter choices admit periodic cycling? The target compares ordered root-of-unity modes, arbitrary pure Fourier modes and mixtures of modes in higher-dimensional cycles.

Why it matters and what counts as progress

Why it matters: The comparison determines whether commonly studied cycle families capture all obstructions to convergence.

What counts as progress: An exact cycling witness outside a previously known region, or an exclusion theorem covering mixed Fourier modes as well as pure modes.

How the problem developed

21 Jul 2023 · Cycling result [1]
Baptiste Goujaud, Adrien Taylor and Aymeric Dieuleveut Showed that heavy-ball parameters either fail to accelerate on quadratics or yield a smooth strongly convex function on which the method cycles and does not converge.
18 Oct 2026 · Reported theorem [2]
Konstantinos Fotopoulos, Michael Helcig, Karl Deck and Jannis Alsbach Reported non-acceleration over one-dimensional smooth strongly convex functions, a strict enlargement of the root-of-unity cycling region, and a higher-dimensional cycle not of root-of-unity type.

AI contribution

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

Konstantinos Fotopoulos, Michael Helcig, Karl Deck, Jannis Alsbach

Two AI-generated heavy-ball counterexamples separate cycling regions for smooth strongly convex functions. Example 3.2 uses κ=4/5, β=17/20 and Lγ=18/5 to admit a primitive (K,m)=(5,2) cycle while excluding all ordered m=1 modes. Section 5 uses L=1, (κ,β,γ)=(531/10000,4879/10000,32829/12500) and a ten-step mixture of modes 2 and 3 with weights 137/500 and 363/500: it supports a four-dimensional cycle but excludes every pure root-of-unity mode.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean + human)

Human understandingFull