Heuristic for choosing the maximum gain in linear gain annealing

Develop a simple, general heuristic to set the maximum gain a_max a priori in linear gain annealing for the soft vector spin dynamical system with time‑dependent gain (used to minimize Ising energies), such that the choice improves optimization performance across arbitrary coupling matrices J without instance‑specific tuning.

Background

The paper studies soft vector spins evolving under a gain-annealed dynamical system and compares several dimensional annealing mechanisms. For linear gain annealing, the gain a(t) is increased from an initial value a0 = −λmax(J) to a maximum value a_max. The authors note that choosing a_max impacts performance but lack a principled way to set it.

They therefore adopt a heuristic choice a_max = a0 + 2 for experiments, acknowledging that a generally applicable, performance-improving rule for choosing a_max across different coupling matrices J is not known.