Instances where subgradient message passing outperforms accelerated smoothing

Determine whether there are quantum Hamiltonians or concrete problem instances for which the subgradient message-passing algorithm achieves better practical performance than the entropy-smoothed Nesterov-type accelerated gradient algorithm.

Background

The paper derives two distributed algorithms for optimizing the dual of local-consistency relaxations. The subgradient method has a worst-case iteration complexity of order $1/\epsilon^2$, whereas accelerated gradient descent applied to an entropy-smoothed objective has a bound of order $1/\epsilon$. Numerical experiments generally favor the smoothed method, although the authors note that the subgradient method can have lower per-iteration costs in some practical settings.

The unresolved issue is whether specific Hamiltonians—particularly adversarially selected or otherwise structured instances—actually make the subgradient method superior overall. This is a concrete comparative algorithmic question rather than a general request for implementation improvements.