Determine the asymptotic measure of hard initializations

Determine whether the hard-initialization probability q_{\varepsilon}(\kappa), defined as the normalized area of initial points in the unit disk whose exact-line-search Frank–Wolfe trajectories require at least \kappa/\sqrt{\varepsilon} iterations to reach objective gap at most \varepsilon, converges to zero as \varepsilon\downarrow0; in particular, determine whether its limit superior is positive or whether q_{\varepsilon}(\kappa) fails to converge.

Background

The paper studies exact-line-search Frank–Wolfe on the unit disk for minimizing the squared distance to a fixed unit vector. It defines H_{\varepsilon}(\kappa) as the set of initializations whose trajectories take at least \kappa/\sqrt{\varepsilon} iterations to attain objective gap at most \varepsilon, and q_{\varepsilon}(\kappa) as the area fraction of this hard set. The paper proves that the set of initializations whose trajectories eventually follow the specified slow envelope has Lebesgue measure zero, but this does not determine the measure of all finite-horizon hard initializations.

The unresolved issue is whether the normalized measure q_{\varepsilon}(\kappa) vanishes as the target accuracy tends to zero. Resolving it would clarify whether hard initializations become asymptotically rare and whether repeatedly sampling uniformly from the disk leads to a diverging expected number of trials before finding a hard initialization. The paper explains that local sensitivity estimates do not settle this global measure question because inverse trajectories can have exponentially many branches whose widths may compensate for their shrinking size.