Extension of the zeroth-order lower-bound construction to the Goldstein Frank-Wolfe gap

Extend the joint lower-bound construction for zeroth-order Goldstein residual optimization to establish a matching lower bound for the Goldstein Frank-Wolfe gap.

Background

The paper notes that the dimension and stationarity dependence of its zeroth-order results is optimal separately, and cites a joint lower bound for the Goldstein residual that matches the projection-based algorithm up to logarithmic factors. The cited construction is restricted to queries in a bounded ball; the authors observe that this restriction may be removable by soft projection, but the corresponding extension to the Goldstein Frank-Wolfe gap is not supplied.