Robustness of the Frank-Wolfe Method under Inexact Oracles and the Cost of Linear Minimization

We investigate the robustness of the Frank-Wolfe method when gradients are computed inexactly and examine the relative computational cost of the linear minimization oracle (LMO) versus projection. For smooth nonconvex functions, we establish a convergence guarantee of order $\mathcal{O}(1/\sqrt{k}+δ)$ for Frank-Wolfe with a $δ$--oracle. Our results strengthen previous analyses for convex objectives and show that the oracle errors do not accumulate asymptotically. We further prove that approximate projections cannot be computationally cheaper than accurate LMOs, thus extending to the case of inexact projections. These findings reinforce the robustness and efficiency of the Frank-Wolfe framework.