Restarting Frank–Wolfe: Faster Rates under Hölderian Error Bounds

Conditional gradient algorithms (aka Frank–Wolfe algorithms) form a classical set of methods for constrained smooth convex minimization due to their simplicity, the absence projection steps, and competitive numerical performance. While vanilla algorithm only ensures worst-case rate $${\mathcal {O}}(1/\epsilon )$$ , various recent results have shown that strongly functions on polytopes, method can be slightly modified achieve linear convergence. However, this still leaves huge gap between sublinear convergence {O}}(\log 1/\epsilon reach an $$\epsilon $$ -approximate solution. Here, we present new variant conditional dynamically adapt function’s geometric properties using restarts smoothly interpolates regimes. These interpolated rates are obtained when optimization problem satisfies type error bounds, which call strong Wolfe primal bounds. They combine information constraint with Hölderian bounds objective function.