Optimal Deterministic Oracle Complexity for Weakly Convex Optimization

Abstract

We study the oracle complexity of finding $\epsilon$-stationary points of $\rho$-weakly convex and $G$-Lipschitz functions, where stationarity is measured by the gradient of the Moreau envelope. We consider a first-order oracle that returns both the function value and the full subdifferential at every query point. We prove that every deterministic first-order algorithm requires $ \Omega({\rho G^2\Delta}/{\epsilon^4})$ oracle queries whenever $\Delta \leq {G^2}/{\rho}$, where $f(\bz)-\inf f \leq \Delta$. This lower bound matches the best known deterministic and stochastic first-order upper bounds, up to universal constants, and establishes the optimal deterministic oracle complexity. The result reveals a fundamental complexity separation between smooth nonconvex and nonsmooth weakly convex optimization. While smooth nonconvex minimization admits a $\Theta(\epsilon^{-2})$ oracle complexity, nonsmooth weakly convex optimization incurs an intrinsic additional $\epsilon^{-2}$ factor arising from nonsmooth geometry rather than stochasticity.