Accelerated first-order methods for uniformly nondegenerate objectives

Establish whether uniform nondegeneracy alone admits a globally convergent first-order method with complexity O(\sqrt{\kappa_\Lambda}\log(g(x_0)/\varepsilon)), where \kappa_\Lambda is the condition parameter governing the Alternating Descent Ascent method and \varepsilon is the target stationarity tolerance.

Background

The paper introduces uniform nondegeneracy for objectives whose Hessian spectrum remains uniformly separated from zero while its positive and negative curvature orientations may vary across the domain. It develops descent--ascent inequalities (DAI) characterizing this class and analyzes Alternating Descent Ascent (ADA), obtaining global convergence with oracle complexity O(\kappa_\Lambda\log(g(x_0)/\varepsilon)).

The authors observe that this linear dependence on \kappa_\Lambda specializes to the classical condition-number dependence in the strongly convex limit, to the O(\kappa\log(1/\varepsilon)) scale for strongly convex--strongly concave problems, and to an accelerated primal--dual scale for linearly constrained strongly convex optimization. The unresolved issue is whether a generally applicable globally convergent first-order method can improve the dependence from \kappa_\Lambda to its square root without requiring a prescribed curvature orientation. A lower bound separating the general uniformly nondegenerate class from classical subclasses is also identified as a possible way to determine what additional structure acceleration requires.