Strict per-instance versus worst-case gap for smooth nonconvex objectives

Determine whether the per-instance minimum-time value remains strictly below the worst-case first-order complexity on smooth nonconvex objective classes, as suggested by the dimension-versus-rate separation established for clustered quadratic spectra, and relate the answer to nonconvex worst-case lower bounds.

Background

The paper defines a clairvoyant per-instance minimum-time value for first-order span methods and contrasts it with worst-case oracle complexity over an objective class. For strongly convex quadratics with clustered spectra, the per-instance count depends on the number of distinct eigenvalues, whereas rate-based worst-case complexity depends on the condition number, producing an arbitrarily large separation.

The authors state that it is unresolved whether an analogous strict separation persists for smooth nonconvex objective classes. They identify the question as being connected to the development of nonconvex worst-case lower bounds, so resolving it would clarify whether the dimension-versus-rate mechanism is specific to the quadratic setting or extends more broadly.