Papers › Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond

Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond

27 Jun 2019arXiv:1906.11985archive 2025-07-28

Oliver Hinder, Aaron Sidford, Nimit S. Sohoni

In this paper, we provide near-optimal accelerated first-order methods for minimizing a broad class of smooth nonconvex functions that are strictly unimodal on all lines through a minimizer. This function class, which we call the class of smooth quasar-convex functions, is parameterized by a constant γ∈(0,1], where γ= 1 encompasses the classes of smooth convex and star-convex functions, and smaller values of γ indicate that the function can be "more nonconvex." We develop a variant of accelerated gradient descent that computes an ϵ-approximate minimizer of a smooth γ-quasar-convex function with at most O(γ⁻¹ ϵ^(-1/2) log(γ⁻¹ ϵ⁻¹)) total function and gradient evaluations. We also derive a lower bound of Ω(γ⁻¹ ϵ^(-1/2)) on the worst-case number of gradient evaluations required by any deterministic first-order method, showing that, up to a logarithmic factor, no deterministic first-order method can improve upon ours.

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