Lower Complexity Bounds of First-order Methods for Affinely Constrained Composite Non-convex Problems

TL;DR

This paper establishes the first known lower complexity bounds for first-order methods solving affinely constrained composite non-convex non-smooth problems, revealing fundamental limits and the impact of regularizers.

Contribution

It provides the first lower bounds for FOMs in affine constrained non-convex non-smooth optimization, highlighting the influence of regularizers on problem difficulty.

Findings

Lower bounds for FOMs to find ε-stationary points are established.

Non-smooth convex regularizers increase problem complexity.

The lower bound with the second oracle is nearly tight, differing by a logarithmic factor.

Abstract

Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these methods. However, little can be claimed about their optimality as no lower bound is known, except for a few special \emph{smooth non-convex} cases. In this paper, we make the first attempt to establish lower complexity bounds of FOMs for solving a class of composite non-convex non-smooth optimization with linear constraints. Assuming two different first-order oracles, we establish lower complexity bounds of FOMs to produce a (near) $\epsilon$-stationary point of a problem (and its reformulation) in the considered problem class, for any given tolerance $\epsilon>0$. Our lower bounds indicate that the existence of a non-smooth convex regularizer can evidently increase the difficulty of an affinely constrained regularized problem over its nonregularized counterpart. In addition, we show that our lower bound of FOMs with the second oracle is tight, with a difference of up to a logarithmic factor from an upper complexity bound established in the extended arXiv version of this paper.