[Paper Review] A stroll in the jungle of error bounds

Trong Phong Nguyen|arXiv (Cornell University)|Apr 23, 2017

Sparse and Compressive Sensing Techniques38 references3 citations

TL;DR

This paper provides a unified framework for error bounds in optimization by establishing the central role of the Łojasiewicz gradient inequality. It characterizes global and local error bounds for convex and nonconvex polynomial functions, deriving Hölder-type bounds critical for convergence rate analysis and complexity theory in optimization algorithms.

ABSTRACT

The aim of this paper is to give a short overview on error bounds and to provide the first bricks of a unified theory. Inspired by the works of [8, 15, 13, 16, 10], we show indeed the centrality of the Lojasiewicz gradient inequality. For this, we review some necessary and sufficient conditions for global/local error bounds, both in the convex and nonconvex case. We also recall some results on quantitative error bounds which play a major role in convergence rate analysis and complexity theory of many optimization methods.

Motivation & Objective

• To establish a unified theoretical foundation for error bounds in optimization by identifying the Łojasiewicz gradient inequality as central.
• To characterize necessary and sufficient conditions for global and local error bounds in both convex and nonconvex settings.
• To derive quantitative error bounds for polynomial and piecewise convex polynomial functions, especially those of Hölder type.
• To connect error bounds with Kurdyka–Łojasiewicz inequality and complexity analysis of descent methods.
• To extend classical results (e.g., Hoffman, Łojasiewicz) to broader classes of functions, including non-differentiable and nonconvex systems.

Proposed method

• Uses the Łojasiewicz gradient inequality as a unifying principle to characterize error bounds in metric spaces.
• Applies results from Azé and Corvellec on strong slope to derive characterizations of error bounds in general Banach spaces.
• Employs the subanalytic and semi-algebraic structure of functions to establish quantitative error bounds via Łojasiewicz-type inequalities.
• Derives Hölder-type error bounds for convex and piecewise convex polynomial functions using degree-dependent exponents.
• Applies the recession cone condition and compactness arguments to extend global error bounds to constrained sets.
• Uses the maximum of convex polynomial functions and their recession behavior to derive error bounds under mild assumptions.

Experimental results

Research questions

• RQ1 How can the Łojasiewicz gradient inequality be used to unify the theory of error bounds across convex and nonconvex optimization?
• RQ2 What are the necessary and sufficient conditions for a global or local error bound in terms of the strong slope and metric subregularity?
• RQ3 What is the precise form of the quantitative error bound for convex polynomial functions of degree d?
• RQ4 Under what conditions does a global error bound hold for systems of convex polynomial inequalities, especially without the Slater condition?
• RQ5 How do error bounds for piecewise convex polynomial functions depend on the structure of the underlying polyhedral partition and function degrees?

Key findings

• For a convex polynomial function f of degree d, there exists τ > 0 such that dist(x, [f ≤ 0]) ≤ τ([f(x)]₊ + [f(x)]₊¹ᐟᵏ(ⁿ,ᵈ)), where k(n,d) is a function of dimension and degree.
• For a piecewise convex polynomial function f of degree d satisfying a coercivity or convexity condition, the same Hölder-type bound holds globally.
• For convex quadratic function systems, a global error bound exists with exponent 1/2, even without the Slater condition, as shown by Pang and Wang.
• For a system of convex polynomial functions of degree d on a convex polyhedral set K, if the recession cone condition holds, then dist(x,S) ≤ c([f(x)]₊ + [f(x)]₊¹ᐟᵏ(ⁿ,²ᵈ)) for x ∈ K.
• The result of Li [44] and Yang [73] shows that for convex polynomial functions, the exponent can be improved to 1/d, giving tighter bounds.
• The connection between error bounds and the Kurdyka–Łojasiewicz inequality is formalized, enabling complexity analysis of descent methods via error bound exponents.