Regional Complexity Analysis of Algorithms for Nonconvex Smooth Optimization

TL;DR

This paper introduces a novel regional complexity analysis strategy for nonconvex smooth optimization algorithms, enabling more realistic worst-case performance characterizations based on search space regions rather than conservative bounds.

Contribution

The paper proposes a new regional analysis method that characterizes algorithm performance over search space regions, independent of specific objective functions, and extends to higher-order derivatives.

Findings

Analyzes regions defined by derivatives for complexity bounds.

Provides example complexity analyses for first- and second-order algorithms.

Discusses generalization to higher-order derivatives and algorithms.

Abstract

A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition is approximately satisfied. This arguably leads to conservative characterizations based on anomalous objectives rather than on ones that are typically encountered in practice. By contrast, the strategy proposed in this paper characterizes worst-case performance separately over regions comprising a search space. These regions are defined generically based on properties of derivative values. In this manner, one can analyze the worst-case performance of an algorithm independently from any particular class of objectives. Then, once given a class of objectives, one can obtain an informative, fine-tuned complexity analysis merely by delineating the types of regions that comprise the search spaces for functions in the class. Regions defined by first- and second-order derivatives are discussed in detail and example complexity analyses are provided for a few fundamental first- and second-order algorithms when employed to minimize convex and nonconvex objectives of interest. It is also explained how the strategy can be generalized to regions defined by higher-order derivatives and for analyzing the behavior of higher-order algorithms.