Non-asymptotic Global Convergence Rates of BFGS with Exact Line Search (2404.01267v1)

Abstract: In this paper, we explore the non-asymptotic global convergence rates of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method implemented with exact line search. Notably, due to Dixon's equivalence result, our findings are also applicable to other quasi-Newton methods in the convex Broyden class employing exact line search, such as the Davidon-Fletcher-Powell (DFP) method. Specifically, we focus on problems where the objective function is strongly convex with Lipschitz continuous gradient and Hessian. Our results hold for any initial point and any symmetric positive definite initial Hessian approximation matrix. The analysis unveils a detailed three-phase convergence process, characterized by distinct linear and superlinear rates, contingent on the iteration progress. Additionally, our theoretical findings demonstrate the trade-offs between linear and superlinear convergence rates for BFGS when we modify the initial Hessian approximation matrix, a phenomenon further corroborated by our numerical experiments.

• The paper establishes explicit global linear convergence rates for BFGS using exact line search, linking the rates to the condition number and Hessian initialization.
• The analysis extends to proving global superlinear convergence through a unified potential function framework that maps the transition from linear to superlinear phases.
• The study reveals trade-offs in initial Hessian approximations, providing actionable insights for optimizing large-scale, strongly convex problems.

Overview of "Non-asymptotic Global Convergence Rates of BFGS with Exact Line Search"

This paper presents a non-asymptotic analysis of the global convergence rates of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method when exact line search is employed. It addresses a significant gap in the convergence analysis of quasi-Newton methods, which traditionally focused on asymptotic results without providing explicit rates. The approach is applicable not only to BFGS but also to other quasi-Newton methods in the convex Broyden class, such as the Davidon-Fletcher-Powell (DFP) method, by virtue of Dixon's equivalence result.

Key Contributions

1. Global Linear Convergence Rates: The paper establishes explicit global linear convergence rates for BFGS under exact line search. This is one of the first rigorous demonstrations of non-asymptotic global convergence for BFGS with this line search scheme. The derived rates depend on the condition number of the objective function and the properties of the initial Hessian approximation matrix.
2. Global Superlinear Convergence Rates: The analysis further extends to demonstrate global superlinear convergence rates. Employing a potential function-based framework, initially introduced in prior works, the paper provides a unified methodology for proving both linear and superlinear convergence.
3. Three-Phase Convergence Process: The paper delineates a detailed three-phase convergence process characterized by changes in the convergence rate from linear to superlinear. This indicates a deeper understanding of the iterates' behavior as the BFGS method progresses.
4. Trade-offs in Initial Hessian Approximation: The paper highlights the influence of the initial Hessian approximation matrix on the convergence behavior. It reveals that there is a trade-off between quicker linear convergence and faster superlinear rates based on matrix initialization, either $B_0 = L I$ or $B_0 = \mu I$.

Theoretical Framework

The paper builds upon two pivotal propositions that connect the potential function of the Hessian approximations to key convergence quantities. The potential function, which elegantly combines the trace and determinant of matrices, aids in relating computational progress to the iterates and their alignment with search directions. The crafted framework provides not only bounds on function value decrease but also an intricate understanding of the convergence mechanisms at play in the BFGS updates.

Practical Implications

Practically, this expansion of our understanding of BFGS efficacy, especially when coupled with exact line search, broadens the potential applications of BFGS in optimizing large-scale, strongly convex problems. The explicit rates provided can inform algorithmic choices, particularly around initial matrix selection, maximizing overall convergence efficiency.

Future Prospects

The findings encourage further exploration into potential function generalizations and the applicability of these techniques to other popular optimization frameworks such as stochastic and decentralized optimization methods. Moreover, examining how different line search conditions or initializations can affect other families of quasi-Newton methods remains an open area for fruitful research.

Conclusion

In sum, this paper presents a substantial leap in the theoretical analysis of BFGS, offering precise and actionable insights into its global convergence properties. By extending the understanding from asymptotic guarantees to exact rates, it empowers researchers and practitioners to deploy BFGS with enhanced confidence in its optimization capabilities, potentially inspiring adaptations and new applications across a variety of domains.