On the Condition Number Upper Bound of the L-BFGS Inverse Hessian Approximation Matrix with a Two-Sided Geometric Envelope Safeguarding Mechanism

2026-07-07 · Source: Takara TLDR - Daily AI Papers · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Two-Sided L-BFGS is a novel variant of the limited-memory BFGS (L-BFGS) algorithm designed to enhance stability in large-scale optimization, particularly within ill-conditioned or non-convex problem landscapes. It addresses the issue of an exploding condition number in the implicit inverse Hessian approximation, which typically causes numerical instability and degraded convergence. The proposed method employs a two-sided geometric envelope to dynamically constrain this condition number, ensuring well-conditioned inverse Hessian approximations. Two-Sided L-BFGS maintains the original algorithm's O(mn) memory and per-iteration time complexities and preserves accumulated curvature information. Theoretical proofs demonstrate a uniform bound on the condition number, explicitly defined by memory depth, problem dimension, and envelope hyperparameters. Furthermore, it retains asymptotic global convergence in non-convex regimes under standard smoothness and strong Wolfe line-search assumptions, matching other L-BFGS variants. Numerical experiments confirm improved robustness and convergence on ill-conditioned benchmarks.

Key takeaway

For research scientists developing or applying large-scale optimization algorithms, Two-Sided L-BFGS offers a robust solution for ill-conditioned or non-convex problems. If you encounter numerical instability or slow convergence with standard L-BFGS, consider implementing this variant to maintain well-conditioned inverse Hessian approximations. This approach ensures improved robustness and convergence behavior without sacrificing O(mn) complexity, providing a theoretically sound and practically effective alternative.

Key insights

Two-Sided L-BFGS stabilizes optimization by bounding the inverse Hessian's condition number using a geometric envelope.

Principles

• Dynamic condition number constraint improves numerical stability.
• Preserving curvature information is crucial for convergence.
• Theoretical bounds can guide algorithm design.

Method

The method dynamically constrains the inverse Hessian's condition number via a two-sided geometric envelope, tracking extreme eigenvalues through m quasi-Newton updates from a scaled identity matrix.

In practice

• Apply to high-dimensional, ill-conditioned optimization problems.
• Consider for non-convex optimization tasks requiring stability.

Topics

• L-BFGS
• Optimization Algorithms
• Condition Number
• Inverse Hessian Approximation
• Non-convex Optimization
• Numerical Stability

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.