Affine-Invariant Global Non-Asymptotic Convergence Analysis of BFGS under Self-Concordance
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915575610998784 |
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| author | Jin, Qiujiang Mokhtari, Aryan |
| author_facet | Jin, Qiujiang Mokhtari, Aryan |
| contents | In this paper, we establish global non-asymptotic convergence guarantees for the BFGS quasi-Newton method without requiring strong convexity or the Lipschitz continuity of the gradient or Hessian. Instead, we consider the setting where the objective function is strictly convex and strongly self-concordant. For an arbitrary initial point and any arbitrary positive-definite initial Hessian approximation, we prove global linear and superlinear convergence guarantees for BFGS when the step size is determined using a line search scheme satisfying the weak Wolfe conditions. Moreover, all our global guarantees are affine-invariant, with the convergence rates depending solely on the initial error and the strongly self-concordant constant. Our results extend the global non-asymptotic convergence theory of BFGS beyond traditional assumptions and, for the first time, establish affine-invariant convergence guarantees aligning with the inherent affine invariance of the BFGS method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine-Invariant Global Non-Asymptotic Convergence Analysis of BFGS under Self-Concordance Jin, Qiujiang Mokhtari, Aryan Optimization and Control Numerical Analysis In this paper, we establish global non-asymptotic convergence guarantees for the BFGS quasi-Newton method without requiring strong convexity or the Lipschitz continuity of the gradient or Hessian. Instead, we consider the setting where the objective function is strictly convex and strongly self-concordant. For an arbitrary initial point and any arbitrary positive-definite initial Hessian approximation, we prove global linear and superlinear convergence guarantees for BFGS when the step size is determined using a line search scheme satisfying the weak Wolfe conditions. Moreover, all our global guarantees are affine-invariant, with the convergence rates depending solely on the initial error and the strongly self-concordant constant. Our results extend the global non-asymptotic convergence theory of BFGS beyond traditional assumptions and, for the first time, establish affine-invariant convergence guarantees aligning with the inherent affine invariance of the BFGS method. |
| title | Affine-Invariant Global Non-Asymptotic Convergence Analysis of BFGS under Self-Concordance |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2507.00361 |