What does this research mean for the field?

An online learning-guided quasi-Newton method achieves global non-asymptotic convergence for smooth and monotone nonlinear equations, provably surpassing the convergence rate of the extragradient method without querying the Jacobian. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish a quasi-Newton optimization method for smooth, monotone nonlinear equations that achieves provable global non-asymptotic convergence without querying the Jacobian matrix.

August 22, 2026Mathematical ProgrammingOpen Access

Online learning guided quasi-Newton methods with global non-asymptotic convergence

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Authors

RJRuichen JiangBeijing University of Chinese Medicine AMAryan MokhtariGoogle (United States)

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Implication

Theoretical analysis demonstrates global non-asymptotic convergence in quasi-Newton optimization, revealing faster convergence than standard extragradient methods without Jacobian queries.

Key Points

To establish a quasi-Newton optimization method for smooth, monotone nonlinear equations that achieves provable global non-asymptotic convergence without querying the Jacobian matrix.
Combined the hybrid proximal extragradient framework with an online convex optimization formulation over non-symmetric matrices to guide Jacobian approximation updates.
Formulated a specialized online learning algorithm using an approximate separation oracle to preserve structural properties like symmetry and sparsity in Jacobian matrices.
Achieved a linear convergence rate and an explicit global superlinear rate surpassing linear convergence within O(d) iterations for strongly monotone settings.
Established a global convergence rate of O(min{1/k, sqrt(d)/k^1.25}) for the duality gap in monotone settings, provably exceeding the extragradient method when k = Omega(d^2).

Cite This Study

Jiang et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c7ef https://doi.org/10.1007/s10107-026-02396-4

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August 22, 2026Mathematical ProgrammingOpen Access

Online learning guided quasi-Newton methods with global non-asymptotic convergence

View Full Paper

Ask AI

Bookmark

Authors

RJRuichen JiangBeijing University of Chinese Medicine AMAryan MokhtariGoogle (United States)

Discussion

Member takes

Implication

Theoretical analysis demonstrates global non-asymptotic convergence in quasi-Newton optimization, revealing faster convergence than standard extragradient methods without Jacobian queries.

Key Points

To establish a quasi-Newton optimization method for smooth, monotone nonlinear equations that achieves provable global non-asymptotic convergence without querying the Jacobian matrix.
Combined the hybrid proximal extragradient framework with an online convex optimization formulation over non-symmetric matrices to guide Jacobian approximation updates.
Formulated a specialized online learning algorithm using an approximate separation oracle to preserve structural properties like symmetry and sparsity in Jacobian matrices.
Achieved a linear convergence rate and an explicit global superlinear rate surpassing linear convergence within O(d) iterations for strongly monotone settings.
Established a global convergence rate of O(min{1/k, sqrt(d)/k^1.25}) for the duality gap in monotone settings, provably exceeding the extragradient method when k = Omega(d^2).

Cite This Study

Jiang et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c7ef https://doi.org/10.1007/s10107-026-02396-4

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