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Quasi-Newton algorithms are among the most popular iterative methods for solving unconstrained minimization problems, largely due to their favorable superlinear convergence property. However, existing results for these algorithms are limited as they provide either (i) a global convergence guarantee with an asymptotic superlinear convergence rate, or (ii) a local non-asymptotic superlinear rate for the case that the initial point and the initial Hessian approximation are chosen properly. In particular, no current analysis for quasi-Newton methods guarantees global convergence with an explicit superlinear convergence rate. In this paper, we close this gap and present the first globally convergent quasi-Newton method with an explicit non asymptotic superlinear convergence rate. Unlike classical quasi-Newton methods, we build our algorithm upon the hybrid proximal extragradient method and propose a novel online learning framework for updating the Hessian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Hessian approximation update as an online convex optimization problem in the space of matrices, and we relate the bounded regret of the online problem to the superlinear convergence of our method.
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This content will become publicly available on December 10, 2025
Stochastic Newton Proximal Extragradient Method
Stochastic second-order methods are known to achieve fast local convergence in strongly convex optimization by relying on noisy Hessian estimates to precondition the gradient. Yet, most of these methods achieve superlinear convergence only when the stochastic Hessian noise diminishes, requiring an increase in the per-iteration cost as time progresses. Recent work in \cite{na2022hessian} addressed this issue via a Hessian averaging scheme that achieves a superlinear convergence rate without increasing the per-iteration cost. However, the considered method exhibits a slow global convergence rate, requiring up to ~O(κ^2) iterations to reach the superlinear rate of ~O((1/t)^{t/2}), where κ is the problem's condition number. In this paper, we propose a novel stochastic Newton proximal extragradient method that significantly improves these bounds, achieving a faster global linear rate and reaching the same fast superlinear rate in ~O(κ) iterations. We achieve this by developing a novel extension of the Hybrid Proximal Extragradient (HPE) framework, which simultaneously achieves fast global and local convergence rates for strongly convex functions with access to a noisy Hessian oracle.
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- Award ID(s):
- 2338655
- PAR ID:
- 10582041
- Publisher / Repository:
- Advances in Neural Information Processing Systems (NeurIPS 2024)
- Date Published:
- Volume:
- 37
- Page Range / eLocation ID:
- 90818--90852
- Format(s):
- Medium: X
- Location:
- Vancouver, Canada
- Sponsoring Org:
- National Science Foundation
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