Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization

• URL: http://arxiv.org/abs/2306.02212v1
• Date: Sat, 3 Jun 2023 23:31:27 GMT
• Title: Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization
• Authors: Ruichen Jiang and Aryan Mokhtari
• Abstract summary: We prove that our method can achieve a convergence rate of $Obigl(minfrac1k2, fracsqrtdlog kk2.5bigr)$, where $d$ is the problem dimension and $k$ is the number of iterations. To the best of our knowledge, this result is the first to demonstrate a provable gain of a quasi-Newton-type method over Nesterov's accelerated gradient.
• Score: 26.328847475942894
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we propose an accelerated quasi-Newton proximal extragradient (A-QPNE) method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective, we prove that our method can achieve a convergence rate of ${O}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr)$, where $d$ is the problem dimension and $k$ is the number of iterations. In particular, in the regime where $k = {O}(d)$, our method matches the optimal rate of ${O}(\frac{1}{k^2})$ by Nesterov's accelerated gradient (NAG). Moreover, in the the regime where $k = \Omega(d \log d)$, it outperforms NAG and converges at a faster rate of ${O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr)$. To the best of our knowledge, this result is the first to demonstrate a provable gain of a quasi-Newton-type method over NAG in the convex setting. To achieve such results, we build our method on a recent variant of the Monteiro-Svaiter acceleration framework and adopt an online learning perspective to update the Hessian approximation matrices, in which we relate the convergence rate of our method to the dynamic regret of a specific online convex optimization problem in the space of matrices.

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