Improved Gradient Complexity for Quasi-Newton Proximal Extragradient Methods via Certified Curvature Reuse

Abstract

We study how curvature estimated from gradient differences can accelerate proximal-extragradient quasi-Newton methods. Existing global guarantees for QNPE, A-QNPE, and OL-OQN learn Hessian approximations online and incur dimension dependence through regret bounds. We introduce certified curvature reuse: one finite-difference probe per iteration refreshes a group of Hessian columns, while a path-length certificate controls the error of stale columns and determines the step. Each iteration uses at most three gradient queries and one regularized linear solve (or trust-region step), without a gradient line search or an inner oracle-querying loop. In the convex and strongly convex settings, bisection on stored scalar quantities selects a near-maximal certified stepsize; in the nonconvex setting, a uniform certificate gives an explicit learning rate. With other problem parameters fixed, the convex method attains $O(\min\{k^{-2},dk^{-3}\})$ in a single run, with no logarithmic factor or additive dimension term, matching accelerated gradient descent (AGD) for $k\lesssim d$ and improving on A-QNPE thereafter. In the strongly convex setting with condition number $κ$, a single run matches AGD's $(1-Θ(κ^{-1/2}))^k$ rate for $k\lesssim d\logκ$ and achieves a superlinear upper envelope, eventually scaling as $(1+k^2/B^2)^{-k}$ with $B=\tilde O(dκ)$, improving on QNPE. In the nonconvex setting, the method finds an $\varepsilon$-stationary point in $O(\min\{d^{1/3}\varepsilon^{-3/2},\varepsilon^{-7/4}\})$ gradient queries, improving on OL-OQN's $O(d^{1/4}\varepsilon^{-13/8})$. For sparse Hessians, $d$ is replaced by an effective dimension $χ\le d$. We also compare our bounds with lazy-Hessian, double-loop, and cutting-plane methods.