This paper shows that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and establishes matching oracle complexity lower bounds for both setups.
Abstract
In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $\delta$-inexact Hessians. We show that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_3/\epsilon)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.
We study the last iterate of the projected subGradient Method (sGM) for convex Lipschitz objectives defined on $\mathbb{R}^d$. We prove that, for a finite horizon $n$ and a constant stepsize $\eta=\Theta(1/\sqrt n)$, the last iterate achieves an optimization error of order $d/\sqrt n$, showing that the extra $\log n$ f...
A variant of the cubic-regularized Newton method for nonconvex optimization that is parameter-free in that it requires no prior knowledge of problem-dependent parameters is analyzed, and an oracle complexity bound is derived for finding an $(varepsilon, \delta)-second-order stationary point.
By using a large-step inexact Halpern iteration, a novel Halpern-NPE method is proposed that achieves an even faster rate of $\tilde{\mathcal{O}}(T^{-2})$ for solving MVIs and improves all prior results for $p \ge 2$ and matches the classical extragradient method for p=1.
Le-Si Chen, Xin-Liang Zhang, He Wang et al.· 0 citations
These results are the first global convergence results to demonstrate a provable advantage of a quasi-Newton method over the extragradient method, without querying the Jacobian of the operator.
Three inexact AL schemes are developed that preserve the standard AL subproblem structure and attain the optimal primal-dual complexity in the convex setting, improving prior AL bounds of $\mathcal O(\epsilon^{-4/3})$, $\mathcal O(\epsilon^{-7/4})$, and $\mathcal O(\epsilon^{-2})$, and removing the logarithmic factor f...
Arnesh Sujanani, Saeed Ghadimi, Henry Wolkowicz· 0 citations
The anytime lower bound shows that the rate of non-anytime silver schedules (Altschuler and Parrilo, 2025; Grimmer et al., 2025) is unattainable in the anytime setting, and establishes a strict separation between the two settings.
Yu-Tian Ye, Kai-Zhao Liu· 6 citations
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