Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.10495

An Optimal Second-Order Algorithm for Finite-Sum Optimization under Average Smoothness and Convexity

Artem Tsedenov·Dmitry Kovalev
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we consider a finite-sum optimization problem, where the objective function is convex, the $n$ component functions are twice continuously differentiable and their Hessians are mean-square Lipschitz. Our contribution is twofold. First, we develop a stochastic algorithm, which requires $\tilde{\mathcal{O}}(n + n^{6/7}/ε^{2/7})$ second-order oracle calls in expectation to find an approximate solution to the problem with the expected accuracy $ε$. Second, we establish a lower complexity bound of $Ω(n + n^{6/7}/ε^{2/7})$ for any stochastic second-order algorithm satisfying a stochastic second-order version of the first-order linear span assumption, which is widely adopted in the optimization literature. Our lower and upper complexity bounds match up to logarithmic factors and thus resolve an important open question of establishing the optimal complexity of this problem class.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Artem Tsedenov, Dmitry Kovalev. 2026-10-07. An Optimal Second-Order Algorithm for Finite-Sum Optimization under Average Smoothness and Convexity. https://arxiv.org/abs/2610.10495

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Combining additivity and active subspaces for high-dimensional Gaussian process modeling

Gaussian processes are a widely embraced technique for regression due to their good prediction accuracy, analytical tractability, and built-in capabilities for uncertainty quantification. However, they suffer from the curse of dimensionality whenever the number of variables increases. This challenge is generally addressed by assuming additional structure in the problem, the preferred options being either additivity or low intrinsic dimensionality. After discussing their relative merits, our contribution for high-dimensional Gaussian process modeling is to combine them with a multi-fidelity strategy. We detail the corresponding construction and showcase the advantages through experiments on synthetic functions and datasets.

math.OC↗
arXiv2026

Constrained portfolio game with heterogeneous agents

We investigate stochastic utility maximization games under relative performance concerns in both finite-agent and infinite-agent (graphon) settings. An incomplete market model is considered where agents with power (CRRA) utility functions trade in a common risk-free bond and individual stocks driven by both common and idiosyncratic noise. The Nash equilibrium for both settings is characterized by forward-backward stochastic differential equations (FBSDEs) with a quadratic growth generator, where the solution of the graphon game leads to a novel form of infinite-dimensional McKean-Vlasov FBSDEs. Under mild conditions, we prove the existence of Nash equilibrium for both the graphon game and the $n$-agent game without common noise. Furthermore, we establish a convergence result showing that, with modest assumptions on the sensitivity matrix, as the number of agents increases, the Nash equilibrium and associated equilibrium value of the finite-agent game converge to those of the graphon game.

math.OC↗
arXiv2026

Controllability of Forward Stochastic Reaction--Convection--Diffusion Systems with Cascade Structure

We investigate the null and approximate controllability of coupled linear forward stochastic reaction--convection--diffusion systems under suitable cascade coupling conditions. The model consists of two forward stochastic parabolic equations governed by general second-order differential operators with time-, space-, and random-dependent coefficients. We consider a localized control acting on the drift term of the first equation, together with controls acting on the diffusion terms. By means of a duality argument, the controllability problem is reduced to an observability problem for the associated adjoint backward stochastic parabolic system. We establish a new global Carleman estimate for coupled backward stochastic parabolic systems whose drift terms belong to a negative Sobolev space. As an application of this estimate, we derive the required observability properties and establish the null and approximate controllability of the original forward system.

math.OC↗