Optimal Oracle Complexity for Finite-Sum Monotone Inclusions

Abstract:We present an oracle-optimal method for finite-sum monotone inclusions under mean-square Lipschitz continuity. Our switching regularization method finds a point $y$ and a certificate $g\in G(y)$ with $(\mathbb{E}\|F(y)+g\|^2)^{1/2}\le\varepsilon$ using $\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$ expected component evaluations and resolvent evaluations. It removes the additive $n\log n$ cost of restarting a variance-reduced solver at every regularization stage by switching to a centered stochastic proximal iteration at regularization strength $L/\sqrt{n}$. Carrying an operator estimate between the remaining stages limits their total cost to $\mathcal{O}(n)$. A matching $\Omega(n+\sqrt{n}LR/\varepsilon)$ lower bound holds for randomized linear-span component-oracle algorithms with adaptive stopping and expected query budgets. Thus, for $0<\varepsilon\le LR/2$, our method attains the optimal worst-case expected component complexity in this oracle model, up to universal constants.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2610.05038 [math.OC]

(or arXiv:2610.05038v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2610.05038 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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