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Primal-Dual Adaptive Subgradient Methods

Karthik Prakhya, Zimeng Wang, Alp Yurtsever · 2026 · arXiv

DOI: 10.48550/arXiv.2610.28300 · open-access PDF

Korean

우리의 첫 번째 방법은 이중 문제에 AdaGrad 스타일의 단계 크기를 적용하고, 연관된 Fenchel-type 오라클 출력을 평균화함으로써 원시 솔루션을 복구합니다. 이 방법은 $\widetilde{\mathcal{O}}(T^{-1/2})$ 회로를 달성하며, 이중 렌즈가 부드럽을 때 자동으로 $\mathcal{O}(T^^-1})$로 향상합니다. 우리는 더 가속화된 변형을 개발하여 $\widetilde{mathO}}(T^^-1/2})$를 유지합니다.

Machine translation (Meta M2M100); the English original follows.

Abstract

We propose adaptive primal-dual subgradient methods for constrained convex optimization. Our first method applies an AdaGrad-style step-sizes to the dual problem and recovers primal solutions by averaging the associated Fenchel-type oracle outputs. It achieves $\widetilde{\mathcal{O}}(T^{-1/2})$ convergence for the dual objective residual, primal objective residual, and feasibility gap in the nonsmooth regime, while automatically improving to $\mathcal{O}(T^{-1})$ when the dual objective is smooth. We further develop an accelerated variant, which retains the $\widetilde{\mathcal{O}}(T^{-1/2})$

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