We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness proposed recently by [CST21].
(1) Separable minimax optimization. We study separable minimax optimization problems $\min_x \max_y f(x) - g(y) + h(x, y)$, where $f$ and $g$ have smoothness and strong convexity parameters $(L^x, μ^x)$, $(L^y, μ^y)$, and $h$ is convex-concave with a $(Λ^{xx}, Λ^{xy}, Λ^{yy})$-blockwise operator norm bounded Hessian. We provide an algorithm with gradient query complexity $\tilde{O}\left(\sqrt{\frac{L^{x}}{μ^{x}}} + \sqrt{\frac{L^{y}}{μ^{y}}} + \frac{Λ^{xx}}{μ^{x}} + \frac{Λ^{xy}}{\sqrt{μ^{x}μ^{y}}} + \frac{Λ^{yy}}{μ^{y}}\right)$. Notably, for convex-concave minimax problems with bilinear coupling (e.g.\ quadratics), where $Λ^{xx} = Λ^{yy} = 0$, our rate matches a lower bound of [ZHZ19].
(2) Finite sum optimization. We study finite sum optimization problems $\min_x \frac{1}{n}\sum_{i\in[n]} f_i(x)$, where each $f_i$ is $L_i$-smooth and the overall problem is $μ$-strongly convex. We provide an algorithm with gradient query complexity $\tilde{O}\left(n + \sum_{i\in[n]} \sqrt{\frac{L_i}{nμ}} \right)$. Notably, when the smoothness bounds $\{L_i\}_{i\in[n]}$ are non-uniform, our rate improves upon accelerated SVRG [LMH15, FGKS15] and Katyusha [All17] by up to a $\sqrt{n}$ factor.
(3) Minimax finite sums. We generalize our algorithms for minimax and finite sum optimization to solve a natural family of minimax finite sum optimization problems at an accelerated rate, encapsulating both above results up to a logarithmic factor.
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）；Beijing Key Laboratory of Big Data Management and Analysis Methods（北京市大数据管理与分析方法重点实验室）
本文研究极小极大问题的泛化界，通过均匀局部收敛获得了非凸-强凹设定下更尖锐的高概率泛化误差界，并基于此推导出比现有结果快 n 倍的超额原风险界。
Primal Methods for Constrained Variational Inequalities: Optimal Rates, Feasibility Trade-offs, and Lower Bounds
Tiangong University（天津工业大学）；University of Hertfordshire（赫特福德大学）；Sefako Makgatho Health Sciences University（塞法科·马卡托健康科学大学）；Zhejiang Normal University（浙江师范大学）；China Medical University（中国医科大学）；Academy of Romanian Scientists（罗马尼亚科学院）
Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA（赫尔曼·米勒工程与系统工程学院，佐治亚理工学院，美国亚特兰大，GA）；Sauder School of Business, University of British Columbia, Vancouver, BC, Canada（萨德勒商学院，不列颠哥伦比亚大学，加拿大温哥华，BC）
Solving stochastic weak Minty variational inequalities without increasing batch size
Laboratory for Information and Inference Systems (LIONS), EPFL（洛桑联邦理工学院信息与推理系统实验室（LIONS））；Laboratoire Traitement et Communication d’Information, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院信息处理与通信实验室）；Department of Electrical Engineering (ESAT-STADIUS), KU Leuven（荷语鲁汶大学电气工程系（ESAT-STADIUS））
Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
Last-Iterate Convergence of Single-Loop Stochastic Methods for Constrained Convex-Concave Minimax Problems
RIKEN AIP（理化学研究所先进智能计算中心）；Sauder School of Business, University of British Columbia（不列颠哥伦比亚大学 Sauder 商学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理系）
