Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods

• URL: http://arxiv.org/abs/2202.04640v1
• Date: Wed, 9 Feb 2022 18:57:47 GMT
• Title: Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods
• Authors: Yujia Jin, Aaron Sidford, Kevin Tian
• Abstract summary: 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 $(Lx, mux)$, $(Ly, muy)$, and $h$ is convex-concave with a $(Lambdaxx, Lambdaxy, Lambdayymuyright)$.
• Score: 39.87865197769047
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $(L^x, \mu^x)$, $(L^y, \mu^y)$, and $h$ is convex-concave with a $(\Lambda^{xx}, \Lambda^{xy}, \Lambda^{yy})$-blockwise operator norm bounded Hessian. We provide an algorithm with gradient query complexity $\tilde{O}\left(\sqrt{\frac{L^{x}}{\mu^{x}}} + \sqrt{\frac{L^{y}}{\mu^{y}}} + \frac{\Lambda^{xx}}{\mu^{x}} + \frac{\Lambda^{xy}}{\sqrt{\mu^{x}\mu^{y}}} + \frac{\Lambda^{yy}}{\mu^{y}}\right)$. Notably, for convex-concave minimax problems with bilinear coupling (e.g.\ quadratics), where $\Lambda^{xx} = \Lambda^{yy} = 0$, 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 $\mu$-strongly convex. We provide an algorithm with gradient query complexity $\tilde{O}\left(n + \sum_{i\in[n]} \sqrt{\frac{L_i}{n\mu}} \right)$. 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.

Related papers

• Dueling Optimization with a Monotone Adversary [35.850072415395395]
  We study the problem of dueling optimization with a monotone adversary, which is a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer $mathbfx*$ for a function $fcolon X to mathbbRd.
  arXiv  Detail & Related papers  (2023-11-18T23:55:59Z)
• An Optimal Algorithm for Strongly Convex Min-min Optimization [79.11017157526815]
  Existing optimal first-order methods require $mathcalO(sqrtmaxkappa_x,kappa_y log 1/epsilon)$ of computations of both $nabla_x f(x,y)$ and $nabla_y f(x,y)$. We propose a new algorithm that only requires $mathcalO(sqrtkappa_x log 1/epsilon)$ of computations of $nabla_x f(x,
  arXiv  Detail & Related papers  (2022-12-29T19:26:12Z)
• Optimal Gradient Sliding and its Application to Distributed Optimization Under Similarity [121.83085611327654]
  We structured convex optimization problems with additive objective $r:=p + q$, where $r$ is $mu$-strong convex similarity. We proposed a method to solve problems master to agents' communication and local calls. The proposed method is much sharper than the $mathcalO(sqrtL_q/mu)$ method.
  arXiv  Detail & Related papers  (2022-05-30T14:28:02Z)
• The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization [88.91190483500932]
  In this paper, we revisit the smooth and strongly-strongly-concave minimax optimization problem. Existing state-of-the-art methods do not match lower bound $Omegaleft(sqrtkappa_xkappa_ylog3 (kappa_xkappa_y)logfrac1epsilonright)$. We fix this fundamental issue by providing the first algorithm with $mathcalOleft( sqrtkappa_xkappa_ylog
  arXiv  Detail & Related papers  (2022-05-11T17:33:07Z)
• Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization [47.27237492375659]
  We study the bilinearly coupled minimax problem: $min_x max_y f(x) + ytop A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions. No known first-order algorithms have hitherto achieved the lower complexity bound of $Omega(sqrtfracL_xmu_x + frac|A|sqrtmu_x,mu_y) log(frac1vareps
  arXiv  Detail & Related papers  (2022-01-19T05:56:19Z)
• Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization [41.432757205864796]
  This paper considers first-order algorithms for convex-con minimax problems of the form $min_bf xmax_yf(bfbf y) simultaneously. Our methods can be used to solve more general unbalanced minimax problems and are also near optimal.
  arXiv  Detail & Related papers  (2021-06-03T11:30:32Z)
• DIPPA: An improved Method for Bilinear Saddle Point Problems [18.65143269806133]
  This paper deals with point dependency problems $min_bfx max_bfy g(fracx) + bfxtop bfbftop fracbfA kappa_x kappa_x (kappa_x + kappa_y) kappa_y (kappa_x + kappa_y) kappa_y (kappa_x + kappa_y)
  arXiv  Detail & Related papers  (2021-03-15T10:55:30Z)
• Improved Algorithms for Convex-Concave Minimax Optimization [10.28639483137346]
  This paper studies minimax optimization problems $min_x max_y f(x,y)$, where $f(x,y)$ is $m_x$-strongly convex with respect to $x$, $m_y$-strongly concave with respect to $y$ and $(L_x,L_xy,L_y)$-smooth.
  arXiv  Detail & Related papers  (2020-06-11T12:21:13Z)
• Revisiting EXTRA for Smooth Distributed Optimization [70.65867695317633]
  We give a sharp complexity analysis for EXTRA with the improved $Oleft(left(fracLmu+frac11-sigma_2(W)right)logfrac1epsilon (1-sigma_2(W))right)$. Our communication complexities of the accelerated EXTRA are only worse by the factors of $left(logfracLmu (1-sigma_2(W))right)$ and $left(logfrac1epsilon (1-
  arXiv  Detail & Related papers  (2020-02-24T08:07:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.