Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions

• URL: http://arxiv.org/abs/2405.18577v4
• Date: Thu, 14 Nov 2024 18:27:48 GMT
• Title: Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions
• Authors: Quanqi Hu, Qi Qi, Zhaosong Lu, Tianbao Yang,
• Abstract summary: We show a class of non-smooth non-asymptotic fairness problems in the form of $min_x[yin Yphi(x, y) - max_zin Zpsix(x, z)]$. We propose an envelope approximate gradient SMAG, the first method for solving these problems, provide a state-of-the-art non-asymptotic convergence rate.
• Score: 41.43895948769255
• License:
• Abstract: In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in Y}\phi(x, y) - \max_{z\in Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i.e., difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.

Related papers

• Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization [77.3396841985172]
  We provide a unified analysis of two-timescale gradient ascent (TTGDA) for solving structured non minimax optimization problems. Our contribution is to design TTGDA algorithms are effective beyond the setting.
  arXiv  Detail & Related papers  (2024-08-21T20:14:54Z)
• Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization [116.89941263390769]
  We consider the smooth convex-concave bilinearly-coupled saddle-point problem, $min_mathbfxmax_mathbfyF(mathbfx) + H(mathbfx,mathbfy)$, where one has access to first-order oracles for $F$, $G$ as well as the bilinear coupling function $H$. We present a emphaccelerated gradient-extragradient (AG-EG) descent-ascent algorithm that combines extragrad
  arXiv  Detail & Related papers  (2022-06-17T06:10:20Z)
• Black-Box Min--Max Continuous Optimization Using CMA-ES with Worst-case Ranking Approximation [22.576922942465142]
  A popular approach updates $x$ and $y$ simultaneously or alternatingly. Existing approaches fail if $f$ is not Lipschitz smooth and strongly convex-concave around the optimal solution. We propose minimizing the worst-case objective function $F(x)=max_yf(x,y)$ directly using the covariance matrix adaptation evolution strategy.
  arXiv  Detail & Related papers  (2022-04-06T08:03:39Z)
• Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling [84.47780064014262]
  We study a linear convex-concave saddle-point problem $min_xmax_y f(x) ytopmathbfA x - g(y)
  arXiv  Detail & Related papers  (2021-12-30T20:31:46Z)
• A first-order primal-dual method with adaptivity to local smoothness [64.62056765216386]
  We consider the problem of finding a saddle point for the convex-concave objective $min_x max_y f(x) + langle Ax, yrangle - g*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vu algorithm, which alternates between primal gradient steps and dual steps.
  arXiv  Detail & Related papers  (2021-10-28T14:19:30Z)
• Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms [0.0]
  Main results are based on a novel characterization of $epsilon$-approximate local minimum for nonsmooth functions. We show that under standard assumptions, the perturbed proximal point, perturbed proximal gradient and perturbed proximal linear algorithms find $epsilon$-approximate local minimum for nonsmooth weakly convex functions.
  arXiv  Detail & Related papers  (2021-02-04T19:17:13Z)
• The Complexity of Constrained Min-Max Optimization [29.57458485068705]
  We show that an approximate local point large enough min-max is guaranteed to exist. More importantly, we show an approximate fixed gradient Descent/Ascent approximation complete. Our result is the first to show an exponential approximation of two fundamental optimization problems.
  arXiv  Detail & Related papers  (2020-09-21T05:54:12Z)
• A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems [8.797831153231664]
  We develop an efficient algorithm for solving minimax problems with theoretical general convexnon objective guarantees. We show that the proposed algorithm can be used to solve both noncaveepsilon concave (strongly) and (strongly) convexnonconcave minimax problems.
  arXiv  Detail & Related papers  (2020-06-03T04:00:52Z)
• Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions [84.49087114959872]
  We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonsmooth functions. In particular, we study Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions.
  arXiv  Detail & Related papers  (2020-02-10T23:23:04Z)

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.