Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization

• URL: http://arxiv.org/abs/2404.09438v1
• Date: Mon, 15 Apr 2024 03:50:47 GMT
• Title: Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization
• Authors: Nachuan Xiao, Kuangyu Ding, Xiaoyin Hu, Kim-Chuan Toh,
• Abstract summary: We develop a framework for developing Lagrangian-based methods. We show that our framework can be extended to solve constrained problems with expectation constraints.
• Score: 2.7998963147546148
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we consider the minimization of a nonsmooth nonconvex objective function $f(x)$ over a closed convex subset $\mathcal{X}$ of $\mathbb{R}^n$, with additional nonsmooth nonconvex constraints $c(x) = 0$. We develop a unified framework for developing Lagrangian-based methods, which takes a single-step update to the primal variables by some subgradient methods in each iteration. These subgradient methods are ``embedded'' into our framework, in the sense that they are incorporated as black-box updates to the primal variables. We prove that our proposed framework inherits the global convergence guarantees from these embedded subgradient methods under mild conditions. In addition, we show that our framework can be extended to solve constrained optimization problems with expectation constraints. Based on the proposed framework, we show that a wide range of existing stochastic subgradient methods, including the proximal SGD, proximal momentum SGD, and proximal ADAM, can be embedded into Lagrangian-based methods. Preliminary numerical experiments on deep learning tasks illustrate that our proposed framework yields efficient variants of Lagrangian-based methods with convergence guarantees for nonconvex nonsmooth constrained optimization problems.

Related papers

• Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity [50.25258834153574]
  We focus on the class of (strongly) convex $(L0)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with smoothnessed Gradient Clipping and for Gradient Descent with Polyak Stepsizes.
  arXiv  Detail & Related papers  (2024-09-23T13:11:37Z)
• Strictly Low Rank Constraint Optimization -- An Asymptotically $\mathcal{O}(\frac{1}{t^2})$ Method [5.770309971945476]
  We propose a class of non-text and non-smooth problems with textitrank regularization to promote sparsity in optimal solution. We show that our algorithms are able to achieve a singular convergence of $Ofrac(t2)$, which is exactly same as Nesterov's optimal convergence for first-order methods on smooth convex problems.
  arXiv  Detail & Related papers  (2023-07-04T16:55:41Z)
• Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching [55.28394191394675]
  We develop an adaptive inexact Newton method for equality-constrained nonlinear, nonIBS optimization problems. We demonstrate the superior performance of our method on benchmark nonlinear problems, constrained logistic regression with data from LVM, and a PDE-constrained problem.
  arXiv  Detail & Related papers  (2023-05-28T06:33:37Z)
• Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization [88.0031283949404]
  Many real-world problems have complicated non functional constraints and use a large number of data points. Our proposed method outperforms an existing method with the previously best-known result.
  arXiv  Detail & Related papers  (2022-12-19T14:48:54Z)
• Faster Projection-Free Augmented Lagrangian Methods via Weak Proximal Oracle [16.290192687098383]
  This paper considers a convex composite optimization problem with affine constraints. Motivated by high-dimensional applications in which exact projection/proximal computations are not tractable, we propose a textitprojection-free augmented Lagrangian-based method.
  arXiv  Detail & Related papers  (2022-10-25T12:51:43Z)
• Multiblock ADMM for nonsmooth nonconvex optimization with nonlinear coupling constraints [3.2815423235774634]
  We propose a method of multipliers for solving a class of multiblock nonsmooth alternating optimization problem with nonlinear constraints. We employ a major sequenceization procedure in update of each block of the primal variables.
  arXiv  Detail & Related papers  (2022-01-19T15:31:30Z)
• Faster Algorithm and Sharper Analysis for Constrained Markov Decision Process [56.55075925645864]
  The problem of constrained decision process (CMDP) is investigated, where an agent aims to maximize the expected accumulated discounted reward subject to multiple constraints. A new utilities-dual convex approach is proposed with novel integration of three ingredients: regularized policy, dual regularizer, and Nesterov's gradient descent dual. This is the first demonstration that nonconcave CMDP problems can attain the lower bound of $mathcal O (1/epsilon)$ for all complexity optimization subject to convex constraints.
  arXiv  Detail & Related papers  (2021-10-20T02:57:21Z)
• A Stochastic Composite Augmented Lagrangian Method For Reinforcement Learning [9.204659134755795]
  We consider the linear programming (LP) formulation for deep reinforcement learning. The augmented Lagrangian method suffers the double-sampling obstacle in solving the LP. A deep parameterized augment Lagrangian method is proposed.
  arXiv  Detail & Related papers  (2021-05-20T13:08:06Z)
• Softmax Policy Gradient Methods Can Take Exponential Time to Converge [60.98700344526674]
  The softmax policy gradient (PG) method is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. We demonstrate that softmax PG methods can take exponential time -- in terms of $mathcalS|$ and $frac11-gamma$ -- to converge.
  arXiv  Detail & Related papers  (2021-02-22T18:56:26Z)
• IDEAL: Inexact DEcentralized Accelerated Augmented Lagrangian Method [64.15649345392822]
  We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds.
  arXiv  Detail & Related papers  (2020-06-11T18:49:06Z)

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.