Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations

• URL: http://arxiv.org/abs/2304.10063v1
• Date: Thu, 20 Apr 2023 03:03:30 GMT
• Title: Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations
• Authors: Penghui Fu and Zhiqiang Tan
• Abstract summary: We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex functions. We establish sufficient conditions, via new discrete Lyapunov analyses, for achieving accelerated convergence rates which match Nesterov's methods in the strongly and general convex settings. We propose a novel class of discrete algorithms, called the Hamiltonian assisted gradient method, directly based on a Hamiltonian function and several interpretable operations.
• Score: 1.0152838128195465
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex or, respectively, smooth and convex functions. We establish sufficient conditions, via new discrete Lyapunov analyses, for achieving accelerated convergence rates which match Nesterov's methods in the strongly and general convex settings. Next, we study the convergence of limiting ordinary differential equations (ODEs) and point out currently notable gaps between the convergence properties of the corresponding algorithms and ODEs. Finally, we propose a novel class of discrete algorithms, called the Hamiltonian assisted gradient method, directly based on a Hamiltonian function and several interpretable operations, and then demonstrate meaningful and unified interpretations of our acceleration conditions.

Related papers

• Linearization Algorithms for Fully Composite Optimization [61.20539085730636]
  This paper studies first-order algorithms for solving fully composite optimization problems convex compact sets. We leverage the structure of the objective by handling differentiable and non-differentiable separately, linearizing only the smooth parts.
  arXiv  Detail & Related papers  (2023-02-24T18:41:48Z)
• Robust empirical risk minimization via Newton's method [9.797319790710711]
  A new variant of Newton's method for empirical risk minimization is studied. The gradient and Hessian of the objective function are replaced by robust estimators. An algorithm for obtaining robust Newton directions based on the conjugate gradient method is also proposed.
  arXiv  Detail & Related papers  (2023-01-30T18:54:54Z)
• First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems [88.58409977434269]
  We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) Our contribution is to provide two simple first-order algorithmic frameworks based on the quadratic penalty method and the augmented Lagrangian method. We provide nonasymptotic theoretical guarantees for these algorithms.
  arXiv  Detail & Related papers  (2022-04-07T00:11:05Z)
• Whiplash Gradient Descent Dynamics [2.0508733018954843]
  We introduce the symplectic convergence analysis for the Whiplash system for convex functions. We study the algorithm's performance for various costs and provide a practical methodology for analyzing convergence rates.
  arXiv  Detail & Related papers  (2022-03-04T05:47:26Z)
• Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems [2.0305676256390934]
  We characterize the finite-time performance of the continuous-time variant of simultaneous gradient descent-ascent algorithm. Our results on the behavior of continuous-time algorithm may be used to enhance the convergence properties of its discrete-time counterpart.
  arXiv  Detail & Related papers  (2021-12-17T15:51:04Z)
• Acceleration Methods [57.202881673406324]
  We first use quadratic optimization problems to introduce two key families of acceleration methods. We discuss momentum methods in detail, starting with the seminal work of Nesterov. We conclude by discussing restart schemes, a set of simple techniques for reaching nearly optimal convergence rates.
  arXiv  Detail & Related papers  (2021-01-23T17:58:25Z)
• A Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints [10.578409461429626]
  In this work, we adapt the integral quadratic constraints theory to first-order methods for smooth and strongly-varying games and iteration. We provide emphfor the first time a global convergence rate for the negative momentum method(NM) with an complexity $mathcalO(kappa1.5)$, which matches its known lower bound. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch.
  arXiv  Detail & Related papers  (2020-09-23T20:02:00Z)
• Differentially Private Accelerated Optimization Algorithms [0.7874708385247353]
  We present two classes of differentially private optimization algorithms. The first algorithm is inspired by Polyak's heavy ball method. The second class of algorithms are based on Nesterov's accelerated gradient method.
  arXiv  Detail & Related papers  (2020-08-05T08:23:01Z)
• Conditional gradient methods for stochastically constrained convex minimization [54.53786593679331]
  We propose two novel conditional gradient-based methods for solving structured convex optimization problems. The most important feature of our framework is that only a subset of the constraints is processed at each iteration. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees.
  arXiv  Detail & Related papers  (2020-07-07T21:26:35Z)
• 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)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

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.