Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity

• URL: http://arxiv.org/abs/2409.14989v1
• Date: Mon, 23 Sep 2024 13:11:37 GMT
• Title: Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity
• Authors: Eduard Gorbunov, Nazarii Tupitsa, Sayantan Choudhury, Alen Aliev, Peter Richtárik, Samuel Horváth, Martin Takáč,
• Abstract summary: 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.
• Score: 50.25258834153574
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness (Zhang et al., 2020). In this paper, we focus on the class of (strongly) convex $(L_0,L_1)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with (Smoothed) Gradient Clipping and for Gradient Descent with Polyak Stepsizes. In contrast to the existing results, our rates do not rely on the standard smoothness assumption and do not suffer from the exponential dependency from the initial distance to the solution. We also extend these results to the stochastic case under the over-parameterization assumption, propose a new accelerated method for convex $(L_0,L_1)$-smooth optimization, and derive new convergence rates for Adaptive Gradient Descent (Malitsky and Mishchenko, 2020).

Related papers

• Gradient Methods with Online Scaling [19.218484733179356]
  We introduce a framework to accelerate the convergence of gradient-based methods with online learning. We show for the first time that the widely-used hypergradient descent improves on the convergence of gradient descent.
  arXiv  Detail & Related papers  (2024-11-04T05:04:18Z)
• A Stochastic Quasi-Newton Method for Non-convex Optimization with Non-uniform Smoothness [4.097563258332958]
  We propose a fast quasi-Newton method when there exists non-uniformity in smoothness. Our algorithm can achieve the best-known $mathcalO(epsilon-3)$ sample complexity and enjoys convergence speedup. Our numerical experiments show that our proposed algorithm outperforms the state-of-the-art approaches.
  arXiv  Detail & Related papers  (2024-03-22T14:40:29Z)
• Variance-reduced Clipping for Non-convex Optimization [24.765794811146144]
  Gradient clipping is a technique used in deep learning applications such as large-scale language modeling. Recent experimental training have a fairly special behavior in that it mitigates order complexity.
  arXiv  Detail & Related papers  (2023-03-02T00:57:38Z)
• High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance [59.211456992422136]
  We propose algorithms with high-probability convergence results under less restrictive assumptions. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes in optimization.
  arXiv  Detail & Related papers  (2023-02-02T10:37:23Z)
• 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)
• Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization [50.83356836818667]
  gradient Langevin Dynamics is one of the most fundamental algorithms to solve non-eps optimization problems. In this paper, we show two variants of this kind, namely the Variance Reduced Langevin Dynamics and the Recursive Gradient Langevin Dynamics.
  arXiv  Detail & Related papers  (2022-03-30T11:39:00Z)
• Adaptive extra-gradient methods for min-max optimization and games [35.02879452114223]
  We present a new family of minmax optimization algorithms that automatically exploit the geometry of the gradient data observed at earlier iterations. Thanks to this adaptation mechanism, the proposed method automatically detects whether the problem is smooth or not. It converges to an $varepsilon$-optimal solution within $mathcalO (1/varepsilon)$ iterations in smooth problems, and within $mathcalO (1/varepsilon)$ iterations in non-smooth ones.
  arXiv  Detail & Related papers  (2020-10-22T22:54:54Z)
• Convergence of adaptive algorithms for weakly convex constrained optimization [59.36386973876765]
  We prove the $mathcaltilde O(t-1/4)$ rate of convergence for the norm of the gradient of Moreau envelope. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly smooth optimization domains.
  arXiv  Detail & Related papers  (2020-06-11T17:43:19Z)
• Adaptive First-and Zeroth-order Methods for Weakly Convex Stochastic Optimization Problems [12.010310883787911]
  We analyze a new family of adaptive subgradient methods for solving an important class of weakly convex (possibly nonsmooth) optimization problems. Experimental results indicate how the proposed algorithms empirically outperform its zerothorder gradient descent and its design variant.
  arXiv  Detail & Related papers  (2020-05-19T07:44:52Z)
• Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets [71.05306664267832]
  Adaptive algorithms perform gradient updates using the history of gradients and are ubiquitous in training deep neural networks. In this paper we analyze a variant of OptimisticOA algorithm for nonconcave minmax problems. Our experiments show that adaptive GAN non-adaptive gradient algorithms can be observed empirically.
  arXiv  Detail & Related papers  (2019-12-26T22:10:10Z)

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.