Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance

• URL: http://arxiv.org/abs/2404.01436v2
• Date: Wed, 3 Apr 2024 21:08:40 GMT
• Title: Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance
• Authors: Qi Zhang, Yi Zhou, Shaofeng Zou,
• Abstract summary: We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. We develop a new upper bound first-order term in the descent lemma, which is also a function of the gradient norm. Our results for both RMSProp and Adam match with the complexity established in citearvani2023lower.
• Score: 23.112775335244258
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an $\epsilon$-stationary point with an iteration complexity of $\mathcal O(\epsilon^{-4})$. We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an $\epsilon$-stationary point with an iteration complexity of $\mathcal O(\epsilon^{-4})$. Our complexity results for both RMSProp and Adam match with the complexity lower bound established in \cite{arjevani2023lower}.

Related papers

• Convergence Rate Analysis of LION [54.28350823319057]
  LION converges iterations of $cal(sqrtdK-)$ measured by gradient Karush-Kuhn-T (sqrtdK-)$. We show that LION can achieve lower loss and higher performance compared to standard SGD.
  arXiv  Detail & Related papers  (2024-11-12T11:30:53Z)
• High Probability Convergence of Adam Under Unbounded Gradients and Affine Variance Noise [4.9495085874952895]
  We show that Adam could converge to the stationary point in high probability with a rate of $mathcalOleft(rm poly(log T)/sqrtTright)$ under coordinate-wise "affine" noise variance. It is also revealed that Adam's confines within an order of $mathcalOleft(rm poly(left T)right)$ are adaptive to the noise level.
  arXiv  Detail & Related papers  (2023-11-03T15:55:53Z)
• Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity [51.96093077151991]
  We derive a new convergence guarantee of Adam, with only an $L$-smooth condition and a bounded noise variance assumption. Our proof utilizes novel techniques to handle the entanglement between momentum and adaptive learning rate.
  arXiv  Detail & Related papers  (2023-10-27T09:16:58Z)
• Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case [0.0]
  This paper focuses on exploring the convergence of vanilla Adam and the challenges of non-ergodic convergence. These findings build a solid theoretical foundation for Adam to solve non-godic optimization problems.
  arXiv  Detail & Related papers  (2023-07-20T12:02:17Z)
• Convergence of Adam Under Relaxed Assumptions [72.24779199744954]
  We show that Adam converges to $epsilon$-stationary points with $O(epsilon-4)$ gradient complexity under far more realistic conditions. We also propose a variance-reduced version of Adam with an accelerated gradient complexity of $O(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-04-27T06:27:37Z)
• Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization [11.017632675093628]
  Coordinate-type subgradient methods for addressing nonsmooth problems are relatively underexplored due to the set of properties of the Lipschitz-type assumption.
  arXiv  Detail & Related papers  (2022-06-30T02:17:11Z)
• 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)
• A Simple Convergence Proof of Adam and Adagrad [74.24716715922759]
  We show a proof of convergence between the Adam Adagrad and $O(d(N)/st)$ algorithms. Adam converges with the same convergence $O(d(N)/st)$ when used with the default parameters.
  arXiv  Detail & Related papers  (2020-03-05T01:56:17Z)
• 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.