Non-Convergence and Limit Cycles in the Adam optimizer

• URL: http://arxiv.org/abs/2210.02070v1
• Date: Wed, 5 Oct 2022 07:44:33 GMT
• Title: Non-Convergence and Limit Cycles in the Adam optimizer
• Authors: Sebastian Bock and Martin Georg Wei{\ss}
• Abstract summary: We show that limit cycles of period 2 exist in batch mode for simple quadratic objective functions. We analyze the stability of these limit cycles and relate our analysis to other results where approximate convergence was shown.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: One of the most popular training algorithms for deep neural networks is the Adaptive Moment Estimation (Adam) introduced by Kingma and Ba. Despite its success in many applications there is no satisfactory convergence analysis: only local convergence can be shown for batch mode under some restrictions on the hyperparameters, counterexamples exist for incremental mode. Recent results show that for simple quadratic objective functions limit cycles of period 2 exist in batch mode, but only for atypical hyperparameters, and only for the algorithm without bias correction. %More general there are several more adaptive gradient methods which try to estimate a fitting learning rate and / or search direction from the training data to improve the learning process compared to pure gradient descent with fixed learningrate. We extend the convergence analysis for Adam in the batch mode with bias correction and show that even for quadratic objective functions as the simplest case of convex functions 2-limit-cycles exist, for all choices of the hyperparameters. We analyze the stability of these limit cycles and relate our analysis to other results where approximate convergence was shown, but under the additional assumption of bounded gradients which does not apply to quadratic functions. The investigation heavily relies on the use of computer algebra due to the complexity of the equations.

Related papers

• From Gradient Clipping to Normalization for Heavy Tailed SGD [19.369399536643773]
  Recent empirical evidence indicates that machine learning applications involve heavy-tailed noise, which challenges the standard assumptions of bounded variance in practice. In this paper, we show that it is possible to achieve tightness of the gradient-dependent noise convergence problem under tailed noise.
  arXiv  Detail & Related papers  (2024-10-17T17:59:01Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• 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)
• Unified Convergence Analysis for Adaptive Optimization with Moving Average Estimator [75.05106948314956]
  We show that an increasing large momentum parameter for the first-order moment is sufficient for adaptive scaling. We also give insights for increasing the momentum in a stagewise manner in accordance with stagewise decreasing step size.
  arXiv  Detail & Related papers  (2021-04-30T08:50:24Z)
• 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)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Path Sample-Analytic Gradient Estimators for Stochastic Binary Networks [78.76880041670904]
  In neural networks with binary activations and or binary weights the training by gradient descent is complicated. We propose a new method for this estimation problem combining sampling and analytic approximation steps. We experimentally show higher accuracy in gradient estimation and demonstrate a more stable and better performing training in deep convolutional models.
  arXiv  Detail & Related papers  (2020-06-04T21:51:21Z)
• Stochastic Proximal Gradient Algorithm with Minibatches. Application to Large Scale Learning Models [2.384873896423002]
  We develop and analyze minibatch variants of gradient algorithm for general composite objective functions with nonsmooth components. We provide complexity for constant and variable stepsize iteration policies obtaining that, for minibatch size $N$, after $mathcalO(frac1Nepsilon)$ $epsilon-$subity is attained in expected quadratic distance to optimal solution.
  arXiv  Detail & Related papers  (2020-03-30T10:43:56Z)
• 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)
• Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem [27.09339991866556]
  We show that ODE searches for optimal control for an unknown computation system by directly searching over the corresponding space of controllers. We take a step towards demystifying the performance and efficiency of such methods by focusing on the gradient-flow dynamics set of stabilizing feedback gains and a similar result holds for the forward disctization of the ODE.
  arXiv  Detail & Related papers  (2019-12-26T16:56:59Z)

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.