Adaptive Gradient Methods Converge Faster with Over-Parameterization (but you should do a line-search)

• URL: http://arxiv.org/abs/2006.06835v3
• Date: Thu, 18 Feb 2021 22:44:26 GMT
• Title: Adaptive Gradient Methods Converge Faster with Over-Parameterization (but you should do a line-search)
• Authors: Sharan Vaswani, Issam Laradji, Frederik Kunstner, Si Yi Meng, Mark Schmidt, Simon Lacoste-Julien
• Abstract summary: We study a simplistic setting -- smooth, convex losses with models over- parameterized enough to interpolate the data. We prove that AMSGrad with constant step-size and momentum converges to the minimizer at a faster $O(1/T)$ rate. We show that these techniques improve the convergence and generalization of adaptive gradient methods across tasks.
• Score: 32.24244211281863
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Adaptive gradient methods are typically used for training over-parameterized models. To better understand their behaviour, we study a simplistic setting -- smooth, convex losses with models over-parameterized enough to interpolate the data. In this setting, we prove that AMSGrad with constant step-size and momentum converges to the minimizer at a faster $O(1/T)$ rate. When interpolation is only approximately satisfied, constant step-size AMSGrad converges to a neighbourhood of the solution at the same rate, while AdaGrad is robust to the violation of interpolation. However, even for simple convex problems satisfying interpolation, the empirical performance of both methods heavily depends on the step-size and requires tuning, questioning their adaptivity. We alleviate this problem by automatically determining the step-size using stochastic line-search or Polyak step-sizes. With these techniques, we prove that both AdaGrad and AMSGrad retain their convergence guarantees, without needing to know problem-dependent constants. Empirically, we demonstrate that these techniques improve the convergence and generalization of adaptive gradient methods across tasks, from binary classification with kernel mappings to multi-class classification with deep networks.

Related papers

• Adaptive Federated Learning Over the Air [108.62635460744109]
  We propose a federated version of adaptive gradient methods, particularly AdaGrad and Adam, within the framework of over-the-air model training. Our analysis shows that the AdaGrad-based training algorithm converges to a stationary point at the rate of $mathcalO( ln(T) / T 1 - frac1alpha ).
  arXiv  Detail & Related papers  (2024-03-11T09:10:37Z)
• Directional Smoothness and Gradient Methods: Convergence and Adaptivity [16.779513676120096]
  We develop new sub-optimality bounds for gradient descent that depend on the conditioning of the objective along the path of optimization. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. We prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness.
  arXiv  Detail & Related papers  (2024-03-06T22:24:05Z)
• Aiming towards the minimizers: fast convergence of SGD for overparametrized problems [25.077446336619378]
  We propose a regularity regime which endows the gradient method with the same worst-case complexity as the gradient method. All existing guarantees require the gradient method to take small steps, thereby resulting in a much slower linear rate of convergence. We demonstrate that our condition holds when training sufficiently wide feedforward neural networks with a linear output layer.
  arXiv  Detail & Related papers  (2023-06-05T05:21:01Z)
• An Adaptive Incremental Gradient Method With Support for Non-Euclidean Norms [19.41328109094503]
  We propose and analyze several novel adaptive variants of the popular SAGA algorithm. We establish its convergence guarantees under general settings. We improve the analysis of SAGA to support non-Euclidean norms.
  arXiv  Detail & Related papers  (2022-04-28T09:43:07Z)
• The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance [46.15915820243487]
  We show that AdaGrad-Norm exhibits an order optimal convergence of $mathcalOleft. We show that AdaGrad-Norm exhibits an order optimal convergence of $mathcalOleft.
  arXiv  Detail & Related papers  (2022-02-11T17:37:54Z)
• STORM+: Fully Adaptive SGD with Momentum for Nonconvex Optimization [74.1615979057429]
  We investigate non-batch optimization problems where the objective is an expectation over smooth loss functions. Our work builds on the STORM algorithm, in conjunction with a novel approach to adaptively set the learning rate and momentum parameters.
  arXiv  Detail & Related papers  (2021-11-01T15:43:36Z)
• Minibatch vs Local SGD with Shuffling: Tight Convergence Bounds and Beyond [63.59034509960994]
  We study shuffling-based variants: minibatch and local Random Reshuffling, which draw gradients without replacement. For smooth functions satisfying the Polyak-Lojasiewicz condition, we obtain convergence bounds which show that these shuffling-based variants converge faster than their with-replacement counterparts. We propose an algorithmic modification called synchronized shuffling that leads to convergence rates faster than our lower bounds in near-homogeneous settings.
  arXiv  Detail & Related papers  (2021-10-20T02:25:25Z)
• Adaptive Gradient Methods for Constrained Convex Optimization and Variational Inequalities [32.51470158863247]
  Main algorithms AdaACSA and AdaAGD+ are accelerated methods for constrained convex optimization. We complement them with a simpler algorithm AdaGrad+ which enjoys the same features, and achieves the standard non-accelerated convergence rate.
  arXiv  Detail & Related papers  (2020-07-17T09:10:21Z)
• 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)
• Balancing Rates and Variance via Adaptive Batch-Size for Stochastic Optimization Problems [120.21685755278509]
  In this work, we seek to balance the fact that attenuating step-size is required for exact convergence with the fact that constant step-size learns faster in time up to an error. Rather than fixing the minibatch the step-size at the outset, we propose to allow parameters to evolve adaptively.
  arXiv  Detail & Related papers  (2020-07-02T16:02:02Z)
• On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization [80.03647903934723]
  We prove adaptive gradient methods in expectation of gradient convergence methods. Our analyses shed light on better adaptive gradient methods in optimizing non understanding gradient bounds.
  arXiv  Detail & Related papers  (2018-08-16T20:25:28Z)

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.