Stochastic Polyak Step-sizes and Momentum: Convergence Guarantees and Practical Performance

• URL: http://arxiv.org/abs/2406.04142v1
• Date: Thu, 6 Jun 2024 15:08:06 GMT
• Title: Stochastic Polyak Step-sizes and Momentum: Convergence Guarantees and Practical Performance
• Authors: Dimitris Oikonomou, Nicolas Loizou,
• Abstract summary: We propose and explore new Polyak-type variants suitable for the update rule of the Heavy Ball method (SHB) For MomSPS$_max$, we provide guarantees for SHB to a neighborhood of the solution for convex and smooth problems (without assuming) The other two variants, MomDecSPS and MomAdaSPS, are the first adaptive step-sizes for SHB that guarantee convergence to the exact minimizer without prior knowledge.
• Score: 10.11126899274029
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic gradient descent with momentum, also known as Stochastic Heavy Ball method (SHB), is one of the most popular algorithms for solving large-scale stochastic optimization problems in various machine learning tasks. In practical scenarios, tuning the step-size and momentum parameters of the method is a prohibitively expensive and time-consuming process. In this work, inspired by the recent advantages of stochastic Polyak step-size in the performance of stochastic gradient descent (SGD), we propose and explore new Polyak-type variants suitable for the update rule of the SHB method. In particular, using the Iterate Moving Average (IMA) viewpoint of SHB, we propose and analyze three novel step-size selections: MomSPS$_{\max}$, MomDecSPS, and MomAdaSPS. For MomSPS$_{\max}$, we provide convergence guarantees for SHB to a neighborhood of the solution for convex and smooth problems (without assuming interpolation). If interpolation is also satisfied, then using MomSPS$_{\max}$, SHB converges to the true solution at a fast rate matching the deterministic HB. The other two variants, MomDecSPS and MomAdaSPS, are the first adaptive step-sizes for SHB that guarantee convergence to the exact minimizer without prior knowledge of the problem parameters and without assuming interpolation. The convergence analysis of SHB is tight and obtains the convergence guarantees of SGD with stochastic Polyak step-sizes as a special case. We supplement our analysis with experiments that validate the theory and demonstrate the effectiveness and robustness of the new algorithms.

Related papers

• A Stochastic Approach to Bi-Level Optimization for Hyperparameter Optimization and Meta Learning [74.80956524812714]
  We tackle the general differentiable meta learning problem that is ubiquitous in modern deep learning. These problems are often formalized as Bi-Level optimizations (BLO) We introduce a novel perspective by turning a given BLO problem into a ii optimization, where the inner loss function becomes a smooth distribution, and the outer loss becomes an expected loss over the inner distribution.
  arXiv  Detail & Related papers  (2024-10-14T12:10:06Z)
• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• Adaptive SGD with Polyak stepsize and Line-search: Robust Convergence and Variance Reduction [26.9632099249085]
  We propose two new variants of SPS and SLS, called AdaSPS and AdaSLS, which guarantee convergence in non-interpolation settings. We equip AdaSPS and AdaSLS with a novel variance reduction technique and obtain algorithms that require $smashwidetildemathcalO(n+1/epsilon)$ gradient evaluations.
  arXiv  Detail & Related papers  (2023-08-11T10:17:29Z)
• Sharper Analysis for Minibatch Stochastic Proximal Point Methods: Stability, Smoothness, and Deviation [41.082982732100696]
  We study a minibatch variant of proximal point (SPP) methods, namely M-SPP, for solving convex composite risk minimization problems. We show that M-SPP with minibatch-size $n$ and quadratic count $T$ enjoys an in-expectation fast rate of convergence. In the small-$n$-large-$T$ setting, this result substantially improves the best known results of SPP-type approaches.
  arXiv  Detail & Related papers  (2023-01-09T00:13:34Z)
• Formal guarantees for heuristic optimization algorithms used in machine learning [6.978625807687497]
  Gradient Descent (SGD) and its variants have become the dominant methods in the large-scale optimization machine learning (ML) problems. We provide formal guarantees of a few convex optimization methods and proposing improved algorithms.
  arXiv  Detail & Related papers  (2022-07-31T19:41:22Z)
• 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)
• Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize [20.376216873620763]
  We investigate the convergence of mirror descent (SMD) under in relatively smooth and smooth convex optimization. We propose a new adaptive stepsize scheme -- the mirror Polyak stepsize (mSPS)
  arXiv  Detail & Related papers  (2021-10-28T19:49:40Z)
• 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)
• Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity [49.66890309455787]
  We introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size. Our convergence guarantees hold under the arbitrary sampling paradigm, and we give insights into the complexity of minibatching.
  arXiv  Detail & Related papers  (2021-06-30T18:32:46Z)
• The Role of Momentum Parameters in the Optimal Convergence of Adaptive Polyak's Heavy-ball Methods [12.93796690939018]
  We prove that the adaptive Polyak's Heavy-ball (HB) method attains an optimal individual convergence rate of $O(frac1sqrtt)$. Our new analysis shows how the HB momentum and its time-varying weight help us to achieve the acceleration in convex optimization.
  arXiv  Detail & Related papers  (2021-02-15T02:57:14Z)
• 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)

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.