Understanding Lookahead Dynamics Through Laplace Transform

• URL: http://arxiv.org/abs/2506.13712v1
• Date: Mon, 16 Jun 2025 17:20:40 GMT
• Title: Understanding Lookahead Dynamics Through Laplace Transform
• Authors: Aniket Sanyal, Tatjana Chavdarova,
• Abstract summary: We introduce a frequency-domain framework for convergence analysis of hyper parameters in games.<n>We transform the discrete-time dynamics of bilinear games into the frequency domain to derive precise convergence criteria.<n> Empirical validation in discrete-time settings demonstrates the effectiveness of our approach.
• Score: 4.204990010424083
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a frequency-domain framework for convergence analysis of hyperparameters in game optimization, leveraging High-Resolution Differential Equations (HRDEs) and Laplace transforms. Focusing on the Lookahead algorithm--characterized by gradient steps $k$ and averaging coefficient $\alpha$--we transform the discrete-time oscillatory dynamics of bilinear games into the frequency domain to derive precise convergence criteria. Our higher-precision $O(\gamma^2)$-HRDE models yield tighter criteria, while our first-order $O(\gamma)$-HRDE models offer practical guidance by prioritizing actionable hyperparameter tuning over complex closed-form solutions. Empirical validation in discrete-time settings demonstrates the effectiveness of our approach, which may further extend to locally linear operators, offering a scalable framework for selecting hyperparameters for learning in games.

Related papers

• More Optimal Fractional-Order Stochastic Gradient Descent for Non-Convex Optimization Problems [2.5971517743176915]
  We propose 2SED Fractional-Order Gradient Descent (2FOSGD), which integrates the Two-Scale Effective Dimension (2SED) with FOSGD.<n>By tracking sensitivity and effective dimensionality, 2SEDFOSGD dynamically modulates the exponent to mitigate sluggish oscillations and hasten convergence.
  arXiv  Detail & Related papers  (2025-05-05T19:27:36Z)
• SDEs for Minimax Optimization [11.290653315174382]
  In this paper, we pioneer the use of differential equations (SDEs) to analyze and compare Minimax convergences. Our SDE models for Gradient Descent-Ascent, Extragradient, and Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts. This perspective also allows for a unified and simplified analysis strategy based on the principles of Ito calculus.
  arXiv  Detail & Related papers  (2024-02-19T20:18:29Z)
• PROMISE: Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates [17.777466668123886]
  We introduce PROMISE ($textbfPr$econditioned $textbfO$ptimization $textbfM$ethods by $textbfI$ncorporating $textbfS$calable Curvature $textbfE$stimates), a suite of sketching-based preconditioned gradient algorithms. PROMISE includes preconditioned versions of SVRG, SAGA, and Katyusha.
  arXiv  Detail & Related papers  (2023-09-05T07:49:10Z)
• NAG-GS: Semi-Implicit, Accelerated and Robust Stochastic Optimizer [45.47667026025716]
  We propose a novel, robust and accelerated iteration that relies on two key elements. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively. We show that NAG-arity is competitive with state-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models.
  arXiv  Detail & Related papers  (2022-09-29T16:54:53Z)
• 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)
• Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations [83.3201889218775]
  Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. We adopt a framework studied in fluid dynamics to design differential equation models for several saddle-point optimization methods.
  arXiv  Detail & Related papers  (2021-12-27T18:31:34Z)
• A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors [60.489902135153415]
  This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement.
  arXiv  Detail & Related papers  (2021-11-26T06:33:29Z)
• Momentum Accelerates the Convergence of Stochastic AUPRC Maximization [80.8226518642952]
  We study optimization of areas under precision-recall curves (AUPRC), which is widely used for imbalanced tasks. We develop novel momentum methods with a better iteration of $O (1/epsilon4)$ for finding an $epsilon$stationary solution. We also design a novel family of adaptive methods with the same complexity of $O (1/epsilon4)$, which enjoy faster convergence in practice.
  arXiv  Detail & Related papers  (2021-07-02T16:21:52Z)
• Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition [54.07797071198249]
  We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections. Notably, our approach achieves state-of-the-art results on CIFAR-10 among pure GP models.
  arXiv  Detail & Related papers  (2021-06-10T18:17:57Z)
• Dynamics of Stochastic Momentum Methods on Large-scale, Quadratic Models [0.2741266294612776]
  We analyze a class of gradient algorithms with momentum on a high-dimensional random least squares problem. We show that (small-batch) momentum with a fixed momentum parameter provides no actual performance improvement over SGD when step sizes are adjusted correctly. In the non-strongly convex setting, it is possible to get a large improvement over SGD using momentum.
  arXiv  Detail & Related papers  (2021-06-07T15:08:24Z)
• Pushing the Envelope of Rotation Averaging for Visual SLAM [69.7375052440794]
  We propose a novel optimization backbone for visual SLAM systems. We leverage averaging to improve the accuracy, efficiency and robustness of conventional monocular SLAM systems. Our approach can exhibit up to 10x faster with comparable accuracy against the state-art on public benchmarks.
  arXiv  Detail & Related papers  (2020-11-02T18:02:26Z)

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.