New Tight Bounds for SGD without Variance Assumption: A Computer-Aided Lyapunov Analysis

• URL: http://arxiv.org/abs/2505.17965v1
• Date: Fri, 23 May 2025 14:34:46 GMT
• Title: New Tight Bounds for SGD without Variance Assumption: A Computer-Aided Lyapunov Analysis
• Authors: Daniel Cortild, Lucas Ketels, Juan Peypouquet, Guillaume Garrigos,
• Abstract summary: This paper contributes to a recent line of works which attempt to provide guarantees without making any variance assumption.<n>We prove new theoretical bounds derived from the monotonicity of a simple Lyapunov energy.
• Score: 0.3937354192623676
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The analysis of Stochastic Gradient Descent (SGD) often relies on making some assumption on the variance of the stochastic gradients, which is usually not satisfied or difficult to verify in practice. This paper contributes to a recent line of works which attempt to provide guarantees without making any variance assumption, leveraging only the (strong) convexity and smoothness of the loss functions. In this context, we prove new theoretical bounds derived from the monotonicity of a simple Lyapunov energy, improving the current state-of-the-art and extending their validity to larger step-sizes. Our theoretical analysis is backed by a Performance Estimation Problem analysis, which allows us to claim that, empirically, the bias term in our bounds is tight within our framework.

Related papers

• Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization [0.40964539027092917]
  We prove strong convergence of reg-SGD to the minimum-norm solution of the original problem without additional boundedness assumptions.<n>Our analysis reveals how vanishing Tikhonov regularization controls the flow of SGD and yields stable learning dynamics.
  arXiv  Detail & Related papers  (2025-05-16T16:53:49Z)
• 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)
• On the Dynamics Under the Unhinged Loss and Beyond [104.49565602940699]
  We introduce the unhinged loss, a concise loss function, that offers more mathematical opportunities to analyze closed-form dynamics. The unhinged loss allows for considering more practical techniques, such as time-vary learning rates and feature normalization.
  arXiv  Detail & Related papers  (2023-12-13T02:11:07Z)
• On the Importance of Gradient Norm in PAC-Bayesian Bounds [92.82627080794491]
  We propose a new generalization bound that exploits the contractivity of the log-Sobolev inequalities. We empirically analyze the effect of this new loss-gradient norm term on different neural architectures.
  arXiv  Detail & Related papers  (2022-10-12T12:49:20Z)
• Benign Underfitting of Stochastic Gradient Descent [72.38051710389732]
  We study to what extent may gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit training data. We analyze the closely related with-replacement SGD, for which an analogous phenomenon does not occur and prove that its population risk does in fact converge at the optimal rate.
  arXiv  Detail & Related papers  (2022-02-27T13:25:01Z)
• A general sample complexity analysis of vanilla policy gradient [101.16957584135767]
  Policy gradient (PG) is one of the most popular reinforcement learning (RL) problems. "vanilla" theoretical understanding of PG trajectory is one of the most popular methods for solving RL problems.
  arXiv  Detail & Related papers  (2021-07-23T19:38:17Z)
• Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent [55.85456985750134]
  We introduce a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting. To our best knowledge, this gives the firstever-known stability and generalization for SGD with even non-differentiable loss functions.
  arXiv  Detail & Related papers  (2020-06-15T06:30:19Z)
• Convergence rates and approximation results for SGD and its continuous-time counterpart [16.70533901524849]
  This paper proposes a thorough theoretical analysis of convex Gradient Descent (SGD) with non-increasing step sizes. First, we show that the SGD can be provably approximated by solutions of inhomogeneous Differential Equation (SDE) using coupling. Recent analyses of deterministic and optimization methods by their continuous counterpart, we study the long-time behavior of the continuous processes at hand and non-asymptotic bounds.
  arXiv  Detail & Related papers  (2020-04-08T18:31:34Z)

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.