mlsanityData to 14 Oct 2026

Step Size Matters in Deep Learning

arXiv 1805.08890 ↗First posted 22 May 2018cs.LG math.OC stat.MLcited by 19

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Abstract (arXiv)

Training a neural network with the gradient descent algorithm gives rise to a discrete-time nonlinear dynamical system. Consequently, behaviors that are typically observed in these systems emerge during training, such as convergence to an orbit but not to a fixed point or dependence of convergence on the initialization. Step size of the algorithm plays a critical role in these behaviors: it determines the subset of the local optima that the algorithm can converge to, and it specifies the magnitude of the oscillations if the algorithm converges to an orbit. To elucidate the effects of the step size on training of neural networks, we study the gradient descent algorithm as a discrete-time dynamical system, and by analyzing the Lyapunov stability of different solutions, we show the relationship between the step size of the algorithm and the solutions that can be obtained with this algorithm. The results provide an explanation for several phenomena observed in practice, including the deterioration in the training error with increased depth, the hardness of estimating linear mappings with large singular values, and the distinct performance of deep residual networks.

Key facts. Cited by 19 papers in the map (data through 14 Oct 2026). First posted on arXiv 22 May 2018.

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Citing papers per year: 2021: 4; 2022: 2; 2023: 2; 2024: 5; 2025: 3; 2026: 3.

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  1. 2021Gradient Descent on Neural Networks Typically Occurs at the Edge of Stabilitycited by 427
  2. 2022Adaptive Gradient Methods at the Edge of Stabilitycited by 101
  3. 2022On the Implicit Bias in Deep-Learning Algorithmscited by 86
  4. 2024Why is SAM Robust to Label Noise?cited by 21
  5. 2023The Implicit Bias of Minima Stability in Multivariate Shallow ReLU Networkscited by 15
  6. 2021AutoLRS: Automatic Learning-Rate Schedule by Bayesian Optimization on the Flycited by 14
  7. 2023Exact Mean Square Linear Stability Analysis for SGDcited by 11
  8. 2025Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Regioncited by 7
  9. 2025Generalization Below the Edge of Stability: The Role of Data Geometrycited by 3
  10. 2025Sharpness of Minima in Deep Matrix Factorizationcited by 2
  11. 2024Faster Convergence for Transformer Fine-tuning with Line Search Methodscited by 1
  12. 2024No learning rates needed: Introducing SALSA -- Stable Armijo Line Search Adaptationcited by 1
7 more
  1. 2024Lecture Notes on Linear Neural Networks: A Tale of Optimization and Generalization in Deep Learningcited by 1
  2. 2026Does Sparse Connectivity Improve Generalization? Convolutional Networks Below the Edge of Stabilitycited by 1
  3. 2021LRTuner: A Learning Rate Tuner for Deep Neural Networkscited by 0
  4. 2021Boosting Resource-Constrained Federated Learning Systems with Guessed Updatescited by 0
  5. 2024GeoAdaLer: Geometric Insights into Adaptive Stochastic Gradient Descent Algorithmscited by 0
  6. 2026Meta-Learning and Meta-Reinforcement Learning -- Tracing the Path towards DeepMind's Adaptive Agentcited by 0
  7. 2026Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimizationcited by 0