# Cumulative Learning Rate Adaptation: Revisiting Path-Based Schedules for SGD and Adam

arXiv: https://arxiv.org/abs/2508.05408 · HTML: https://mlsanity.com/paper/2508.05408/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 21 references that are in the map.
- First posted on arXiv 7 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [COCO: A Platform for Comparing Continuous Optimizers in a Black-Box Setting](https://mlsanity.com/paper/1603.08785/) (2016)
- [Prodigy: An Expeditiously Adaptive Parameter-Free Learner](https://mlsanity.com/paper/2306.06101/) (2023)
- [Learning-Rate-Free Learning by D-Adaptation](https://mlsanity.com/paper/2301.07733/) (2023)
- [Selecting and Composing Learning Rate Policies for Deep Neural Networks](https://mlsanity.com/paper/2210.12936/) (2022)
- [Stochastic Gradient Descent: Going As Fast As Possible But Not Faster](https://mlsanity.com/paper/1709.01427/) (2017)
- [Tuning Learning Rates with the Cumulative-Learning Constant](https://mlsanity.com/paper/2505.13457/) (2025)

## Abstract (arXiv)

The learning rate is a crucial hyperparameter in deep learning, with its ideal value depending on the problem and potentially changing during training. In this paper, we investigate the practical utility of adaptive learning rate mechanisms that adjust step sizes dynamically in response to the loss landscape. We revisit a cumulative path-based adaptation scheme proposed in 2017, which adjusts the learning rate based on the discrepancy between the observed path length, computed as a time-discounted sum of normalized gradient steps, and the expected length of a random walk. While the original approach offers a compelling intuition, we show that its adaptation mechanism for Adam is conceptually inconsistent due to the optimizer's internal preconditioning. We propose a corrected variant that better reflects Adam's update dynamics. To assess the practical value of online learning rate adaptation, we benchmark SGD and Adam, with and without cumulative adaptation, and compare them to a recent alternative method. Our results aim to clarify when and why such adaptive strategies offer practical benefits.