# Convergence of Steepest Descent and Adam under Non-Uniform Smoothness

arXiv: https://arxiv.org/abs/2605.30648 · HTML: https://mlsanity.com/paper/2605.30648/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 28 May 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 5 of 5)

- [Gradient methods for convex minimization: better rates under weaker conditions](https://mlsanity.com/paper/1303.4645/) (2013)
- [Elementary Analysis of Policy Gradient Methods](https://mlsanity.com/paper/2404.03372/) (2024)
- [On the Convergence of Adam under Non-uniform Smoothness: Separability from SGDM and Beyond](https://mlsanity.com/paper/2403.15146/) (2024)
- [Why Do We Need Warm-up? A Theoretical Perspective](https://mlsanity.com/paper/2510.03164/) (2025)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)

## Built on by (most cited first, 1 of 1)

- [Convergence Guarantees of Gradient Descent for Neural Networks via Generalized Lipschitz Smoothness](https://mlsanity.com/paper/2608.11479/) (2026)

## Abstract (arXiv)

Recent work has analyzed the convergence of first-order methods under non-uniform smoothness assumptions that better model the loss landscape in machine learning tasks. We generalize this assumption to objectives whose curvature is an affine function of the objective value. This property is satisfied by a broad class of problems, including logistic regression, generalized linear models with a logistic link function, softmax policy gradient in reinforcement learning, and a class of neural networks. Under this assumption and gradient domination conditions, we establish a general convergence rate for the steepest descent method, and deterministic, diagonal variants of RMSProp and Adam. Our results imply that for logistic regression on separable data and the softmax policy gradient objective, sign GD converges linearly and is provably faster than GD. Furthermore, we show that for a class of two-layer neural networks on separable data, RMSProp and Adam can converge at a linear rate with a constant step-size and momentum parameter. Finally, we present a lower bound demonstrating that, under our assumption, RMSProp and Adam are provably faster than AdaGrad, AMSGrad, gradient descent, and heavy-ball momentum.