# The duality structure gradient descent algorithm: analysis and applications to neural networks

arXiv: https://arxiv.org/abs/1708.00523 · HTML: https://mlsanity.com/paper/1708.00523/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2017.

## Built on by (most cited first, 8 of 8)

- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)
- [Convergence Bound and Critical Batch Size of Muon Optimizer](https://mlsanity.com/paper/2507.01598/) (2025)
- [Training Transformers with Enforced Lipschitz Constants](https://mlsanity.com/paper/2507.13338/) (2025)
- [Generalized Gradient Norm Clipping & Non-Euclidean (L0,L1)-Smoothness](https://mlsanity.com/paper/2506.01913/) (2025)
- [On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer](https://mlsanity.com/paper/2603.09952/) (2026)
- [SOAP, Muon, and Beyond: Pushing LLM Pretraining Scales](https://mlsanity.com/paper/2607.20548/) (2026)
- [A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon](https://mlsanity.com/paper/2604.17423/) (2026)

## Abstract (arXiv)

The training of machine learning models is typically carried out using some form of gradient descent, often with great success. However, non-asymptotic analyses of first-order optimization algorithms typically employ a gradient smoothness assumption (formally, Lipschitz continuity of the gradient) that is too strong to be applicable in the case of deep neural networks. To address this, we propose an algorithm named duality structure gradient descent (DSGD) that is amenable to non-asymptotic performance analysis, under mild assumptions on the training set and network architecture. The algorithm can be viewed as a form of layer-wise coordinate descent, where at each iteration the algorithm chooses one layer of the network to update. The decision of what layer to update is done in a greedy fashion, based on a rigorous lower bound on the improvement of the objective function for each choice of layer. In the analysis, we bound the time required to reach approximate stationary points, in both the deterministic and stochastic settings. The convergence is measured in terms of a parameter-dependent family of norms that is derived from the network architecture and designed to confirm a smoothness-like property on the gradient of the training loss function. We empirically demonstrate the behavior of DSGD in several neural network training scenarios.