Regression Descent: A Statistical Framework for Neural Network Optimization

Kamaljeet Singh, Nicolas Hengartner, Hao Zhang, Brian Wesley Bell, James Hyman
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:4078-4086, 2026.

Abstract

We present Regression Descent (RD), a novel optimization algorithm for training deep neural networks that reformulates each gradient step as a regression problem in the span of the Jacobian. By leveraging the implicit function theorem in over-parameterized settings where the number of parameters exceed observations $(p > n)$, we project the optimization onto an $n$-dimensional subspace, enabling the use of statistical techniques and potentially improved conditioning. Our key insight is that in the over-parameterized regime, meaningful parameter updates lie in the row space of the Jacobian matrix, allowing us to solve a lower-dimensional regression problem with explicit regularization control. We establish convergence guarantees for RD under standard smoothness assumptions, showing that it achieves a convergence rate of $O(1/k)$ for smooth non-convex objectives. The algorithm naturally handles the ill-conditioning common in neural network optimization through adaptive regularization and extends seamlessly to multi-output problems and mini-batch settings. Experimental results on Lorenz96, MNIST, and FMNIST datasets demonstrate that RD achieves up to 40% faster convergence compared to SGD and Adam in terms of wall-clock time, with strong performance in the presence of activation function saturation. The computational overhead of solving $m \times m$ linear systems (where $m$ is the batch size) is offset by improved convergence properties and GPU-efficient operations. Our work opens new avenues for understanding neural network optimization through the lens of statistical regression, providing a practical algorithm for scenarios where standard gradient methods struggle.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-singh26b, title = { Regression Descent: A Statistical Framework for Neural Network Optimization }, author = {Singh, Kamaljeet and Hengartner, Nicolas and Zhang, Hao and Bell, Brian Wesley and Hyman, James}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {4078--4086}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/singh26b/singh26b.pdf}, url = {https://proceedings.mlr.press/v300/singh26b.html}, abstract = { We present Regression Descent (RD), a novel optimization algorithm for training deep neural networks that reformulates each gradient step as a regression problem in the span of the Jacobian. By leveraging the implicit function theorem in over-parameterized settings where the number of parameters exceed observations $(p > n)$, we project the optimization onto an $n$-dimensional subspace, enabling the use of statistical techniques and potentially improved conditioning. Our key insight is that in the over-parameterized regime, meaningful parameter updates lie in the row space of the Jacobian matrix, allowing us to solve a lower-dimensional regression problem with explicit regularization control. We establish convergence guarantees for RD under standard smoothness assumptions, showing that it achieves a convergence rate of $O(1/k)$ for smooth non-convex objectives. The algorithm naturally handles the ill-conditioning common in neural network optimization through adaptive regularization and extends seamlessly to multi-output problems and mini-batch settings. Experimental results on Lorenz96, MNIST, and FMNIST datasets demonstrate that RD achieves up to 40% faster convergence compared to SGD and Adam in terms of wall-clock time, with strong performance in the presence of activation function saturation. The computational overhead of solving $m \times m$ linear systems (where $m$ is the batch size) is offset by improved convergence properties and GPU-efficient operations. Our work opens new avenues for understanding neural network optimization through the lens of statistical regression, providing a practical algorithm for scenarios where standard gradient methods struggle. } }
Endnote
%0 Conference Paper %T Regression Descent: A Statistical Framework for Neural Network Optimization %A Kamaljeet Singh %A Nicolas Hengartner %A Hao Zhang %A Brian Wesley Bell %A James Hyman %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-singh26b %I PMLR %P 4078--4086 %U https://proceedings.mlr.press/v300/singh26b.html %V 300 %X We present Regression Descent (RD), a novel optimization algorithm for training deep neural networks that reformulates each gradient step as a regression problem in the span of the Jacobian. By leveraging the implicit function theorem in over-parameterized settings where the number of parameters exceed observations $(p > n)$, we project the optimization onto an $n$-dimensional subspace, enabling the use of statistical techniques and potentially improved conditioning. Our key insight is that in the over-parameterized regime, meaningful parameter updates lie in the row space of the Jacobian matrix, allowing us to solve a lower-dimensional regression problem with explicit regularization control. We establish convergence guarantees for RD under standard smoothness assumptions, showing that it achieves a convergence rate of $O(1/k)$ for smooth non-convex objectives. The algorithm naturally handles the ill-conditioning common in neural network optimization through adaptive regularization and extends seamlessly to multi-output problems and mini-batch settings. Experimental results on Lorenz96, MNIST, and FMNIST datasets demonstrate that RD achieves up to 40% faster convergence compared to SGD and Adam in terms of wall-clock time, with strong performance in the presence of activation function saturation. The computational overhead of solving $m \times m$ linear systems (where $m$ is the batch size) is offset by improved convergence properties and GPU-efficient operations. Our work opens new avenues for understanding neural network optimization through the lens of statistical regression, providing a practical algorithm for scenarios where standard gradient methods struggle.
APA
Singh, K., Hengartner, N., Zhang, H., Bell, B.W. & Hyman, J.. (2026). Regression Descent: A Statistical Framework for Neural Network Optimization . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:4078-4086 Available from https://proceedings.mlr.press/v300/singh26b.html.

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