Toward a Unified Theory of Gradient Descent under Generalized Smoothness

Alexander Tyurin
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:60493-60514, 2025.

Abstract

We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$–smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + ⟨\nabla f(x_k), x_{k+1} - x_k ⟩+ \frac{L}{2} \|\| x_{k+1} - x_k \|\|^2$ is $x_{k+1} = x_k - \gamma_k \nabla f(x_k)$ with step size $\gamma_k = \frac{1}{L}$. Surprisingly, a similar result can be derived under the $\ell$-generalized smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq \ell( \|\| \nabla f(x) \|\| )$). In this case, we derive the step size $\gamma_k = \int_{0}^{1} \frac{d v}{\ell( \|\| \nabla f(x_k) \|\| + \|\| \nabla f(x_k) \|\|{v})}.$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-tyurin25a, title = {Toward a Unified Theory of Gradient Descent under Generalized Smoothness}, author = {Tyurin, Alexander}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {60493--60514}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/tyurin25a/tyurin25a.pdf}, url = {https://proceedings.mlr.press/v267/tyurin25a.html}, abstract = {We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$–smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + ⟨\nabla f(x_k), x_{k+1} - x_k ⟩+ \frac{L}{2} \|\| x_{k+1} - x_k \|\|^2$ is $x_{k+1} = x_k - \gamma_k \nabla f(x_k)$ with step size $\gamma_k = \frac{1}{L}$. Surprisingly, a similar result can be derived under the $\ell$-generalized smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq \ell( \|\| \nabla f(x) \|\| )$). In this case, we derive the step size $\gamma_k = \int_{0}^{1} \frac{d v}{\ell( \|\| \nabla f(x_k) \|\| + \|\| \nabla f(x_k) \|\|{v})}.$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups.} }
Endnote
%0 Conference Paper %T Toward a Unified Theory of Gradient Descent under Generalized Smoothness %A Alexander Tyurin %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-tyurin25a %I PMLR %P 60493--60514 %U https://proceedings.mlr.press/v267/tyurin25a.html %V 267 %X We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$–smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + ⟨\nabla f(x_k), x_{k+1} - x_k ⟩+ \frac{L}{2} \|\| x_{k+1} - x_k \|\|^2$ is $x_{k+1} = x_k - \gamma_k \nabla f(x_k)$ with step size $\gamma_k = \frac{1}{L}$. Surprisingly, a similar result can be derived under the $\ell$-generalized smoothness assumption ($\|\| \nabla^2 f(x) \|\| \leq \ell( \|\| \nabla f(x) \|\| )$). In this case, we derive the step size $\gamma_k = \int_{0}^{1} \frac{d v}{\ell( \|\| \nabla f(x_k) \|\| + \|\| \nabla f(x_k) \|\|{v})}.$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups.
APA
Tyurin, A.. (2025). Toward a Unified Theory of Gradient Descent under Generalized Smoothness. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:60493-60514 Available from https://proceedings.mlr.press/v267/tyurin25a.html.

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