Toward a Unified Theory of Gradient Descent under Generalized Smoothness
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915360540721152 |
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| author | Tyurin, Alexander |
| author_facet | Tyurin, Alexander |
| contents | 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 $f(x_k) + \langle\nabla f(x_k), x_{k+1} - x_k\rangle + \frac{L}{2} \|x_{k+1} - x_k\|^2$ is $x_{k+1} = x_k - γ_k \nabla f(x_k)$ with step size $γ_k = \frac{1}{L}$. 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 $$γ_k = \int_{0}^{1} \frac{d v}{\ell(\|\nabla f(x_k)\| + \|\nabla f(x_k)\| v)}.$$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward a Unified Theory of Gradient Descent under Generalized Smoothness Tyurin, Alexander Optimization and Control 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 $f(x_k) + \langle\nabla f(x_k), x_{k+1} - x_k\rangle + \frac{L}{2} \|x_{k+1} - x_k\|^2$ is $x_{k+1} = x_k - γ_k \nabla f(x_k)$ with step size $γ_k = \frac{1}{L}$. 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 $$γ_k = \int_{0}^{1} \frac{d v}{\ell(\|\nabla f(x_k)\| + \|\nabla f(x_k)\| v)}.$$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups. |
| title | Toward a Unified Theory of Gradient Descent under Generalized Smoothness |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.11773 |