Linear Convergence Rate in Convex Setup is Possible! Gradient Descent Method Variants under $(L_0,L_1)$-Smoothness
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912237323550720 |
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| author | Lobanov, Aleksandr Gasnikov, Alexander Gorbunov, Eduard Takáč, Martin |
| author_facet | Lobanov, Aleksandr Gasnikov, Alexander Gorbunov, Eduard Takáč, Martin |
| contents | The gradient descent (GD) method -- is a fundamental and likely the most popular optimization algorithm in machine learning (ML), with a history traced back to a paper in 1847 (Cauchy, 1847). It was studied under various assumptions, including so-called $(L_0,L_1)$-smoothness, which received noticeable attention in the ML community recently. In this paper, we provide a refined convergence analysis of gradient descent and its variants, assuming generalized smoothness. In particular, we show that $(L_0,L_1)$-GD has the following behavior in the convex setup: as long as $\|\nabla f(x^k)\| \geq \frac{L_0}{L_1}$ the algorithm has linear convergence in function suboptimality, and when $\|\nabla f(x^k)\| < \frac{L_0}{L_1}$ is satisfied, $(L_0,L_1)$-GD has standard sublinear rate. Moreover, we also show that this behavior is common for its variants with different types of oracle: Normalized Gradient Descent as well as Clipped Gradient Descent (the case when the full gradient $\nabla f(x)$ is available); Random Coordinate Descent (when the gradient component $\nabla_{i} f(x)$ is available); Random Coordinate Descent with Order Oracle (when only $\text{sign} [f(y) - f(x)]$ is available). In addition, we also extend our analysis of $(L_0,L_1)$-GD to the strongly convex case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17050 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Convergence Rate in Convex Setup is Possible! Gradient Descent Method Variants under $(L_0,L_1)$-Smoothness Lobanov, Aleksandr Gasnikov, Alexander Gorbunov, Eduard Takáč, Martin Optimization and Control The gradient descent (GD) method -- is a fundamental and likely the most popular optimization algorithm in machine learning (ML), with a history traced back to a paper in 1847 (Cauchy, 1847). It was studied under various assumptions, including so-called $(L_0,L_1)$-smoothness, which received noticeable attention in the ML community recently. In this paper, we provide a refined convergence analysis of gradient descent and its variants, assuming generalized smoothness. In particular, we show that $(L_0,L_1)$-GD has the following behavior in the convex setup: as long as $\|\nabla f(x^k)\| \geq \frac{L_0}{L_1}$ the algorithm has linear convergence in function suboptimality, and when $\|\nabla f(x^k)\| < \frac{L_0}{L_1}$ is satisfied, $(L_0,L_1)$-GD has standard sublinear rate. Moreover, we also show that this behavior is common for its variants with different types of oracle: Normalized Gradient Descent as well as Clipped Gradient Descent (the case when the full gradient $\nabla f(x)$ is available); Random Coordinate Descent (when the gradient component $\nabla_{i} f(x)$ is available); Random Coordinate Descent with Order Oracle (when only $\text{sign} [f(y) - f(x)]$ is available). In addition, we also extend our analysis of $(L_0,L_1)$-GD to the strongly convex case. |
| title | Linear Convergence Rate in Convex Setup is Possible! Gradient Descent Method Variants under $(L_0,L_1)$-Smoothness |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.17050 |