A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908965117362176 |
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| author | Davis, Damek Drusvyatskiy, Dmitriy |
| author_facet | Davis, Damek Drusvyatskiy, Dmitriy |
| contents | Davis, Drusvyatskiy, and Jiang showed that gradient descent with an adaptive stepsize converges locally at a nearly-linear rate for smooth functions that grow at least quartically away from their minimizers. The argument is intricate, relying on monitoring the performance of the algorithm relative to a certain manifold of slow growth -- called the ravine. In this work, we provide a direct Lyapunov-based argument that bypasses these difficulties when the objective is in addition convex and a has a unique minimizer. As a byproduct of the argument, we obtain a more adaptive variant than the original algorithm with encouraging numerical performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13393 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity Davis, Damek Drusvyatskiy, Dmitriy Optimization and Control Machine Learning Davis, Drusvyatskiy, and Jiang showed that gradient descent with an adaptive stepsize converges locally at a nearly-linear rate for smooth functions that grow at least quartically away from their minimizers. The argument is intricate, relying on monitoring the performance of the algorithm relative to a certain manifold of slow growth -- called the ravine. In this work, we provide a direct Lyapunov-based argument that bypasses these difficulties when the objective is in addition convex and a has a unique minimizer. As a byproduct of the argument, we obtain a more adaptive variant than the original algorithm with encouraging numerical performance. |
| title | A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexity |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2604.13393 |