Armijo Line-search Can Make (Stochastic) Gradient Descent Provably Faster
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arXiv
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| Format: | Preprint |
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2025
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| author | Vaswani, Sharan Babanezhad, Reza |
| author_facet | Vaswani, Sharan Babanezhad, Reza |
| contents | Armijo line-search (Armijo-LS) is a standard method to set the step-size for gradient descent (GD). For smooth functions, Armijo-LS alleviates the need to know the global smoothness constant L and adapts to the ``local'' smoothness, enabling GD to converge faster. Existing theoretical analyses show that GD with Armijo-LS (GD-LS) can result in constant factor improvements over GD with a 1/L step-size (denoted as GD(1/L)). We strengthen these results and show that if the objective function satisfies a certain non-uniform smoothness condition, GD-LS can result in a faster convergence rate than GD(1/L). In particular, we prove that for convex objectives corresponding to logistic regression and multi-class classification, GD-LS can converge to the optimum at a linear rate, and hence improves over the sublinear convergence of GD(1/L). Furthermore, for non-convex objectives satisfying gradient domination (e.g., those corresponding to the softmax policy gradient in RL or generalized linear models with a logistic link function), GD-LS can match the fast convergence of algorithms tailored for these specific settings. Finally, we analyze the convergence of stochastic GD with a stochastic line-search on convex losses under the interpolation assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Armijo Line-search Can Make (Stochastic) Gradient Descent Provably Faster Vaswani, Sharan Babanezhad, Reza Machine Learning Optimization and Control Armijo line-search (Armijo-LS) is a standard method to set the step-size for gradient descent (GD). For smooth functions, Armijo-LS alleviates the need to know the global smoothness constant L and adapts to the ``local'' smoothness, enabling GD to converge faster. Existing theoretical analyses show that GD with Armijo-LS (GD-LS) can result in constant factor improvements over GD with a 1/L step-size (denoted as GD(1/L)). We strengthen these results and show that if the objective function satisfies a certain non-uniform smoothness condition, GD-LS can result in a faster convergence rate than GD(1/L). In particular, we prove that for convex objectives corresponding to logistic regression and multi-class classification, GD-LS can converge to the optimum at a linear rate, and hence improves over the sublinear convergence of GD(1/L). Furthermore, for non-convex objectives satisfying gradient domination (e.g., those corresponding to the softmax policy gradient in RL or generalized linear models with a logistic link function), GD-LS can match the fast convergence of algorithms tailored for these specific settings. Finally, we analyze the convergence of stochastic GD with a stochastic line-search on convex losses under the interpolation assumption. |
| title | Armijo Line-search Can Make (Stochastic) Gradient Descent Provably Faster |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2503.00229 |