Armijo Line-search Makes (Stochastic) Gradient Descent Go Fast

• URL: http://arxiv.org/abs/2503.00229v1
• Date: Fri, 28 Feb 2025 22:26:33 GMT
• Title: Armijo Line-search Makes (Stochastic) Gradient Descent Go Fast
• Authors: Sharan Vaswani, Reza Babanezhad,
• Abstract summary: Armijo line-search convex (Armijo-LS) is a standard method to set the step-size for gradient descent (GD)<n>We show that if a certain non-uniform condition, GD-LS converges provably faster than GD with a constant $1/L$ step-size.
• Score: 7.974134340935598
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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. However, existing theoretical analyses of GD with Armijo-LS (GD-LS) do not characterize this fast convergence. We show that if the objective function satisfies a certain non-uniform smoothness condition, GD-LS converges provably faster than GD with a constant $1/L$ step-size (denoted as GD(1/L)). Our results imply that for convex losses corresponding to logistic regression and multi-class classification, GD-LS can converge to the optimum at a linear rate and, hence, improve over the sublinear convergence of GD(1/L). Furthermore, for non-convex losses satisfying gradient domination (for example, 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 prove that under the interpolation assumption, for convex losses, stochastic GD with a stochastic line-search can match the fast convergence of GD-LS.

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