SAPPHIRE: Preconditioned Stochastic Variance Reduction for Faster Large-Scale Statistical Learning

• URL: http://arxiv.org/abs/2501.15941v1
• Date: Mon, 27 Jan 2025 10:36:45 GMT
• Title: SAPPHIRE: Preconditioned Stochastic Variance Reduction for Faster Large-Scale Statistical Learning
• Authors: Jingruo Sun, Zachary Frangella, Madeleine Udell,
• Abstract summary: Ill-conditioned objectives and nonsmooth regularizers undermine the performance of traditional convex methods. We propose a variance-free solution for ill-conditioned composite large-scale machine learning problems.
• Score: 18.055120576191204
• License:
• Abstract: Regularized empirical risk minimization (rERM) has become important in data-intensive fields such as genomics and advertising, with stochastic gradient methods typically used to solve the largest problems. However, ill-conditioned objectives and non-smooth regularizers undermine the performance of traditional stochastic gradient methods, leading to slow convergence and significant computational costs. To address these challenges, we propose the $\texttt{SAPPHIRE}$ ($\textbf{S}$ketching-based $\textbf{A}$pproximations for $\textbf{P}$roximal $\textbf{P}$reconditioning and $\textbf{H}$essian $\textbf{I}$nexactness with Variance-$\textbf{RE}$educed Gradients) algorithm, which integrates sketch-based preconditioning to tackle ill-conditioning and uses a scaled proximal mapping to minimize the non-smooth regularizer. This stochastic variance-reduced algorithm achieves condition-number-free linear convergence to the optimum, delivering an efficient and scalable solution for ill-conditioned composite large-scale convex machine learning problems. Extensive experiments on lasso and logistic regression demonstrate that $\texttt{SAPPHIRE}$ often converges $20$ times faster than other common choices such as $\texttt{Catalyst}$, $\texttt{SAGA}$, and $\texttt{SVRG}$. This advantage persists even when the objective is non-convex or the preconditioner is infrequently updated, highlighting its robust and practical effectiveness.

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