Adaptive Coordinate-Wise Step Sizes for Quasi-Newton Methods: A Learning-to-Optimize Approach

Tuning effective step sizes is crucial for the stability and efficiency of optimization algorithms. While adaptive coordinate-wise step sizes tuning methods have been explored in first-order methods, second-order methods still lack efficient techniques. Current approaches, including hypergradient descent and cutting plane methods, offer limited improvements or encounter difficulties in second-order contexts. To address these challenges, we introduce a novel Learning-to-Optimize (L2O) model within the Broyden-Fletcher-Goldfarb-Shanno (BFGS) framework, which leverages neural networks to predict optimal coordinate-wise step sizes. Our model integrates a theoretical foundation that establishes conditions for the stability and convergence of these step sizes. Extensive experiments demonstrate that our approach achieves substantial improvements over traditional backtracking line search and hypergradient descent-based methods, offering up to 7$\times$ faster and stable performance across diverse optimization tasks.