Jane J. Ye

Lucy Gao, Jane J. Ye, Haian Yin et al.

Gradient-based optimization methods for hyperparameter tuning guarantee theoretical convergence to stationary solutions when for fixed upper-level variable values, the lower level of the bilevel program is strongly convex (LLSC) and smooth (LLS). This condition is not satisfied for bilevel programs arising from tuning hyperparameters in many machine learning algorithms. In this work, we develop a sequentially convergent Value Function based Difference-of-Convex Algorithm with inexactness (VF-iDCA). We show that this algorithm achieves stationary solutions without LLSC and LLS assumptions for bilevel programs from a broad class of hyperparameter tuning applications. Our extensive experiments confirm our theoretical findings and show that the proposed VF-iDCA yields superior performance when applied to tune hyperparameters.

Lucy L. Gao, Jane J. Ye, Haian Yin et al.

Bilevel programming has emerged as a valuable tool for hyperparameter selection, a central concern in machine learning. In a recent study by Ye et al. (2023), a value function-based difference of convex algorithm was introduced to address bilevel programs. This approach proves particularly powerful when dealing with scenarios where the lower-level problem exhibits convexity in both the upper-level and lower-level variables. Examples of such scenarios include support vector machines and $\ell_1$ and $\ell_2$ regularized regression. In this paper, we significantly expand the range of applications, now requiring convexity only in the lower-level variables of the lower-level program. We present an innovative single-level difference of weakly convex reformulation based on the Moreau envelope of the lower-level problem. We further develop a sequentially convergent Inexact Proximal Difference of Weakly Convex Algorithm (iP-DwCA). To evaluate the effectiveness of the proposed iP-DwCA, we conduct numerical experiments focused on tuning hyperparameters for kernel support vector machines on simulated data.