Gradient-based bilevel optimization for multi-penalty Ridge regression
  through matrix differential calculus
Gabriele Maroni Loris Cannelli Dario Piga
Common regularization algorithms for linear regression, such as LASSO and
Ridge regression, rely on a regularization hyperparameter that balances the
tradeoff between minimizing the fitting error and the norm of the learned model
coefficients. As this hyperparameter is scalar, it can be easily selected via
random or grid search optimizing a cross-validation criterion. However, using a
scalar hyperparameter limits the algorithm's flexibility and potential for
better generalization. In this paper, we address the problem of linear
regression with l2-regularization, where a different regularization
hyperparameter is associated with each input variable. We optimize these
hyperparameters using a gradient-based approach, wherein the gradient of a
cross-validation criterion with respect to the regularization hyperparameters
is computed analytically through matrix differential calculus. Additionally, we
introduce two strategies tailored for sparse model learning problems aiming at
reducing the risk of overfitting to the validation data. Numerical examples
demonstrate that our multi-hyperparameter regularization approach outperforms
LASSO, Ridge, and Elastic Net regression. Moreover, the analytical computation
of the gradient proves to be more efficient in terms of computational time
compared to automatic differentiation, especially when handling a large number
of input variables. Application to the identification of over-parameterized
Linear Parameter-Varying models is also presented.