Optimal ridge regularization revisited

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where $i)$ we let the estimator's regularization parameter grow proportionally to the sample size; and $ii)$ we treat the population regression coefficients as \emph{local} to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.

A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. In this paper, we analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal distributional assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with any learning rate schedule and finite time horizon, the early stopped solution is equivalent to the minimum norm solution for a generalized ridge regression problem. We also prove that early stopping is beneficial for generic data with arbitrary spectrum and for a wide variety of learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate.

Given a high-dimensional covariate matrix and a response vector, ridge-regularized sparse linear regression selects a subset of features that explains the relationship between covariates and the response in an interpretable manner. To choose hyperparameters that control the sparsity level and amount of regularization, practitioners commonly use k-fold cross-validation. However, cross-validation substantially increases the computational cost of sparse regression as it requires solving many mixed-integer optimization problems (MIOs) for each hyperparameter combination. To address this computational burden, we derive computationally tractable relaxations of the k-fold cross-validation loss, facilitating hyperparameter selection while solving $50$--$80\%$ fewer MIOs in practice. Our computational results demonstrate, across eleven real-world UCI datasets, that exact MIO-based cross-validation can be competitive with mature software packages such as glmnet and L0Learn.