Gain Confidence, Reduce Disappointment: A New Approach to Cross-Validation for Sparse Regression

Ridge regularized sparse linear regression involves selecting a subset of features that explains the relationship between a high-dimensional design matrix and an output vector in an interpretable manner. To select the sparsity and robustness of linear regressors, techniques like leave-one-out cross-validation are commonly used for hyperparameter tuning. However, cross-validation typically increases the cost of sparse regression by several orders of magnitude, because it requires solving multiple mixed-integer optimization problems (MIOs) for each hyperparameter combination. Additionally, validation metrics are noisy estimators of the test-set error, with different hyperparameter combinations leading to models with different amounts of noise. Therefore, optimizing over these metrics is vulnerable to out-of-sample disappointment, especially in underdetermined settings. To address this state of affairs, we make two contributions. First, we leverage the generalization theory literature to propose confidence-adjusted variants of the leave-one-out error that display less propensity to out-of-sample disappointment. Second, we leverage ideas from the mixed-integer optimization literature to obtain computationally tractable relaxations of the confidence-adjusted leave-one-out error, thereby minimizing it without solving as many MIOs. Our relaxations give rise to an efficient cyclic coordinate descent scheme which allows us to obtain significantly lower leave-one-out errors than via other methods in the literature. We validate our theory by demonstrating that we obtain significantly sparser and comparably accurate solutions than via popular methods like GLMNet and suffer from less out-of-sample disappointment. On synthetic datasets, our confidence adjustment procedure generates significantly fewer false discoveries, and improves out-of-sample performance by 2%–5% compared to cross-validating without confidence adjustment. Across a suite of 13 real datasets, a calibrated version of our confidence adjustment improves the test set error by an average of 4% compared to cross-validating without confidence adjustment.



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