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An important unresolved challenge in the theory of regularization is to setthe regularization coefficients of popular techniques like the ElasticNet withgeneral provable guarantees. We consider the problem of tuning theregularization parameters of Ridge regression, LASSO, and the ElasticNet acrossmultiple problem instances, a setting that encompasses both cross-validationand multi-task hyperparameter optimization. We obtain a novel structural resultfor the ElasticNet which characterizes the loss as a function of the tuningparameters as a piecewise-rational function with algebraic boundaries. We usethis to bound the structural complexity of the regularized loss functions andshow generalization guarantees for tuning the ElasticNet regressioncoefficients in the statistical setting. We also consider the more challengingonline learning setting, where we show vanishing average expected regretrelative to the optimal parameter pair. We further extend our results to tuningclassification algorithms obtained by thresholding regression fits regularizedby Ridge, LASSO, or ElasticNet. Our results are the first generallearning-theoretic guarantees for this important class of problems that avoidstrong assumptions on the data distribution. Furthermore, our guarantees holdfor both validation and popular information criterion objectives.

Provably tuning the ElasticNet across instances