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When training a model a train, a validation and test set are used. I was wondering if there is any paper or example that proves that the use of an independent validation set increase the performance of the lasso estimator. I am particularly interested in situations where the penalty value is chosen through cross validation

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The independent validation is not supposed to increase the performance: it is supposed to measure the performance of the final model (and detect/monitor the optimistic bias introduced to model selection during data-driven optimization such as cross validating for the "optimal" penalty).

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