Toward Theoretical Guidance for Two Common Questions in Practical Cross-Validation based Hyperparameter Selection

TL;DR

This paper provides theoretical insights into two practical questions in cross-validation hyperparameter tuning: whether to retrain on all data after selection and how to set optimization tolerance, introducing new risks and heuristics.

Contribution

It introduces the hold-in risk and model class mis-specification risk, offering a simple, general theoretical framework and heuristics for hyperparameter selection and optimization in cross-validation.

Findings

Heuristics can outperform or match simple retraining strategies.

Using the heuristics can halve computational costs without sacrificing accuracy.

Theoretical quantities guide practical decisions in hyperparameter tuning.

Abstract

We show, to our knowledge, the first theoretical treatments of two common questions in cross-validation based hyperparameter selection: (1) After selecting the best hyperparameter using a held-out set, we train the final model using {\em all} of the training data -- since this may or may not improve future generalization error, should one do this? (2) During optimization such as via SGD (stochastic gradient descent), we must set the optimization tolerance $\rho$ -- since it trades off predictive accuracy with computation cost, how should one set it? Toward these problems, we introduce the {\em hold-in risk} (the error due to not using the whole training data), and the {\em model class mis-specification risk} (the error due to having chosen the wrong model class) in a theoretical view which is simple, general, and suggests heuristics that can be used when faced with a dataset instance. In proof-of-concept studies in synthetic data where theoretical quantities can be controlled, we show that these heuristics can, respectively, (1) always perform at least as well as always performing retraining or never performing retraining, (2) either improve performance or reduce computational overhead by $2\times$ with no loss in predictive performance.