Fast Computation of Leave-One-Out Cross-Validation for $k$-NN Regression
This provides a significant speed-up for model selection in k-NN regression, though it is incremental as it builds on existing methods with a specific optimization.
The paper tackles the computational inefficiency of leave-one-out cross-validation (LOOCV) for k-nearest neighbors regression by proving that the LOOCV estimate equals the mean square error of (k+1)-NN regression on training data scaled by a factor, enabling fast computation with a single model fit.
We describe a fast computation method for leave-one-out cross-validation (LOOCV) for $k$-nearest neighbours ($k$-NN) regression. We show that, under a tie-breaking condition for nearest neighbours, the LOOCV estimate of the mean square error for $k$-NN regression is identical to the mean square error of $(k+1)$-NN regression evaluated on the training data, multiplied by the scaling factor $(k+1)^2/k^2$. Therefore, to compute the LOOCV score, one only needs to fit $(k+1)$-NN regression only once, and does not need to repeat training-validation of $k$-NN regression for the number of training data. Numerical experiments confirm the validity of the fast computation method.