Sparse Approximate Cross-Validation for High-Dimensional GLMs

Leave-one-out cross validation (LOOCV) can be particularly accurate among CV variants for estimating out-of-sample error. Unfortunately, LOOCV requires re-fitting a model $N$ times for a dataset of size $N$. To avoid this prohibitive computational expense, a number of authors have proposed approximations to LOOCV. These approximations work well when the unknown parameter is of small, fixed dimension but suffer in high dimensions; they incur a running time roughly cubic in the dimension, and, in fact, we show their accuracy significantly deteriorates in high dimensions. We demonstrate that these difficulties can be surmounted in $\ell_1$-regularized generalized linear models when we assume that the unknown parameter, while high dimensional, has a small support. In particular, we show that, under interpretable conditions, the support of the recovered parameter does not change as each datapoint is left out. This result implies that the previously proposed heuristic of only approximating CV along the support of the recovered parameter has running time and error that scale with the (small) support size even when the full dimension is large. Experiments on synthetic and real data support the accuracy of our approximations.