  Models like LASSO and ridge regression are extensively used in practice due
to their interpretability, ease of use, and strong theoretical guarantees.
Cross-validation (CV) is widely used for hyperparameter tuning in these models,
but do practical optimization methods minimize the true out-of-sample loss? A
recent line of research promises to show that the optimum of the CV loss
matches the optimum of the out-of-sample loss (possibly after simple
corrections). It remains to show how tractable it is to minimize the CV loss.
In the present paper, we show that, in the case of ridge regression, the CV
loss may fail to be quasiconvex and thus may have multiple local optima. We can
guarantee that the CV loss is quasiconvex in at least one case: when the
spectrum of the covariate matrix is nearly flat and the noise in the observed
responses is not too high. More generally, we show that quasiconvexity status
is independent of many properties of the observed data (response norm,
covariate-matrix right singular vectors and singular-value scaling) and has a
complex dependence on the few that remain. We empirically confirm our theory
using simulated experiments.
