  Recent empirical and theoretical analyses of several commonly used prediction
procedures reveal a peculiar risk behavior in high dimensions, referred to as
double/multiple descent, in which the asymptotic risk is a non-monotonic
function of the limiting aspect ratio of the number of features or parameters
to the sample size. To mitigate this undesirable behavior, we develop a general
framework for risk monotonization based on cross-validation that takes as input
a generic prediction procedure and returns a modified procedure whose
out-of-sample prediction risk is, asymptotically, monotonic in the limiting
aspect ratio. As part of our framework, we propose two data-driven
methodologies, namely zero- and one-step, that are akin to bagging and
boosting, respectively, and show that, under very mild assumptions, they
provably achieve monotonic asymptotic risk behavior. Our results are applicable
to a broad variety of prediction procedures and loss functions, and do not
require a well-specified (parametric) model. We exemplify our framework with
concrete analyses of the minimum $\ell_2$, $\ell_1$-norm least squares
prediction procedures. As one of the ingredients in our analysis, we also
derive novel additive and multiplicative forms of oracle risk inequalities for
split cross-validation that are of independent interest.
