  We establish precise structural and risk equivalences between subsampling and
ridge regularization for ensemble ridge estimators. Specifically, we prove that
linear and quadratic functionals of subsample ridge estimators, when fitted
with different ridge regularization levels $\lambda$ and subsample aspect
ratios $\psi$, are asymptotically equivalent along specific paths in the
$(\lambda,\psi)$-plane (where $\psi$ is the ratio of the feature dimension to
the subsample size). Our results only require bounded moment assumptions on
feature and response distributions and allow for arbitrary joint distributions.
Furthermore, we provide a data-dependent method to determine the equivalent
paths of $(\lambda,\psi)$. An indirect implication of our equivalences is that
optimally tuned ridge regression exhibits a monotonic prediction risk in the
data aspect ratio. This resolves a recent open problem raised by Nakkiran et
al. for general data distributions under proportional asymptotics, assuming a
mild regularity condition that maintains regression hardness through linearized
signal-to-noise ratios.
