  Overparameterization in deep learning is powerful: Very large models fit the
training data perfectly and yet often generalize well. This realization brought
back the study of linear models for regression, including ordinary least
squares (OLS), which, like deep learning, shows a "double-descent" behavior:
(1) The risk (expected out-of-sample prediction error) can grow arbitrarily
when the number of parameters $p$ approaches the number of samples $n$, and (2)
the risk decreases with $p$ for $p>n$, sometimes achieving a lower value than
the lowest risk for $p<n$. The divergence of the risk for OLS can be avoided
with regularization. In this work, we show that for some data models it can
also be avoided with a PCA-based dimensionality reduction (PCA-OLS, also known
as principal component regression). We provide non-asymptotic bounds for the
risk of PCA-OLS by considering the alignments of the population and empirical
principal components. We show that dimensionality reduction improves robustness
while OLS is arbitrarily susceptible to adversarial attacks, particularly in
the overparameterized regime. We compare PCA-OLS theoretically and empirically
with a wide range of projection-based methods, including random projections,
partial least squares (PLS), and certain classes of linear two-layer neural
networks. These comparisons are made for different data generation models to
assess the sensitivity to signal-to-noise and the alignment of regression
coefficients with the features. We find that methods in which the projection
depends on the training data can outperform methods where the projections are
chosen independently of the training data, even those with oracle knowledge of
population quantities, another seemingly paradoxical phenomenon that has been
identified previously. This suggests that overparameterization may not be
necessary for good generalization.
