  Overparametrization often helps improve the generalization performance. This
paper presents a dual view of overparametrization suggesting that downsampling
may also help generalize. Focusing on the proportional regime $m\asymp n \asymp
p$, where $m$ represents the sketching size, $n$ is the sample size, and $p$ is
the feature dimensionality, we investigate two out-of-sample prediction risks
of the sketched ridgeless least square estimator. Our findings challenge
conventional beliefs by showing that downsampling does not always harm
generalization but can actually improve it in certain cases. We identify the
optimal sketching size that minimizes out-of-sample prediction risks and
demonstrate that the optimally sketched estimator exhibits stabler risk curves,
eliminating the peaks of those for the full-sample estimator. To facilitate
practical implementation, we propose an empirical procedure to determine the
optimal sketching size. Finally, we extend our analysis to cover central limit
theorems and misspecified models. Numerical studies strongly support our
theory.
