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

Sketched Ridgeless Linear Regression: The Role of Downsampling