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In this work we establish an algorithm and distribution independentnon-asymptotic trade-off between the model size, excess test loss, and trainingloss of linear predictors. Specifically, we show that models that perform wellon the test data (have low excess loss) are either "classical" -- have trainingloss close to the noise level, or are "modern" -- have a much larger number ofparameters compared to the minimum needed to fit the training data exactly.  We also provide a more precise asymptotic analysis when the limiting spectraldistribution of the whitened features is Marchenko-Pastur. Remarkably, whilethe Marchenko-Pastur analysis is far more precise near the interpolation peak,where the number of parameters is just enough to fit the training data, itcoincides exactly with the distribution independent bound as the level ofoverparametrization increases.

A Universal Trade-off Between the Model Size, Test Loss, and Training Loss of Linear Predictors