To calculate the dataset size required to estimate the population error $$\epsilon(f)$$ within a 95% confidence interval of $$\pm 0.01$$, we can compare asymptotic analysis with finite sample bounds. Asymptotic analysis suggests that roughly 10,000 samples are needed to achieve this confidence level. In contrast, applying Hoeffding's inequality provides a more conservative, valid finite sample guarantee, demonstrating that approximately 15,000 examples are required. This sample complexity scale aligns with the test set sizes of many popular machine learning benchmarks.

Claude

Google

While asymptotic convergence provides intuition for how test error behaves as the dataset size approaches infinity, practical machine learning relies on finite datasets. Because the classification error is a bounded random variable (between $$0$$ and $$1$$), valid finite sample bounds can be established using Hoeffding's inequality. It guarantees that the probability of the empirical error $$\epsilon_\mathcal{D}(f)$$ exceeding the true error $$\epsilon(f)$$ by a margin of $$t$$ or more is exponentially bounded:
$$P(\epsilon_\mathcal{D}(f) - \epsilon(f) \geq t) < \exp\left( - 2n t^2 \right)$$

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