some notes on generalization bounds in machine learning with an emphasis on stability

1.13. generalization#

1.13.1. basics#

Risk: \(R\left(A_{S}\right)=\mathbb{E}_{(X, Y) \sim P} \ell\left(A_{S}(X), Y\right)\)
Empirical risk: \(R_{\mathrm{emp}}\left(A_{S}\right)=\frac{1}{n} \sum_{i=1}^{n} \ell\left(A_{S}\left(X_{i}\right), Y_{i}\right)\)
Generalization error: \(R\left(A_{S}\right)-R_{\mathrm{emp}}\left(A_{S}\right)\)

Sharper Bounds for Uniformly Stable Algorithms
- uniformly stable with prameter \(\gamma\) if \(\left|\ell\left(A_{S}(x), y\right)-\ell\left(A_{S^{i}}(x), y\right)\right| \leq \gamma\)
  - where \(A_{S^i}\) can alter one point in the training data
- stability approach can provide bounds even when empirical risk is 0 (e.g. 1-nearest-neighbor, devroye & wagner 1979)
- \(n\underbrace{\left(R\left(A_{S}\right)-R_{\mathrm{emp}}\left(A_{S}\right)\right)}_{\text{generalization error}} \lesssim(n \sqrt{n} \gamma+L \sqrt{n}) \sqrt{\log \left(\frac{1}{\delta}\right)}\) (bousegeuet & elisseef, 2002)