AdaBoost training error bound (${\hat{R}}_{01}\le {\prod }_{m}2\sqrt{{ϵ}_m(1-{ϵ}_m)}\le e^{-2{\sum }_m{\gamma }_m^2}$)

For AdaBoost with initial uniform weights, labels and base predictions in $\{-1,1\}$, and edges ${\gamma }_m=1/2-{ϵ}_m>0$, the training exponential classification risk equals the product of stage normalizers. A wrongly classified point has exponential loss at least one, so empirical classification error is bounded by this product. The final inequality uses $\sqrt{1-4{\gamma }_m^2}\le e^{-2{\gamma }_m^2}$. The result concerns training error, not a guarantee against overfitting or mislabeled observations.

// Target: probability-and-statistics.bigb