 Exponential-loss derivation of AdaBoost

At score $F$, the exponential classification risk weights observation $i$ proportionally to $e^{-y_{i}F(x_i)}$. A stage $\alpha G$ multiplies this weight by $e^{-\alpha y_{i}G(x_i)}$. For weighted error $ϵ$, the loss multiplier is $(1-ϵ)e^{-\alpha }+ϵe^{\alpha }$, minimized at the displayed value. Its minimum is $2\sqrt{ϵ(1-ϵ)}$. Thus errors gain weight relative to correct observations in the next stage.

// Target: foundations-of-mathematics.bigb