= Exponential-loss derivation of AdaBoost
{title2=$\alpha_m=\tfrac12\log[(1-\epsilon_m)/\epsilon_m]$}
At score $F$, the <exponential classification risk> weights observation $i$ proportionally to $e^{-y_iF(x_i)}$. A stage $\alpha G$ multiplies this weight by $e^{-\alpha y_iG(x_i)}$. For weighted error $\epsilon$, the loss multiplier is $(1-\epsilon)e^{-\alpha}+\epsilon e^\alpha$, minimized at the displayed value. Its minimum is $2\sqrt{\epsilon(1-\epsilon)}$. Thus errors gain weight relative to correct observations in the next stage.
// Target: foundations-of-mathematics.bigb
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