In a classification problem of K classes (i.e., f!1,!2, ,!K g), assume that we use an ensemble model for each class !k (for all k = 1, 2, , K) as follows:

Fm¹x; !k º = f1¹x; !k º + f2¹x; !k º + + fm¹x; !k º, where each base model fm¹x;!k º is a regression tree. Derive the gradient-tree-boosting procedure to estimate the ensemble models for all K classes by minimizing the following cross-entropy loss functional:

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eF(x, y) 1(F(x), y) = -In K k=1 eF(x;wk) (y = {w1, W2, WK}). ,