This is the loss function of XGBoost.

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This is the Second-order approximation of the loss function. enter image description here


\begin{equation} L^{(t)} \text{: cross entropy loss function.} \end{equation}

\begin{equation} y_i \text{: label. Marked as y, as a matter of convenience.} \end{equation}

\begin{equation} \hat{y_i}^{(t-1)} \text{: probability of previous tree. Marked as p, as a matter of convenience.} \end{equation}

\begin{equation} f_t(x_i) \text{: output value of the tree } \end{equation}

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\begin{equation} g_i = \frac{ ∂l(y, p) }{∂p} = \frac{p-y}{p(p-1)} \end{equation}

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\begin{equation} h_i = \frac{ ∂l(y, p) }{∂^2p} = \frac{\left(y-p\right)p+y\left(p-1\right)}{\left(p-1\right)^2p^2} \end{equation}

Why does everyone in the gi and hi calculation do :

\begin{equation} g_i = \frac{ ∂l(y, p) }{∂x} = p-y \end{equation}

\begin{equation} h_i = \frac{ ∂l(y, p) }{∂^2x} = p(1-p) \end{equation}

NOTE: the x is:

\begin{equation} x = ln(\frac{p}{1 - p}) \end{equation}

and the yi hat (t-1) is sigmoid function:

\begin{equation} p = \frac{e^x}{1+e^x} = \frac{1}{1+e^{-x}} \end{equation}

Why are the derivatives are with respect to x (or y_hat in the code) instead of p? \begin{equation} g_i = \frac{ ∂l(y, p)}{∂p} => \frac{ ∂l(y, p) }{∂x} \end{equation}


1 Answer 1


Since you mention probabilities, I assume you are thinking about a binary classification, in which case the trees all operate in the log-odds space, the $x$s in your notation. So the $\hat{y}$ and $f_t$ are actually log-odds not probabilities.

See also What is the "binary:logistic" objective function in XGBoost?

  • $\begingroup$ Thanks for your reply and help. Now that XGBoost paper says: $$gi=\frac{∂l(y,p)}{∂p}$$, how can we change p to x: $$gi=\frac{∂l(y,p)}{∂x}$$ It's totally not the same equation. $\endgroup$
    – ChrisChu
    May 10, 2022 at 8:52
  • $\begingroup$ What paper do you mean? $\endgroup$ May 10, 2022 at 12:49
  • $\begingroup$ XGBoost original thesis. arxiv.org/abs/1603.02754 I can understand now, thanks a lot. $\endgroup$
    – ChrisChu
    May 11, 2022 at 3:04

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