Output value of a gradient boosting decision tree node that has just a single example in it

The general gradient boosting algorithm for tree-based classifiers is as follows:

Input: training set $$\{(x_{i},y_{i})\}_{i=1}^{n}$$, a differentiable loss function $$L(y,F(x))$$, and a number of iterations $$M$$.

Algorithm:

1. Initialize model with a constant value: $$\displaystyle F_{0}(x)={\underset {\gamma }{\arg \min }}\sum _{i=1}^{n}L(y_{i},\gamma )$$.

2. For m = 1 to M:

1. Compute so-called pseudo-residuals: $$r_{im}=-\left[{\frac {\partial L(y_{i},F(x_{i}))}{\partial F(x_{i})}}\right]_{F(x)=F_{m-1}(x)}\quad {\mbox{for }}i=1,\ldots ,n.$$
2. Fit a base learner (or weak learner, e.g. tree) $$\displaystyle h_{m}(x)$$ to pseudo-residuals, i.e. train it using the training set $$\{(x_{i},r_{im})\}_{i=1}^{n}$$.
3. Compute multipliers and solve for the output values of each leaf. $$\gamma$$ is the output value of a leaf here: $$F_{m}(x)=F_{m-1}(x)+\sum _{j=1}^{J_{m}}\gamma _{jm}\mathbf {1} _{R_{jm}}(x),\quad \gamma _{jm}={\underset {\gamma }{\operatorname {arg\,min} }}\sum _{x_{i}\in R_{jm}}L(y_{i},F_{m-1}(x_{i})+\gamma ).$$
3. Output $$F_{M}(x)$$.

Now the loss function that they input is log-loss: $$\sum_i −(y_ilog(p_i)+(1−y_i)log(1−p_i))$$ where $$i$$ is counter over all the samples in training set. The output of a tree leaf in the algorithm is assumed to be log-odds of samples belonging to the positive class.

In reality, how the algorithm is executed is - The $$\gamma _{jm}$$ is computed by expanding the Loss function in Taylor's series of degree 2 and then differentiating that w.r.t $$\gamma$$ .

I will write the formulation for the case when there is just one sample ($$x1$$) in a terminal node.

$$L(y_1,F_{m-1}(x1)+\gamma)=L(y_1,F_{m-1}(x1)) + \frac{\partial L(y_1,F_{m-1}(x1))}{\partial F_{m-1}}\gamma + \frac{1}{2}\frac{\partial^2 L(y_1,F_{m-1}(x1))}{\partial F_{m-1}^2}\gamma^2$$

Now, this formulation is differentiated w.r.t $$\gamma$$ and set equal to 0 to get the optimal value of gamma

$$\gamma = -\frac{\frac{\partial L(y_1,F_{m-1}(x1))}{\partial F_{m-1}}}{\frac{\partial^2 L(y_1,F_{m-1}(x1))}{\partial F_{m-1}^2}}$$

This computes to $$Residual_1/P_0(1-P_0) \tag 1$$ where $$P_0$$ is the most recent predicted probability.

The example below is taken from this youtube video - linked to exact time 16:43

For an example case where $$F_0(X1) = 0.69$$ (these are log-odds)

(converting the log odds to probability) $$P_0 = 0.67; Residual_1 = y_1-P_0 = 0.33$$

Now they compute $$\gamma$$ (the output value) for the leaf containing $$x1$$ by plugging these values into the equation (1) and get the $$\gamma$$ to be equal to $$\frac{0.33}{(0.67)(0.33)} = 1.49$$

Questions:

1. How can we approximate the loss function $$L$$ for each gamma. We need to know radius of convergence around that $$\gamma$$ I think and then the $$\gamma$$ thus computed at the end should have lied in that radius of convergence?
2. The log-loss (binary cross-entropy) is monotonic in predicted probability value. It increases with $$P$$ if $$y = 1$$ and decreases if $$y = 0$$ and thus it is also monotonic in log-odds since log-odds and probability are monotonic w.r.t each other.In that light why is $$\gamma$$ not equal to as-high-as-numerical-stability-permits when $$y=1$$ and $$0$$ when $$y=0$$?

For Q2, the fact that the selected $$\gamma^*$$ is not infinite is because of the Taylor approximation. And in this context, that's actually a very useful thing to have, to prevent overfitting! Again have a look at the answer to the second question linked above, it has some additional information about other possible descent options.