From 15c6172e09195ec1c12f0b9ca4de4ae98f703e43 Mon Sep 17 00:00:00 2001 From: Valentin Waeselynck Date: Wed, 18 Sep 2024 20:33:49 +0200 Subject: [PATCH] [doc] Improve the model introduction. (#10822) --- doc/tutorials/model.rst | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/doc/tutorials/model.rst b/doc/tutorials/model.rst index aa9e2c1a1..97171fc3c 100644 --- a/doc/tutorials/model.rst +++ b/doc/tutorials/model.rst @@ -3,7 +3,7 @@ Introduction to Boosted Trees ############################# XGBoost stands for "Extreme Gradient Boosting", where the term "Gradient Boosting" originates from the paper *Greedy Function Approximation: A Gradient Boosting Machine*, by Friedman. -The **gradient boosted trees** has been around for a while, and there are a lot of materials on the topic. +The term **gradient boosted trees** has been around for a while, and there are a lot of materials on the topic. This tutorial will explain boosted trees in a self-contained and principled way using the elements of supervised learning. We think this explanation is cleaner, more formal, and motivates the model formulation used in XGBoost. @@ -119,13 +119,16 @@ Let the following be the objective function (remember it always needs to contain .. math:: - \text{obj} = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t)}) + \sum_{i=1}^t\omega(f_i) + \text{obj} = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t)}) + \sum_{k=1}^t\omega(f_k) + +in which :math:`t` is the number of trees in our ensemble. +(Each training step will add one new tree, so that at step :math:`t` the ensemble contains :math:`K=t` trees). Additive Training ================= The first question we want to ask: what are the **parameters** of trees? -You can find that what we need to learn are those functions :math:`f_i`, each containing the structure +You can find that what we need to learn are those functions :math:`f_k`, each containing the structure of the tree and the leaf scores. Learning tree structure is much harder than traditional optimization problem where you can simply take the gradient. It is intractable to learn all the trees at once. Instead, we use an additive strategy: fix what we have learned, and add one new tree at a time. @@ -150,7 +153,7 @@ If we consider using mean squared error (MSE) as our loss function, the objectiv .. math:: - \text{obj}^{(t)} & = \sum_{i=1}^n (y_i - (\hat{y}_i^{(t-1)} + f_t(x_i)))^2 + \sum_{i=1}^t\omega(f_i) \\ + \text{obj}^{(t)} & = \sum_{i=1}^n (y_i - (\hat{y}_i^{(t-1)} + f_t(x_i)))^2 + \sum_{k=1}^t\omega(f_k) \\ & = \sum_{i=1}^n [2(\hat{y}_i^{(t-1)} - y_i)f_t(x_i) + f_t(x_i)^2] + \omega(f_t) + \mathrm{constant} The form of MSE is friendly, with a first order term (usually called the residual) and a quadratic term.