From 07182444d26a77597a5f6ec9001c34d4cdf6749c Mon Sep 17 00:00:00 2001 From: Tong He Date: Thu, 20 Aug 2015 13:53:20 -0700 Subject: [PATCH 1/8] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 299524758..121462d38 100644 --- a/README.md +++ b/README.md @@ -30,7 +30,7 @@ What's New ---------- * XGBoost helps Chenglong Chen to win [Kaggle CrowdFlower Competition](https://www.kaggle.com/c/crowdflower-search-relevance) - Check out the [winning solution](doc/README.md#highlight-links) + Check out the [winning solution](https://github.com/ChenglongChen/Kaggle_CrowdFlower) * XGBoost-0.4 release, see [CHANGES.md](CHANGES.md#xgboost-04) * XGBoost helps three champion teams to win [WWW2015 Microsoft Malware Classification Challenge (BIG 2015)](http://www.kaggle.com/c/malware-classification/forums/t/13490/say-no-to-overfitting-approaches-sharing) Check out the [winning solution](doc/README.md#highlight-links) From 10273a0288854184443c34d8936a6fb4742287ab Mon Sep 17 00:00:00 2001 From: VGuette Date: Sun, 23 Aug 2015 11:01:43 +0200 Subject: [PATCH 2/8] Update setup.py --- python-package/setup.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/python-package/setup.py b/python-package/setup.py index a446983be..d8c0df0ee 100644 --- a/python-package/setup.py +++ b/python-package/setup.py @@ -10,7 +10,7 @@ sys.path.insert(0, '.') build_sh = subprocess.Popen(['sh', 'xgboost/build-python.sh']) build_sh.wait() output = build_sh.communicate() -print output +print (output) import xgboost From 00702dc39b18ec7b74d3b7e8bf1a55b2dee75126 Mon Sep 17 00:00:00 2001 From: sinhrks Date: Sun, 23 Aug 2015 13:37:29 +0900 Subject: [PATCH 3/8] Fix for python 3 --- .travis.yml | 4 +--- python-package/xgboost/plotting.py | 8 ++------ scripts/travis_osx_install.sh | 8 +++++++- scripts/travis_script.sh | 21 ++++++++++++++++++++- tests/python/test_basic.py | 3 +++ 5 files changed, 33 insertions(+), 11 deletions(-) diff --git a/.travis.yml b/.travis.yml index 10f217038..5e10c3360 100644 --- a/.travis.yml +++ b/.travis.yml @@ -12,6 +12,7 @@ env: - TASK=lint LINT_LANG=python - TASK=R-package CXX=g++ - TASK=python-package CXX=g++ + - TASK=python-package3 CXX=g++ - TASK=java-package CXX=g++ - TASK=build CXX=g++ - TASK=build-with-dmlc CXX=g++ @@ -29,9 +30,6 @@ addons: - wget - libcurl4-openssl-dev - unzip - - python-numpy - - python-scipy - - python-nose before_install: - scripts/travis_osx_install.sh diff --git a/python-package/xgboost/plotting.py b/python-package/xgboost/plotting.py index 7c34a11f1..9c9b2a97d 100644 --- a/python-package/xgboost/plotting.py +++ b/python-package/xgboost/plotting.py @@ -8,11 +8,7 @@ import re import numpy as np from .core import Booster -try: - from StringIO import StringIO -except ImportError: - from io import StringIO - +from io import BytesIO def plot_importance(booster, ax=None, height=0.2, xlim=None, title='Feature importance', @@ -217,7 +213,7 @@ def plot_tree(booster, num_trees=0, rankdir='UT', ax=None, **kwargs): g = to_graphviz(booster, num_trees=num_trees, rankdir=rankdir, **kwargs) - s = StringIO() + s = BytesIO() s.write(g.pipe(format='png')) s.seek(0) img = image.imread(s) diff --git a/scripts/travis_osx_install.sh b/scripts/travis_osx_install.sh index 9f3f6e831..adc620a52 100755 --- a/scripts/travis_osx_install.sh +++ b/scripts/travis_osx_install.sh @@ -9,5 +9,11 @@ brew update if [ ${TASK} == "python-package" ]; then brew install python git graphviz easy_install pip - pip install numpy scipy nose + pip install numpy scipy matplotlib nose +fi + +if [ ${TASK} == "python-package3" ]; then + brew install python3 git graphviz + sudo pip3 install --upgrade setuptools + pip3 install numpy scipy matplotlib nose graphviz fi diff --git a/scripts/travis_script.sh b/scripts/travis_script.sh index d58eaa5de..c5708b0c8 100755 --- a/scripts/travis_script.sh +++ b/scripts/travis_script.sh @@ -35,11 +35,30 @@ fi if [ ${TASK} == "python-package" ]; then sudo apt-get install graphviz - sudo pip install matplotlib graphviz + sudo apt-get install python-numpy python-scipy python-matplotlib python-nose + sudo python -m pip install graphviz make all CXX=${CXX} || exit -1 nosetests tests/python || exit -1 fi +if [ ${TASK} == "python-package3" ]; then + sudo apt-get install graphviz + # python3-matplotlib is unavailale on Ubuntu 12.04 + sudo apt-get install python3-dev + sudo apt-get install python3-numpy python3-scipy python3-nose python3-setuptools + + make all CXX=${CXX} || exit -1 + + if [ ${TRAVIS_OS_NAME} != "osx" ]; then + sudo easy_install3 pip + sudo easy_install3 -U distribute + sudo pip install graphviz matplotlib + nosetests3 tests/python || exit -1 + else + nosetests tests/python || exit -1 + fi +fi + # only test java under linux for now if [ ${TASK} == "java-package" ]; then if [ ${TRAVIS_OS_NAME} != "osx" ]; then diff --git a/tests/python/test_basic.py b/tests/python/test_basic.py index 2ed1cc462..93ebaa7fd 100644 --- a/tests/python/test_basic.py +++ b/tests/python/test_basic.py @@ -33,6 +33,9 @@ def test_plotting(): bst2 = xgb.Booster(model_file='xgb.model') # plotting + import matplotlib + matplotlib.use('Agg') + from matplotlib.axes import Axes from graphviz import Digraph From d5d48560a7f400c5baa3ea30e0ce11ba8aa44e2e Mon Sep 17 00:00:00 2001 From: hetong007 Date: Sun, 23 Aug 2015 16:25:28 -0700 Subject: [PATCH 4/8] add model description --- doc/model.md | 188 +++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 188 insertions(+) create mode 100644 doc/model.md diff --git a/doc/model.md b/doc/model.md new file mode 100644 index 000000000..1a67a6962 --- /dev/null +++ b/doc/model.md @@ -0,0 +1,188 @@ +Introduction to the Model of XGBoost +========================= + +## The Origin + +XGBoost is short for "Extreme Gradient Boosting", where the term "Gradient Boosting" is proposed in the paper _Greedy Function Approximation: A Gradient Boosting Machine_, Friedman. Based on this original model, we incoporated several modifications to make it faster and more robust. + +## The General Problem + +### Supervised Model + +XGBoost is used for supervised learning problems, where we use the training data ``$ x_i $`` to predict a target variable ``$ y_i $``. Our model is a mathematical structure that captures the pattern from the training data. Given the structure, we need to learn the best parameters ``$ \Theta $`` in the model. + +### Loss Function + +Based on different understanding or assumption of ``$ y_i $``, we can have different problems as regression, classification, ordering, etc. To model different problems, we use a so-called `loss function` to describe how good is our model's performance. The function usually takes two parameters: the true value ``$ y_i $`` and the prediction ``$ \hat{y}_i $``. For example, we can use Rooted Mean Squared Error (RMSE) + +`` `math +l(y_i, \hat{y}_i) = (y_i-\hat{y}_i)^2 +`` ` + +for a regression problem, and logistic loss function + +`` `math +l(y_i, \hat{y}_i) = y_i\ln (1+e^{-\hat{y}_i}) + (1-y_i)\ln (1+e^{\hat{y}_i}) +`` ` + +for a classification problem. + +### Regularization + +Besides we need to control the complexity of our model. A model achieving a perfect loss function score on the training dataset is overfitting it, which means it not only captures the useful pattern, but also the outliers, noise and the specific pattern in the training data. Controlling the complexity can make the model focus on more important and general pattern rather than the unnecessary details. + +### Optimize the Objective + +Combining the loss function and the regularization, we have our objective for the supervised learning model as + +`` `math +Obj(\Theta) = L(\Theta) + \Omega(\Theta) +`` ` + +where ``$ L $`` is the loss function, and ``$ \Omega $`` is the regularization term. The first one is making our model being accurate, while the second one is preventing our model being overfitting. We want to have a balance between these two parts when optimizing the objective. The optimization algorithm depends on the structure of our model. The following content will introduce the details. + +## Boosting Trees Model + +### Classification and Regression Tree + +The boosting trees model is a set of classification and regression trees. Here's a simple example of such a model: + +![CART]() + +We classify the members in thie family into different leaves, and assign them the score on corresponding leaf. + +### Tree Ensemble + +However a single CART model is not so strong in practice. How about predict with more trees? + +![TwoCART]() + +Now we are predicting with two trees, by predict on each tree individually and then sum the scores up. Mathematically, we can write our model into the form + +`` `math +\hat{y}_i = \sum_{k=1}^K f_k(x_i), f_k \in F +`` ` + +where ``$ f $`` is a function in the functional space ``$ F $``, and ``$ F $`` is the set of all possible CARTs. Therefore our objective to optimize can be written as + +`` `math +obj(\Theta) = \sum_i^n l(y_i, \hat{y}_i) + \sum_{k=1}^K \Omega(f_k) +`` ` + +### Additive Training + +It is not easy to train all the trees at once. Instead, we use the strategy to train them in a sequence so that everytime we train one CART and add it to the model. We note the prediction value at step `t` by ``$ \hat{y}_i^{(t)}$``, so we have + +`` `math +\hat{y}_i^{(0)} = 0\\ +\hat{y}_i^{(1)} = f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i)\\ +\hat{y}_i^{(2)} = f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i)\\ +\dots\\ +\hat{y}_i^{(t)} = \sum_{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i) +`` ` + +Which CART do we want at each step? Of course we want to add the one that minimize our objective. + +`` `math +Obj^{(t)} & = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t)}) + \sum_{i=1}^t\Omega(f_i) \\ + & = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t-1)} + f_t(x_i)) + \Omega(f_t) + constant +`` ` + +Let's consider using RMSE as our loss function + +`` `math +Obj^{(t)} & = \sum_{i=1}^n (y_i - (\hat{y}_i^{(t-1)} + f_t(x_i)))^2 + \sum_{i=1}^t\Omega(f_i) \\ + & = \sum_{i=1}^n [2(\hat{y}_i^{(t-1)} - y_i)f_t(x_i) + f_t(x_i)^2] + \Omega(f_t) + constant +`` ` + +The form of RMSE is friendly. But other loss functions could be tricky to expand. For convenience we calculate the Taylor expansion of the loss function up to the second order + +`` `math +Obj^{(t)} = \sum_{i=1}^n [l(y_i, \hat{y}_i^{(t-1)}) + g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i)] + \Omega(f_t) + constant +`` ` + +where + +`` `math +g_i &= \partial_{\hat{y}_i^{(t)}} l(y_i, \hat{y}_i^{(t-1)})\\ +h_i &= \partial_{\hat{y}_i^{(t)}}^2 l(y_i, \hat{y}_i^{(t-1)}) +`` ` + +So we can remove all the constant at the t-th step and the specific objective is + +`` `math +\sum_{i=1}^n [g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i)] + \Omega(f_t) +`` ` + +One of the benifit of this definition is as long as the loss function has the first and second order derivative, we can optimized every loss function within the same framework. + +### Model Complexity + +We have introduced the details in the loss function, next we talk about the regularization term. We want to control the complexity of a tree, thus we need to define it first. We define a tree ``$ f(x) $`` as + +`` `math +f_t(x) = w_{q(x)}, w\inR^T, q:R^d\rightarrow \{1,2,\cdots,T\} +`` ` + +where ``$ w $`` is the vector of scores on leaves, ``$ q $`` is a function assigning each data point to the corresponding leaf and ``$ T $`` is the number of leaves. In XGBoost, we define the complexity as + +`` `math +\Omega(f) = \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2 +`` ` + +It is possible to define other form of regularization terms, but this one works well in practice. + +### Get the best score on leaf + +Now we have the objective value with the ``$ t $``-th tree added: + +`` `math +Obj^{(t)} &\approx \sum_{i=1}^n [g_i w_q(x_i) + \frac{1}{2} h_i w_{q(x_i)}^2] + \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2\\ +&= \sum^T_{j=1} [(\sum_{i\in I_j} g_i) w_j + \frac{1}{2} (\sum_{i\in I_j} h_i + \lambda) w_j^2 ] + \gamma T +`` ` + +where ``$ I_j = \{i|q(x_i)=j\} $`` is the set of indices of data points assigned to the ``$ j $``-th leaf. Notice that in the second line we have change the index of the summation because all the data points on the same leaf get the same score. We could further compress the expression by defining ``$ G_j = \sum_{i\in I_j} g_i $`` and ``$ H_j = \sum_{i\in I_j} h_i $``: + +`` `math +Obj^{(t)} = \sum^T_{j=1} [G_jw_j + \frac{1}{2} (H_j+\lambda) w_j^2] +\gamma T +`` ` + +In this equation ``$ w_j $`` are independent to each other, the form ``$ G_jw_j+\frac{1}{2}(H_j+\lambda)w_j^2 $`` is quadratic and the best ``$ w_j $`` to minimize it can be solved deterministically: + +`` `math +w_j^\ast = -\frac{G_j}{H_j+\lambda}\\ +Obj = -\frac{1}{2} \sum_{j=1}^T \frac{G_j^2}{H_j+\lambda} + \gamma T +`` ` + +**Therefore, given the parameters, the gradients and the structure of the tree, we know how to set the score on each leaf.** + +### Learn the tree structure + +Our algorithm aims at optimizing the objective, so it also guides us to a good tree structure. We score the structure by ``$ Obj^{(t)} $`` which is mentioned just above. Since we can evaluate the tree, ideally we can enumerate all possible trees and pick the best one. In practice it is impossible, so we enumerate all the trees no deeper than a certain depth greedily. + +Specifically we try to split a leaf into two leaves, and the score it gains is + +`` `math +Gain = \frac{1}{2} [\frac{G_L^2}{H_L+\lambda}+\frac{G_R^2}{H_R+\lambda}-\frac{(G_L+G_R)^2}{H_L+H_R+\lambda}] - \gamma +`` ` + +This formula can be decomposited as 1) the score on the new left leaf 2) the score on the new right leaf 3) The score on the original leaf 4) regularization on the additional leaf. + +The regularization in the end can be seen as the minimum increment from this split. In the end, we will prune out the split with a negative gain. + + + + + + + + + + + + + + + + + From 51964583059f82a442e6065ac5c9d98eca5a981b Mon Sep 17 00:00:00 2001 From: hetong007 Date: Sun, 23 Aug 2015 16:28:24 -0700 Subject: [PATCH 5/8] add plot --- doc/img/cart.png | Bin 0 -> 179620 bytes doc/img/twocart.png | Bin 0 -> 102756 bytes doc/model.md | 4 ++-- 3 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 doc/img/cart.png create mode 100644 doc/img/twocart.png diff --git a/doc/img/cart.png b/doc/img/cart.png new file mode 100644 index 0000000000000000000000000000000000000000..aaaa9ddbec4e35dd97444329f4f21b639e93d714 GIT binary patch literal 179620 zcmV+2Kq9}1P)Px#1ZP1_K>z@;j|==^1poj532;bRa{vG#)c^n?)d2*Yw4DF||D{PpK~#8N?41Qz z9LctT?|t{)mF#A-aTna(-6h1`-EfU7Btb&lHtvGECj_^dVFq``aCaE#p5fKd9RdV` zCLz20Uwu8NtE#Q$^eO3*e|1jq6_`lYufnHfJ#ys8s#UB0^{;<5G&G=I9}yE1Yi(`) z`RAXltgKM4&d%x6r-SVufBXRsv9z>&@ZbRu=FXi9JUu-eQ(} z{qz&M(;INecgT%i@+9wATKX3#0LOk_KXaY6_tj@^XJc5ELKNHM{sa( 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Here's a simple example of such a model: -![CART]() +![CART](img/cart.png) We classify the members in thie family into different leaves, and assign them the score on corresponding leaf. @@ -55,7 +55,7 @@ We classify the members in thie family into different leaves, and assign them th However a single CART model is not so strong in practice. How about predict with more trees? -![TwoCART]() +![TwoCART](img/twocart.png) Now we are predicting with two trees, by predict on each tree individually and then sum the scores up. Mathematically, we can write our model into the form From 30c30d36960f41ab62f2e82c8faf37f2a1733dfa Mon Sep 17 00:00:00 2001 From: hetong007 Date: Sun, 23 Aug 2015 16:56:57 -0700 Subject: [PATCH 6/8] modify model doc --- doc/index.md | 1 + doc/model.md | 70 ++++++++++++++++++++++++++-------------------------- 2 files changed, 36 insertions(+), 35 deletions(-) diff --git a/doc/index.md b/doc/index.md index 7c41d15e2..cdf1b12a2 100644 --- a/doc/index.md +++ b/doc/index.md @@ -52,6 +52,7 @@ User Guide ---------- * [Frequently Asked Questions](faq.md) * [Introduction to Boosted Trees](http://homes.cs.washington.edu/~tqchen/pdf/BoostedTree.pdf) +* [Introduction to the Model of XGBoost](model.md) * [Using XGBoost in Python](python/python_intro.md) * [Using XGBoost in R](../R-package/vignettes/xgboostPresentation.Rmd) * [Learning to use XGBoost by Example](../demo) diff --git a/doc/model.md b/doc/model.md index aa42f7272..d55d262c2 100644 --- a/doc/model.md +++ b/doc/model.md @@ -15,15 +15,15 @@ XGBoost is used for supervised learning problems, where we use the training data Based on different understanding or assumption of ``$ y_i $``, we can have different problems as regression, classification, ordering, etc. To model different problems, we use a so-called `loss function` to describe how good is our model's performance. The function usually takes two parameters: the true value ``$ y_i $`` and the prediction ``$ \hat{y}_i $``. For example, we can use Rooted Mean Squared Error (RMSE) -`` `math +```math l(y_i, \hat{y}_i) = (y_i-\hat{y}_i)^2 -`` ` +``` for a regression problem, and logistic loss function -`` `math +```math l(y_i, \hat{y}_i) = y_i\ln (1+e^{-\hat{y}_i}) + (1-y_i)\ln (1+e^{\hat{y}_i}) -`` ` +``` for a classification problem. @@ -35,9 +35,9 @@ Besides we need to control the complexity of our model. A model achieving a perf Combining the loss function and the regularization, we have our objective for the supervised learning model as -`` `math +```math Obj(\Theta) = L(\Theta) + \Omega(\Theta) -`` ` +``` where ``$ L $`` is the loss function, and ``$ \Omega $`` is the regularization term. The first one is making our model being accurate, while the second one is preventing our model being overfitting. We want to have a balance between these two parts when optimizing the objective. The optimization algorithm depends on the structure of our model. The following content will introduce the details. @@ -59,60 +59,60 @@ However a single CART model is not so strong in practice. How about predict with Now we are predicting with two trees, by predict on each tree individually and then sum the scores up. Mathematically, we can write our model into the form -`` `math +```math \hat{y}_i = \sum_{k=1}^K f_k(x_i), f_k \in F -`` ` +``` where ``$ f $`` is a function in the functional space ``$ F $``, and ``$ F $`` is the set of all possible CARTs. Therefore our objective to optimize can be written as -`` `math +```math obj(\Theta) = \sum_i^n l(y_i, \hat{y}_i) + \sum_{k=1}^K \Omega(f_k) -`` ` +``` ### Additive Training It is not easy to train all the trees at once. Instead, we use the strategy to train them in a sequence so that everytime we train one CART and add it to the model. We note the prediction value at step `t` by ``$ \hat{y}_i^{(t)}$``, so we have -`` `math +```math \hat{y}_i^{(0)} = 0\\ \hat{y}_i^{(1)} = f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i)\\ \hat{y}_i^{(2)} = f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i)\\ \dots\\ \hat{y}_i^{(t)} = \sum_{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i) -`` ` +``` Which CART do we want at each step? Of course we want to add the one that minimize our objective. -`` `math +```math Obj^{(t)} & = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t)}) + \sum_{i=1}^t\Omega(f_i) \\ & = \sum_{i=1}^n l(y_i, \hat{y}_i^{(t-1)} + f_t(x_i)) + \Omega(f_t) + constant -`` ` +``` Let's consider using RMSE as our loss function -`` `math +```math Obj^{(t)} & = \sum_{i=1}^n (y_i - (\hat{y}_i^{(t-1)} + f_t(x_i)))^2 + \sum_{i=1}^t\Omega(f_i) \\ & = \sum_{i=1}^n [2(\hat{y}_i^{(t-1)} - y_i)f_t(x_i) + f_t(x_i)^2] + \Omega(f_t) + constant -`` ` +``` The form of RMSE is friendly. But other loss functions could be tricky to expand. For convenience we calculate the Taylor expansion of the loss function up to the second order -`` `math +```math Obj^{(t)} = \sum_{i=1}^n [l(y_i, \hat{y}_i^{(t-1)}) + g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i)] + \Omega(f_t) + constant -`` ` +``` where -`` `math +```math g_i &= \partial_{\hat{y}_i^{(t)}} l(y_i, \hat{y}_i^{(t-1)})\\ h_i &= \partial_{\hat{y}_i^{(t)}}^2 l(y_i, \hat{y}_i^{(t-1)}) -`` ` +``` So we can remove all the constant at the t-th step and the specific objective is -`` `math +```math \sum_{i=1}^n [g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i)] + \Omega(f_t) -`` ` +``` One of the benifit of this definition is as long as the loss function has the first and second order derivative, we can optimized every loss function within the same framework. @@ -120,15 +120,15 @@ One of the benifit of this definition is as long as the loss function has the fi We have introduced the details in the loss function, next we talk about the regularization term. We want to control the complexity of a tree, thus we need to define it first. We define a tree ``$ f(x) $`` as -`` `math +```math f_t(x) = w_{q(x)}, w\inR^T, q:R^d\rightarrow \{1,2,\cdots,T\} -`` ` +``` where ``$ w $`` is the vector of scores on leaves, ``$ q $`` is a function assigning each data point to the corresponding leaf and ``$ T $`` is the number of leaves. In XGBoost, we define the complexity as -`` `math +```math \Omega(f) = \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2 -`` ` +``` It is possible to define other form of regularization terms, but this one works well in practice. @@ -136,23 +136,23 @@ It is possible to define other form of regularization terms, but this one works Now we have the objective value with the ``$ t $``-th tree added: -`` `math +```math Obj^{(t)} &\approx \sum_{i=1}^n [g_i w_q(x_i) + \frac{1}{2} h_i w_{q(x_i)}^2] + \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2\\ &= \sum^T_{j=1} [(\sum_{i\in I_j} g_i) w_j + \frac{1}{2} (\sum_{i\in I_j} h_i + \lambda) w_j^2 ] + \gamma T -`` ` +``` where ``$ I_j = \{i|q(x_i)=j\} $`` is the set of indices of data points assigned to the ``$ j $``-th leaf. Notice that in the second line we have change the index of the summation because all the data points on the same leaf get the same score. We could further compress the expression by defining ``$ G_j = \sum_{i\in I_j} g_i $`` and ``$ H_j = \sum_{i\in I_j} h_i $``: -`` `math +```math Obj^{(t)} = \sum^T_{j=1} [G_jw_j + \frac{1}{2} (H_j+\lambda) w_j^2] +\gamma T -`` ` +``` In this equation ``$ w_j $`` are independent to each other, the form ``$ G_jw_j+\frac{1}{2}(H_j+\lambda)w_j^2 $`` is quadratic and the best ``$ w_j $`` to minimize it can be solved deterministically: -`` `math +```math w_j^\ast = -\frac{G_j}{H_j+\lambda}\\ Obj = -\frac{1}{2} \sum_{j=1}^T \frac{G_j^2}{H_j+\lambda} + \gamma T -`` ` +``` **Therefore, given the parameters, the gradients and the structure of the tree, we know how to set the score on each leaf.** @@ -162,11 +162,11 @@ Our algorithm aims at optimizing the objective, so it also guides us to a good t Specifically we try to split a leaf into two leaves, and the score it gains is -`` `math +```math Gain = \frac{1}{2} [\frac{G_L^2}{H_L+\lambda}+\frac{G_R^2}{H_R+\lambda}-\frac{(G_L+G_R)^2}{H_L+H_R+\lambda}] - \gamma -`` ` +``` -This formula can be decomposited as 1) the score on the new left leaf 2) the score on the new right leaf 3) The score on the original leaf 4) regularization on the additional leaf. +This formula can be decomposited as 1) the score on the new left leaf, 2) the score on the new right leaf, 3) The score on the original leaf and 4) regularization on the additional leaf. The regularization in the end can be seen as the minimum increment from this split. In the end, we will prune out the split with a negative gain. From cc3c98d9b7631b5cf48450bba6e22aba6a64f57e Mon Sep 17 00:00:00 2001 From: hetong007 Date: Sun, 23 Aug 2015 16:59:29 -0700 Subject: [PATCH 7/8] fix formula --- doc/model.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/doc/model.md b/doc/model.md index d55d262c2..8e1d6477d 100644 --- a/doc/model.md +++ b/doc/model.md @@ -74,11 +74,11 @@ obj(\Theta) = \sum_i^n l(y_i, \hat{y}_i) + \sum_{k=1}^K \Omega(f_k) It is not easy to train all the trees at once. Instead, we use the strategy to train them in a sequence so that everytime we train one CART and add it to the model. We note the prediction value at step `t` by ``$ \hat{y}_i^{(t)}$``, so we have ```math -\hat{y}_i^{(0)} = 0\\ -\hat{y}_i^{(1)} = f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i)\\ -\hat{y}_i^{(2)} = f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i)\\ -\dots\\ -\hat{y}_i^{(t)} = \sum_{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i) +\hat{y}_i^{(0)} &= 0\\ +\hat{y}_i^{(1)} &= f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i)\\ +\hat{y}_i^{(2)} &= f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i)\\ +\dots &\\ +\hat{y}_i^{(t)} &= \sum_{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i) ``` Which CART do we want at each step? Of course we want to add the one that minimize our objective. @@ -121,7 +121,7 @@ One of the benifit of this definition is as long as the loss function has the fi We have introduced the details in the loss function, next we talk about the regularization term. We want to control the complexity of a tree, thus we need to define it first. We define a tree ``$ f(x) $`` as ```math -f_t(x) = w_{q(x)}, w\inR^T, q:R^d\rightarrow \{1,2,\cdots,T\} +f_t(x) = w_{q(x)}, w\in R^T, q:R^d\rightarrow \{1,2,\cdots,T\} ``` where ``$ w $`` is the vector of scores on leaves, ``$ q $`` is a function assigning each data point to the corresponding leaf and ``$ T $`` is the number of leaves. In XGBoost, we define the complexity as @@ -132,7 +132,7 @@ where ``$ w $`` is the vector of scores on leaves, ``$ q $`` is a function assig It is possible to define other form of regularization terms, but this one works well in practice. -### Get the best score on leaf +### The best score on leaf Now we have the objective value with the ``$ t $``-th tree added: @@ -150,13 +150,13 @@ Obj^{(t)} = \sum^T_{j=1} [G_jw_j + \frac{1}{2} (H_j+\lambda) w_j^2] +\gamma T In this equation ``$ w_j $`` are independent to each other, the form ``$ G_jw_j+\frac{1}{2}(H_j+\lambda)w_j^2 $`` is quadratic and the best ``$ w_j $`` to minimize it can be solved deterministically: ```math -w_j^\ast = -\frac{G_j}{H_j+\lambda}\\ -Obj = -\frac{1}{2} \sum_{j=1}^T \frac{G_j^2}{H_j+\lambda} + \gamma T +w_j^\ast &= -\frac{G_j}{H_j+\lambda}\\ +Obj &= -\frac{1}{2} \sum_{j=1}^T \frac{G_j^2}{H_j+\lambda} + \gamma T ``` **Therefore, given the parameters, the gradients and the structure of the tree, we know how to set the score on each leaf.** -### Learn the tree structure +### Learning the tree structure Our algorithm aims at optimizing the objective, so it also guides us to a good tree structure. We score the structure by ``$ Obj^{(t)} $`` which is mentioned just above. Since we can evaluate the tree, ideally we can enumerate all possible trees and pick the best one. In practice it is impossible, so we enumerate all the trees no deeper than a certain depth greedily. From 7294ac4fc96e2d2d22e97be8de02ce0bdf736510 Mon Sep 17 00:00:00 2001 From: hetong007 Date: Sun, 23 Aug 2015 17:04:08 -0700 Subject: [PATCH 8/8] refine model doc --- doc/model.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/model.md b/doc/model.md index 8e1d6477d..3dc59815c 100644 --- a/doc/model.md +++ b/doc/model.md @@ -63,7 +63,7 @@ Now we are predicting with two trees, by predict on each tree individually and t \hat{y}_i = \sum_{k=1}^K f_k(x_i), f_k \in F ``` -where ``$ f $`` is a function in the functional space ``$ F $``, and ``$ F $`` is the set of all possible CARTs. Therefore our objective to optimize can be written as +where ``$ K $`` is the number of trees, ``$ f $`` is a function in the functional space ``$ F $``, and ``$ F $`` is the set of all possible CARTs. Therefore our objective to optimize can be written as ```math obj(\Theta) = \sum_i^n l(y_i, \hat{y}_i) + \sum_{k=1}^K \Omega(f_k) @@ -137,7 +137,7 @@ It is possible to define other form of regularization terms, but this one works Now we have the objective value with the ``$ t $``-th tree added: ```math -Obj^{(t)} &\approx \sum_{i=1}^n [g_i w_q(x_i) + \frac{1}{2} h_i w_{q(x_i)}^2] + \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2\\ +Obj^{(t)} &\approx \sum_{i=1}^n [g_i w_{q(x_i)} + \frac{1}{2} h_i w_{q(x_i)}^2] + \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2\\ &= \sum^T_{j=1} [(\sum_{i\in I_j} g_i) w_j + \frac{1}{2} (\sum_{i\in I_j} h_i + \lambda) w_j^2 ] + \gamma T ```