Introduction to XGBoost
Outline
- The canonical forest model
- Why we need another tree model
- Regression Tree and Ensemble (What are we Learning)
- Gradient Boosting (How do we Learn)
- Experiment
Outline
- The forest model
- Why we need another tree model
- Regression Tree and Ensemble (What are we Learning)
- Gradient Boosting (How do we Learn)
- Experiment
Forest model
- A bunch of trees
- Output the aggregation of the outcome of each tree
Example
Problems in tree model
- How to split the node?
- How to aggregate the outcome?
- How to prevent overfitting?
Outline
- The Forest Model
- Why We Need Another Tree Model
- Gradient Boosting (How do we Learn)
- System Designing Tricks
- Take Home Message
- Tips for Open Source
Common setting of
Classification & Regression Problem
A classification problem:
- Find \(F(\textbf{x}) = f(\textbf{x};\Theta)\), \(\Theta\) is the parameter
- So that \(F\) outputs class label \(\hat{y}\) on input \(x\) (classification)
- \(F\) outputs a real number on input \(x\) (regression)
How to train it
- Increasing the accuracy
- Defining the loss \(L = \sum_i \ell(y_i, \hat{y}_i)\)on the training data
- Minimize the loss \(min \sum_i \ell(y_i, \hat{y}_i)\)
- Preventing over fitting
- Adding regularization \(\sum_i \ell(y_i, \hat{y}_i)+\Omega(\Theta)\)
- Minimize the whole thing
- Objective function that is everywhere
Obj(\Theta) = L(\Theta) + \Omega(\Theta)
- Loss on training data: \(L = \sum^{n}_{i=1} l(y_i, \hat{y}_i)\)
- Square loss: \(l(y_i, \hat{y}_i) = (y_i - \hat{y}_i)^2\)
- Logistic loss: \(l(y_i, \hat{y}_i) = y_i ln(1 + e^{-\hat{y}_i}) + (1-y_i)ln(1+e^{\hat{y}_i})\)
- Regularization: how complicated the model is?
- L2 norm: \(\Omega(w) = \lambda \|w\|^2\)
- L1 norm (lasso): \(\Omega(w) = \lambda \|w\|_1\)
Formal definition of the task
Revisiting Random Forest
- How to split the node?
- Heuristic method, entropy
- How to aggregate the outcome?
- Voting for classification
- Average for regression
- How to prevent overfitting?
- Max depth
- Pruning by impurity
Revisiting Random Forest
- How to split the node?
- Heuristic method, entropy - Bad
- How to aggregate the outcome?
- Voting for classification - Good
- Average for regression - Good
- How to prevent overfitting?
- Max depth - Bad
- Pruning by impurity - Bad
Adapting optimization view to tree problems
- Entropy-> training loss
- Pruning -> regularization defined by #nodes
- Max depth -> constraint on the function space
- Smoothing leaf values -> L2 regularization on leaf weights
Tree in \(Obj(\Theta) = L(\Theta) + \Omega(\Theta)\)
Beautiful!
Outline
- The Forest Model
- Why We Need Another Tree Model
- Gradient Boosting (How do we Learn)
- System Designing Tricks
How do we Learn
- Objective: \(\sum^n_{i=1} l(y_i, \hat{y}_i) + \sum_k\Omega(f_k), f_k \in \textbf{F} \)
- no SGD
- Solution: Additive Training (Boosting) [Friedman 99]
- Starting from constant prediction, add a new function each time
\(\hat{y}^{(0)}_i = 0\)
\(\hat{y}^{(1)}_i = f_1(x_i) = \hat{y}^{(0)}_i + f_1(x_i) \)
\(\hat{y}^{(2)}_i = f_1(x_i) + f_2(x_i)= \hat{y}^{(1)}_i + f_2(x_i) \)
...
\(\hat{y}^{(t)}_i = \sum^t_{k=1} f_k(x_i) = \hat{y}^{(t-1)}_i + f_t(x_i) \)
Boosted Tree Algorithm
- Add a new tree in each iteration
- Grow a tree \(f_t(x)\)
-
Add \(f_t(x)\) to the model \(\hat{y}^{(t)}_i = \hat{y}^{(t-1)}_i + f_t(x_i) \)
- Usually, instead we do \(\hat{y}^{(t)} = \hat{y}^{(t-1)} + \epsilon f_t(x_i)\)
- \(\epsilon\) is called step-size or shrinkage, usually set around 0.1
- This means we do not do full optimization in each step and reserve chance for future rounds which helps prevent overfitting
How do we get \(f_t(x)\)
- Optimize the objective!!
- The prediction at round t is \(\hat{y}_i^{(t)} = \hat{y}_i^{(t-1)} + f_t(x_i)\)
\(Obj^{(t)} = \sum^n_{i=1}l(y_i,\hat{y}_i^{(t)}) + \sum^t_{i=1}\Omega(f_i)\)
\(\ \ \ \ \ \ \ \ \ \ = \sum^n_{i=1}l(y_i,\hat{y}_i^{(t-1)} + f_t(x_i)) + \Omega(f_t) + const\)
What is \(l\)
- Loss function
- Available for user customization
What is \(\Omega\)
- \(\Omega(f_t) = \gamma T + \frac{1}{2} \lambda \sum^T_{j=1}w^{2}_j \)
\( \Omega = 3 \gamma + \frac{1}{2} \lambda (4 + 0.01 + 1) \)
The Structure Score(Obj) Calculation
-
How to split the node?
- Training loss
- How to aggregate the outcome?
- How to prevent overfitting?
Revisiting Random Forest
Searching Algorithm for Single Tree
- Enumerate the possible tree structures q
- Calculate the structure score for the q, using the scoring function.
- Find the best tree structure, and use the optimal leaf weight: \(w^*_j=-\frac{G_j}{H_j+\lambda} \)
- But... there can be infinite possible tree structures.
Greedy Learning of the Tree
- In practice, we grow the tree greedily
- Start from tree with depth 0
- For each leaf node of the tree, try to add a split. The change of objective after adding the split is
Gain = \frac{1}{2}\Big[\frac{G^2_L}{H_L+\lambda} + \frac{G^2_R}{H_R+\lambda} - \frac{(G_L+G^R)^2}{H_L+H_R+\lambda}\Big] - \gamma
the score of the left child
the score of the right child
the score of if we do not split
the complexity cost by introducing additional leaf
- How to split the node?
- How to aggregate the outcome?
-
How to prevent overfitting?
- Pruning By Train Loss
Revisiting Random Forest
Pruning and Regularization
Gain = \frac{1}{2}\Big[\frac{G^2_L}{H_L+\lambda} + \frac{G^2_R}{H_R+\lambda} - \frac{(G_L+G^R)^2}{H_L+H_R+\lambda}\Big] - \gamma
- Recall the gain of split, it can be negative
- When the training loss reduction is smaller than regularization
- Trade-off between simplicity and predictiveness
- Pre-stopping
- Stop split if the best split have negative gain
- But maybe a split can benefit future splits..
- Post-Prunning
- Grow a tree to maximum depth, recursively prune all the leaf splits with negative gain
Outline
- The Forest Model
- Why We Need Another Tree Model
- Gradient Boosting (How do we Learn)
- System Designing Tricks
Approximate Split Algorithm
- Traverse all the data points in each split stage is time-consuming!
- Using quantiles!
- Global quantile
- Local quantile
- Using quantiles!
Sparsity-aware Split Algorithm
- Sparsity can be caused by:
- Missing values
- One-hot encoding
Solution: Skip missing data and give them a default direction in each split
Block Structure
Sorting data in each step of split finding.
Precomputing a data structure that is sorted on each column
Cache miss!
- Bulk fetch data
Outline
- The Forest Model
- Why We Need Another Tree Model
- Gradient Boosting (How do we Learn)
- System Designing Tricks
Take home messages
- Optimizing none smooth functions? Try the trick used for gradient boost tree
- Computer has it's own way to accelerate. Make use of them! (cache, pipeline)
Outline
- The Forest Model
- Why We Need Another Tree Model
- Gradient Boosting (How do we Learn)
- System Designing Tricks
Tips to help your open source project successful
Documents
- User document and developer document
- Doc string and gitbook or readthedoc is helpful
- What it is, what's the use case, how's the performance
Code quality
- Extensible module system
- meaningful function and variable name
- critical part would have document
- Users are also developers and can help you!
Branding your project
- Think about the using scenario
- make it compatible with user's everyday tool.
- for data science, make your tool distributing
- Post news on hacker news and reddit
Introduction to XGBoost
By Weiyüen Wu
