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<!DOCTYPE html>
<html>
<head>
<title>Boosting</title>
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<textarea id="source">
class: center, middle
# Boosting
CS534 - Machine Learning
Yubin Park, PhD
---
class: center, middle
Recall our original problem:
$$ \min_{f} \mathcal{L}(\mathbf{y}, f(\mathbf{X}))$$
As the original problem is difficult to solve,
we limited our search as follows:
$$ \min_{\mathbf{\beta}} \mathcal{L}(\mathbf{y}, f(\mathbf{X};\mathbf{\beta}))$$
(Remember the linear models)
---
class: center, middle
This time, let's try to solve
the difficult one directly.
Perhaps, we can try [Gradient Descent](https://en.wikipedia.org/wiki/Gradient_descent)
on the function itself?
$$ \mathbf{f}\_k = \mathbf{f}\_{k-1} - \rho\_k \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} \Bigr|\_{\mathbf{f}=\mathbf{f}_{k-1}} $$
where `\(\mathbf{f} = [f(\mathbf{x}_1), f(\mathbf{x}_2), \cdots, f(\mathbf{x}_n)]^T\)`
---
class: center, middle
Note that we are looking at
$$ \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} $$
not
$$ \frac{\partial \mathcal{L}(\mathbf{y}, f(\mathbf{X}; \mathbf{\beta}))}{\partial \mathbf{\beta}} $$
---
## Meaning of the Functional Gradient
Consider Squared Loss for example.
$$ \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} = \frac{1}{2}\frac{\partial (\mathbf{y}- \mathbf{f})^T(\mathbf{y}- \mathbf{f})}{\partial \mathbf{f}} = -(\mathbf{y} - \mathbf{f}) $$
Plugging the above into the update form:
$$ \mathbf{f}\_k = \mathbf{f}\_{k-1} + \rho\_k (\mathbf{y} - \mathbf{f}_{k-1})$$
If we initialize `\(\mathbf{f}_0 = \mathbf{0}\)`, then
$$ \mathbf{f}\_1 = \mathbf{f}\_{0} + \rho\_1 (\mathbf{y} - \mathbf{f}_{0}) = \rho\_1 \mathbf{y} $$
$$ \mathbf{f}\_2 = \mathbf{f}\_{1} + \rho\_2 (\mathbf{y} - \mathbf{f}_{1}) = (\rho\_1 + \rho\_2 - \rho_1\rho\_2 )\mathbf{y}$$
$$ \vdots $$
$$ \lim_{k \rightarrow \infty} \mathbf{f}_k = \mathbf{y}$$
---
## Approximating the Functional Gradient (1)
This seems too obvious and not that useful.
In the end, we need a function that can process **unseen** data.
One idea is to **approximate** the gradient with another function:
$$ \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} \Bigr|\_{\mathbf{f}=\mathbf{f}\_{k-1}} \approx h\_{k} (\mathbf{X})$$
where `\(h_{k} (\mathbf{X})\)` is a function that maps `\(\mathbb{R}^{n \times m}\)` to `\(\mathbb{R}^{n }\)`.
For example, if `\(h(\mathbf{X}) = \mathbf{X}\mathbf{\beta}\)` (a linear regression),
$$ \mathbf{y} - \mathbf{f}\_{k-1} \approx \mathbf{X}\mathbf{\beta}_{k}$$
$$ \mathbf{f}\_k = \mathbf{f}\_{k-1} + \rho\_k \mathbf{X}\hat{\mathbf{\beta}}_{k} $$
Similar to **Forward Stagewise Regression**?
---
## Approximating the Functional Gradient (2)
After `\(K\)` iterations, you have
$$ \mathbf{f}\_K = \mathbf{f}\_0 + \sum_{k=1}^K \rho_k h_k(\mathbf{X})$$
For a new dataset, `\( \mathbf{X}^{\prime} \)`, you can use this collection of functions to predict the target variable:
$$ \hat{\mathbf{y}} = \mathbf{f}\_0 + \sum_{k=1}^K \rho_k h_k(\mathbf{X}^{\prime}) $$
---
## Gradient Boosting Machine
1. Initialize `\( \mathbf{f}_0 = \arg \min_{\gamma} \mathcal{L}(\mathbf{y}, \gamma) \)`
2. For `\(k=1\)` to `\(K\)`:
1. Set `\( \mathbf{r}_k = - \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} \Bigr|_{\mathbf{f}=\mathbf{f}_{k-1}} \)`
2. Fit `\(h_k(\mathbf{X})\)` to `\(\mathbf{r}_k\)`
3. Find `\(\rho_k\)` that minimizes the loss function
4. Update `\(\mathbf{f}_k = \mathbf{f}_{k-1} + \rho_k h_k(\mathbf{X})\)`
`\(h(\mathbf{X})\)` can be any parametric or non-parametric function.
Decision Tree is a popular choice for `\(h(\mathbf{X})\)`.
When Decision Tree is used as `\(h(\mathbf{X})\)`, then you can estimate `\(\rho_k\)` for each region (or node) of the estimated decision tree. This variant of Gradient Boosting Machine (GBM) is called **[TreeBoost](https://projecteuclid.org/euclid.aos/1013203451)**.
---
## GBM for Classification
$$ \mathcal{L}(\mathbf{y}, f(\mathbf{X})) = - \sum_{i=1}^{n} [ y_i f(\mathbf{x}_i) - \log (1 + \exp(f(\mathbf{x}_i)))] $$
Note that, for Logistic Regression, we have `\( f(\mathbf{x}_i) = \mathbf{x}_i\mathbf{\beta} \)`.
Taking the first derivative w.r.t. the function:
$$ - \frac{\partial \mathcal{L}(\mathbf{y}, \mathbf{f})}{\partial \mathbf{f}} = [y_1 - p_1, y_2 - p_2, \cdots, y_n - p_n ]^T$$
where `\( p_i = \frac{1}{1 + \exp(-f(\mathbf{x}_i))} \)`.
---
## Stochastic Gradient TreeBoost
GBM can fit the training data very well, sometimes too well. To mitigate overfitting to the training data, [Stochastic Gradient TreeBoost (SGTB)](https://statweb.stanford.edu/~jhf/ftp/stobst.pdf) adds two regularization mechanisms:
- **Shrinkage**: when updating the function, we reduce the contribution of each model by a factor `\(0< \nu < 1\)`. Thus, the update becomes:
$$ \mathbf{f}\_k = \mathbf{f}\_{k-1} + \nu \rho_k h_k(\mathbf{X}) $$
- **Subsampling**: to reduce the variance (in the bias-variance trade-off), at each iteration, we fit a model on randomly subsampled data points. Typically 50% to 70%.
---
class: middle, center
.reference[Chapter 10 of [ESLII](https://web.stanford.edu/~hastie/ElemStatLearn/)]
---
class: middle, center
** IMPORTANT **
Please read the Chapter 10.14 of [ESLII](https://web.stanford.edu/~hastie/ElemStatLearn/).
---
class: center, middle
## Questions?
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