data_mining:xgboost

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data_mining:xgboost [2019/05/03 00:52] – [XGBoost] phreazerdata_mining:xgboost [2020/08/02 16:11] – phreazer
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 $$ $$
  
-$F$ is space of functions containing all regression trees +===== Gradient boosting ===== 
-$K$ is number of trees + 
-$f_k(x_i)$ is regression tree that maps a attribute to a score+  * $F$ is space of functions containing all regression trees 
 +  * $K$ is number of trees 
 +  * $f_k(x_i)$ is regression tree that maps a attribute to a score
  
 Learn functions (trees) instead of weights in $R^d$. Learn functions (trees) instead of weights in $R^d$.
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   * Logistic loss $l(y_i,\hat{y}_i)=y_i \ln(1+e^{-\hat{y}_i})+(1-y_i)\ln(1+e^{\hat{y}_i})$ (LogitBoost)   * Logistic loss $l(y_i,\hat{y}_i)=y_i \ln(1+e^{-\hat{y}_i})+(1-y_i)\ln(1+e^{\hat{y}_i})$ (LogitBoost)
  
-Stochastic Gradient Descent can not be applied, since trees are used.+Stochastic Gradient descent can not be applied, since trees are used.
  
 Solution is **additive training**: Start with constant prediction, add a new function each time. Solution is **additive training**: Start with constant prediction, add a new function each time.
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-Taylor expansion approximation of loss+==== Taylor expansion ====
  
 Use taylor expansion to approximate a function through a power series (polynom). Use taylor expansion to approximate a function through a power series (polynom).
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 $$\sum^n_{i=1} [l(y_i,\hat{y}_i^{(t-1)}) + g_if_t(x_i) + \frac{1}{2}h_if_t^2(x_i)]$$ with $g_i=\delta_{\hat{y}^{(t-1)}} l(y_i,\hat{y}^{(t-1)})$ and $h_i=\delta^2_{\hat{y}^{(t-1)}} l(y_i,\hat{y}^{(t-1)})$ $$\sum^n_{i=1} [l(y_i,\hat{y}_i^{(t-1)}) + g_if_t(x_i) + \frac{1}{2}h_if_t^2(x_i)]$$ with $g_i=\delta_{\hat{y}^{(t-1)}} l(y_i,\hat{y}^{(t-1)})$ and $h_i=\delta^2_{\hat{y}^{(t-1)}} l(y_i,\hat{y}^{(t-1)})$
  
-With removed constants+With removed constants (and square loss)
 $$\sum^n_{i=1} [g_if_t(x_i) + \frac{1}{2}h_if_t^2(x_i)] + \Omega(f_t)$$  $$\sum^n_{i=1} [g_if_t(x_i) + \frac{1}{2}h_if_t^2(x_i)] + \Omega(f_t)$$ 
 So that learning function only influences $g_i$ and $h_i$ while rest stays the same. So that learning function only influences $g_i$ and $h_i$ while rest stays the same.
  • data_mining/xgboost.txt
  • Last modified: 2020/08/02 16:12
  • by phreazer