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align formula

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Tong He 2015-08-23 22:31:00 -07:00
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@ -113,11 +113,11 @@ Instead, we use an additive strategy: fix what we have learned, add a new tree a
We note the prediction value at step `t` by ``$ \hat{y}_i^{(t)}$``, so we have We note the prediction value at step `t` by ``$ \hat{y}_i^{(t)}$``, so we have
```math ```math
\hat{y}_i^{(0)} = 0\\ \hat{y}_i^{(0)} &= 0\\
\hat{y}_i^{(1)} = f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i)\\ \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)\\ \hat{y}_i^{(2)} &= f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i)\\
\dots\\ &\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^{(t)} &= \sum_{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i)
``` ```
It remains to ask Which tree do we want at each step? A natural thing is to add the one that optimizes our objective. It remains to ask Which tree do we want at each step? A natural thing is to add the one that optimizes our objective.