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In boosting, each additional tree is fitted to the unexplained variation in the response that is currently un-modelled. If we are using squared-error loss, this amounts to fitting on the residuals from the aggregation of the trees fitted up to this point. I am not clear on whether it is at this point that the shrinkage (learning rate) is applied? E.g. we fit the first tree, and compute fitted values. Do we now shrink (down weight) these fitted values by the learning rate yielding a new set of fitted values, from which we compute the residuals that are used as the responses for the second tree in the ensemble, and iterate through many trees?

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up vote 4 down vote accepted

Using trees, the shrinkage takes place at the update stage of the algorithm, when the new function $f(x)_k$ is created as the function prior step ($f(x)_{k-1}$) + the new decision tree output ($p(x)_k$). This new tree output ($p(x)_k$) is scaled by the learning rate parameter.

See for example the implementation in R GBM on page 6.

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