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I'm trying to understand the difference between random forests and extremely randomized trees (https://orbi.uliege.be/bitstream/2268/9357/1/geurts-mlj-advance.pdf)

I understand that extratrees uses random splits and no bootstrapping, as covered here: https://stackoverflow.com/questions/22409855/randomforestclassifier-vs-extratreesclassifier-in-scikit-learn

The question I'm struggling with is, if all the splits are randomized, how does a extremely randomized decision tree learn anything about the objective function? Where is the 'optimization' step?

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The splits are random, but the value assigned at each leaf is still the average of the response among training points landing in that leaf. Without pruning, both kinds of trees will perfectly fit their training data; the difference is in how unseen data will pass through the splits (and pruned trees will of course cause a bigger discrepancy).

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