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I'm learning about regression decision trees and I was wondering if there always exists a split in each of the terminal nodes that will reduce the sum of squared errors (assuming there is more than one data point in the node). If there is, is there a proof?

Edit: I don't mean reduced in the strict sense.

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    $\begingroup$ Do you mean strictly reduce or just reduce? The answers differ! $\endgroup$
    – whuber
    Aug 19, 2016 at 12:55

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No. Imagine the case where all the target values in a leaf are the same, there is no way to (striclty, as stressed by @whuber) reduce the error of this leaf.

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