what is robust to the noisy training data
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The robustness is the property that characterizes how effective your algorithm is while being tested on the new independent (but similar) dataset. In the other words, the robust algorithm is the one, the testing error of which is close to the training error.
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To my best knowledge, "robustness to noise" (or "noise robustness") is a slightly different term, that describe the stability of the algorithm performance after adding some noise to your data (sorry for a bit self-evident definition=))
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