What is the definition of the robustness of a machine learning algorithm? Is it different from the definition of the performance?
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Dear El Sayed Mahmoud,
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.
For details see the following presentation:
( http://www.colt2010.org/presentation/rob_colt.pdf )
There you can find the references to the original papers as well.
P.S.:
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 selfevident definition=)) 
As Alexander Lebedev nicely described above, the robust performance of the algorithm is the one which does not deteriorate too much when training and testing with slightly different data (either by adding noise or by taking other dataset), hence, algorithm is prone to overfitting. To my best knowledge, this robustness property is also known as algorithmic stability. See a recent discussion about 'sparsity' vs 'stability' and how feature selection should be taken with caution when trying to improve performance of machine learning algorithms (which already have built in regularization)  Huan Xu and Shie Mannor "Sparse Algorithms are not Stable: A Nofreelunch Theorem":
http://users.ece.utexas.edu/~cmcaram/pubs/XuCaramanisMannor.NFL.pdf
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