Are there formal measures for classifier or regression robustness? Are there performance measures that produce a numerical value of the robustness of a classifier or regression.
By robustness I mean graceful degradation in performance to unexpected input (similar to robust control theory)
 A: Yes you can consider three measure of robustness:

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*Breakdown point what is the proportion of outliers one can tolerate without degrading the performance arbitrarily. In this definition, you have to see whether you want robustness with respect to the featurer $X$ or with respect to the label $Y$. (Huber Regressor from scikit-learn is robust to outliers in $Y$ but not for outliers in $X$). For the classification setting, don't forget to consider the loss that you actually minimize, i.e. not accuracy but typically logarithmic loss it makes no sense to have an arbitrarily bad accuracy because accuracy is at least $0$ but you can have log loss as large as you want


*Influence function this is a finer measure that tries to see what is the influence of moving one point in the dataset. You can see it for instance in regression in the book "Robust Statistics" by Huber and Ronchetti.


*Oracle inequalities a classifier/regressor will be said to be robust if its risk satisfies a sub-gaussian concentration inequality even when the data are heavy-tailed or corrupted. This is rather complicated and still a research topic.
If you are interested I can explain more any of the points. Main references are "Robust Statistics" by Huber and Ronchetti for the first two points and for instance https://arxiv.org/abs/1711.10306 for an idea of what can be done on the last one but this is rather technical.
