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I have read this paper Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning

My question is

What does the arbitrary error functions mean?

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Arbitrary error (or loss) function is what the name says: arbitrary, i.e. "any" or "some" loss function. There are (infinitely) many loss functions, with mean square error and logistic loss being the most popular ones. Formally, as defined by Christian P. Robert in The Bayesian Choice,

Definition 2.1.1 A loss function is any function $\mathrm{L}$ from $\theta\times\mathcal{D}$ in $[0, +\infty)$.

Loss function is a negative of utility function, so when utility function tells us that something is better than something else, then loss function tells us that something is worse than something else. Utility is something we maximize, loss is something we minimize. What follows, loss function penalizes the incorrectness of the results. "Arbitrary loss function" is just some function used for this purpose.

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  • $\begingroup$ ♦ thanks, It also mentioned in the paper arbitrary subquadratic error, what is the difference. $\endgroup$
    – jeza
    Oct 9, 2018 at 14:47
  • $\begingroup$ ♦, why should we care about the arbitrary error function? $\endgroup$
    – jeza
    Oct 9, 2018 at 14:58
  • $\begingroup$ @jeza "subquadratic" function is a function that raises slower then quadratic function. "Why should we care" -- for the same reason as we care about any arbitrary objects in mathematics, to discuss the general cases. Please limit to one question at a time. If you have other questions, please post them as separate questions, since comments are not meant for questions. $\endgroup$
    – Tim
    Oct 9, 2018 at 15:01

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