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Copy file name to clipboardExpand all lines: docs/src/nondiff_points.md
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## What is the sub/super-differential?
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The subderivative is defined only for locally convex functions, where-as the super-deriviative is fined only for locally concave functions.
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For our purpose we, basically never care which we are working with and so write sub/super-derivative.
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The subderivative is defined only for locally convex functions, whereas the super-derivative is defined only for locally concave functions.
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For our purpose we basically never care which we are working with and so write sub/super-derivative.
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For a function $f$ at some point $x_0$ a sub/super-derivative is a real number $c$ such that there exists a open ball $\mathcal{B} \subset \mathrm{dom}(f)$ containing $x_0$,
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and for all points $z \in \mathcal{B}$ the following holds:
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Practically speaking, people are generally using AD to perform some gradient descent like optimization procedure.
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`NaN` and `Inf` do not generally take us to nice places.
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Erroring can be worse, especially if it is a local minima -- the optimization fully converges to that minima and then throws an error rather than reporting the result.
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If the primal function takes a non-finite value or errors on one side, then we are in the case that we are at a boundry of the domain.
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If the primal function takes a non-finite value or errors on one side, then we are in the case that we are at a boundary of the domain.
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Which means we are free to chose *any* value.
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In particular we often want to chose value of the derivative from **the other side where the primal is defined.**
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