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We can enforce some constraints on functions used in deep learning in order to guarantee optimizations. You can find it in Numerical Computation of the deep learning book.

In the context of deep learning, we sometimes gain some guarantees by restricting ourselves to functions that are either Lipschitz continuous or have Lipschitz continuous derivatives.

They include

  1. Lipschitz continuous functions
  2. Having Lipschitz continuous derivatives

The definition given for Lipschitz continuous function is as follows

A Lipschitz continuous function is a function $f$ whose rate of change is bounded by a Lipschitz constant $\mathcal{L}$:

$$\forall x, \forall y, |f(x)-f(y)| \le \mathcal{L} \|x-y\|_2 $$

Now, what is meant by having Lipschitz continuous derivatives?

Does they refer to the derivatives of Lipschitz continuous functions? If yes, then why do they mention it as a separate option?

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Consider a function $f(x) : \mathcal{R}^m\rightarrow\mathcal{R}^n$ defined for $x \in X$. If $f$ is Lipschitz continuous, it has three main properties:

  1. $f(x)$ is continuous for all $x \in X$
  2. $\frac{d f(x)}{d x}$ exists almost everywhere. Meaning, if the derivative is not defined for $x \in \mathcal{B}$, where the set $\mathcal{B} \subset X$, then $\mathcal{B}$ has measure zero.
  1. $\underset{x \in X}{\sup} \left\lVert \frac{d f(x)}{d x} \right\rVert_2 \leq L$, where $L$ is the Lipschitz constant, and the norm indicates the induced matrix norm (or if $f$ is scalar, just the regular 2 norm).

So it follows that a Lipschitz continuous function is continuous and has a bounded jacobian.

Now if $f$ has a lipschitz continuous derivative, then it means $\frac{d f}{d x}$ is Lipschitz continious, i.e. \begin{align*} \left\lVert \dfrac{d f}{d x}\big|_{x = s} - \dfrac{d f}{d x}\big|_{x = t} \right\rVert_2 \leq M \left\lVert s - t \right\rVert_2 \quad s, t \in X \end{align*} where $M$ is the lipschitz constant. So a function with Lipschitz continuous gradient is continuously differentiable and has a bounded hessian.

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