```markdown
### Newton Decrement

**Formula:**
\[
\lambda(x) = \left( \nabla f(x)^T \nabla^2 f(x)^{-1} \nabla f(x) \right)^{1/2}
\]

- A measure of the proximity of \( x \) to \( x^\star \).

**Properties:**

- Provides an estimate of \( f(x) - p^\star \) using quadratic approximation \(\hat{f}\):
  \[
  f(x) - \inf_y \hat{f}(y) = \frac{1}{2} \lambda(x)^2
  \]

- Equal to the norm of the Newton step in the quadratic Hessian norm:
  \[
  \lambda(x) = \left( \Delta x_{nt} \nabla^2 f(x) \Delta x_{nt} \right)^{1/2}
  \]

- Directional derivative in the Newton direction:
  \[
  \nabla f(x)^T \Delta x_{nt} = -\lambda(x)^2
  \]

- Affine invariant (unlike \(\|\nabla f(x)\|_2\)).

*Source: A. d'Aspremont. Convex Optimization M2. Slide 18/53.*
```