```markdown
## Classical Convergence Analysis

### Assumptions
- \( f \) is strongly convex on \( S \) with constant \( m \).
- \( \nabla^2 f \) is Lipschitz continuous on \( S \), with constant \( L > 0 \):
  \[
  \|\nabla^2 f(x) - \nabla^2 f(y)\|_2 \leq L \|x - y\|_2
  \]
  \( (L \) measures how well \( f \) can be approximated by a quadratic function\() \).

### Outline
There exist constants \( \eta \in (0, m^2/L) \), \( \gamma > 0 \) such that:
- If \( \|\nabla f(x)\|_2 \geq \eta \), then \( f(x^{(k+1)}) - f(x^{(k)}) \leq -\gamma \).
- If \( \|\nabla f(x)\|_2 < \eta \), then
  \[
  \frac{L}{2m^2} \|\nabla f(x^{(k+1)})\|_2 \leq \left( \frac{L}{2m^2} \|\nabla f(x^{(k)})\|_2 \right)^2
  \]

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A. d’Aspremont. Convex Optimization M2. 20/53
```
