```markdown
### Convergence Analysis for Self-Concordant Functions

**Summary:** There exist constants \(\eta \in (0, 1/4]\), \(\gamma > 0\) such that:

- If \(\lambda(x) > \eta\), then  
  \[
  f(x^{(k+1)}) - f(x^{(k)}) \leq -\gamma
  \]

- If \(\lambda(x) \leq \eta\), then  
  \[
  2\lambda(x^{(k+1)}) \leq \left(2\lambda(x^{(k)})\right)^2
  \]

(\(\eta\) and \(\gamma\) only depend on backtracking parameters \(\alpha, \beta\))

**Complexity Bound:** Number of Newton iterations bounded by  
\[
\frac{f(x^{(0)}) - p^\star}{\gamma} + \log_2 \log_2(1/\epsilon)
\]

For \(\alpha = 0.1\), \(\beta = 0.8\), \(\epsilon = 10^{-10}\), bound evaluates to  
\[ 
375(f(x^{(0)}) - p^\star) + 6 
\]

*Source: A. d'Aspremont. Convex Optimization M2. Page 29/53*
```
