convergence of self-concordant functions — Jemoka Knowledge Base

constituents Functions is self-concordant if:

\begin{align} \mid f’’’\left(x\right)\mid \leq 2f’’\left(x\right)^{\frac{3}{2}}, \forall x \in \text{dom } f \end{align}

and f is self-concordant if g\left(t\right) = f\left(x+tv\right) is self concordant for all x \in \text{dom } f. requirements Convergence analysis! There exists \eta \in (0, \frac{1}{4}], \gamma > 0 such that: if \lambda \left(x\right) > \eta, then f\left(x^{(k+1)}\right) - f\left(x^{(k)}\right) \leq -y if \lambda \left(x\right) \leq \eta, then 2\lambda \left(x^{(k+1)}\right) \leq \left(2 \lambda \left(x^{(k)}\right)\right)^{2} and \eta, \gamma depends only on backtracking line search parameters. This gives bounds:

\begin{align} \frac{f\left(x^{(0)}\right) -p^{*}}{\gamma} + \log \log \left(\frac{1}{\epsilon}\right) \end{align}

additional information things that are self concordant linear and quadratic functions negative logarithm f\left(x\right) = -\log x negative entropy plus negative logarithm calculus of self-concordant functions perserved under postiive scaling, summing preserved under composition affine function