Suggest edit — convergence of self-concordant functions

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---
title: "convergence of self-concordant functions"
type: concept
source: https://www.jemoka.com/posts/kbhconvergence_of_self_concordant_functions/
confidence: high
status: active
---

constituents Functions is self-concordant if:
\begin{align} \mid f&rsquo;&rsquo;&rsquo;\qty(x)\mid \leq 2f&rsquo;&rsquo;\qty(x)^{\frac{3}{2}}, \forall x \in \text{dom } f \end{align}
and \(f\) is self-concordant if \(g\qty(t) = f\qty(x+tv)\) is self concordant for all \(x \in \text{dom } f\).
requirements Convergence analysis! There exists \(\eta \in (0, \frac{1}{4}]\), \(\gamma &gt; 0\) such that:
if \(\lambda \qty(x) &gt; \eta\), then \(f\qty(x^{(k+1)}) - f\qty(x^{(k)}) \leq -y\) if \(\lambda \qty(x) \leq \eta\), then \(2\lambda \qty(x^{(k+1)}) \leq \qty(2 \lambda \qty(x^{(k)}))^{2}\) and \(\eta, \gamma\) depends only on backtracking line search parameters. This gives bounds:
\begin{align} \frac{f\qty(x^{(0)}) -p^{*}}{\gamma} + \log \log \qty(\frac{1}{\epsilon}) \end{align}
additional information things that are self concordant linear and quadratic functions negative logarithm \(f\qty(x) = -\log x\) negative entropy plus negative logarithm calculus of self-concordant functions perserved under postiive scaling, summing preserved under composition affine function