  In this paper, we consider comparison-based adaptive stochastic algorithms
for solving numerical optimisation problems. We consider a specific subclass of
algorithms that we call comparison-based step-size adaptive randomized search
(CB-SARS), where the state variables at a given iteration are a vector of the
search space and a positive parameter, the step-size, typically controlling the
overall standard deviation of the underlying search distribution.We investigate
the linear convergence of CB-SARS on\emph{scaling-invariant} objective
functions. Scaling-invariantfunctions preserve the ordering of points with
respect to their functionvalue when the points are scaled with the same
positive parameter (thescaling is done w.r.t. a fixed reference point). This
class offunctions includes norms composed with strictly increasing functions
aswell as many non quasi-convex and non-continuousfunctions. On
scaling-invariant functions, we show the existence of ahomogeneous Markov
chain, as a consequence of natural invarianceproperties of CB-SARS (essentially
scale-invariance and invariance tostrictly increasing transformation of the
objective function). We thenderive sufficient conditions for \emph{global
linear convergence} ofCB-SARS, expressed in terms of different stability
conditions of thenormalised homogeneous Markov chain (irreducibility,
positivity, Harrisrecurrence, geometric ergodicity) and thus define a general
methodologyfor proving global linear convergence of CB-SARS algorithms
onscaling-invariant functions. As a by-product we provide aconnexion between
comparison-based adaptive stochasticalgorithms and Markov chain Monte Carlo
algorithms.
