  The design and complexity analysis of randomized coordinate descent methods,
and in particular of variants which update a random subset (sampling) of
coordinates in each iteration, depends on the notion of expected separable
overapproximation (ESO). This refers to an inequality involving the objective
function and the sampling, capturing in a compact way certain smoothness
properties of the function in a random subspace spanned by the sampled
coordinates. ESO inequalities were previously established for special classes
of samplings only, almost invariably for uniform samplings. In this paper we
develop a systematic technique for deriving these inequalities for a large
class of functions and for arbitrary samplings. We demonstrate that one can
recover existing ESO results using our general approach, which is based on the
study of eigenvalues associated with samplings and the data describing the
function.
