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The design and complexity analysis of randomized coordinate descent methods,and in particular of variants which update a random subset (sampling) ofcoordinates in each iteration, depends on the notion of expected separableoverapproximation (ESO). This refers to an inequality involving the objectivefunction and the sampling, capturing in a compact way certain smoothnessproperties of the function in a random subspace spanned by the sampledcoordinates. ESO inequalities were previously established for special classesof samplings only, almost invariably for uniform samplings. In this paper wedevelop a systematic technique for deriving these inequalities for a largeclass of functions and for arbitrary samplings. We demonstrate that one canrecover existing ESO results using our general approach, which is based on thestudy of eigenvalues associated with samplings and the data describing thefunction.

Coordinate Descent with Arbitrary Sampling II: Expected Separable Overapproximation