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In this paper we introduce two novel generalizations of the theory forgradient descent type methods in the proximal setting. First, we introduce theproportion function, which we further use to analyze all known (and many new)block-selection rules for block coordinate descent methods under a singleframework. This framework includes randomized methods with uniform, non-uniformor even adaptive sampling strategies, as well as deterministic methods withbatch, greedy or cyclic selection rules. Second, the theory of strongly-convexoptimization was recently generalized to a specific class of non-convexfunctions satisfying the so-called Polyak-{\L}ojasiewicz condition. To mirrorthis generalization in the weakly convex case, we introduce the WeakPolyak-{\L}ojasiewicz condition, using which we give global convergenceguarantees for a class of non-convex functions previously not considered intheory. Additionally, we establish (necessarily somewhat weaker) convergenceguarantees for an even larger class of non-convex functions satisfying acertain smoothness assumption only. By combining the two abovementionedgeneralizations we recover the state-of-the-art convergence guarantees for alarge class of previously known methods and setups as special cases of ourgeneral framework. Moreover, our frameworks allows for the derivation of newguarantees for many new combinations of methods and setups, as well as a largeclass of novel non-convex objectives. The flexibility of our approach offers alot of potential for future research, as a new block selection procedure willhave a convergence guarantee for all objectives considered in our framework,while a new objective analyzed under our approach will have a whole fleet ofblock selection rules with convergence guarantees readily available.

Global Convergence of Arbitrary-Block Gradient Methods for Generalized Polyak-{\L}ojasiewicz Functions