Proximal random reshuffling under local Lipschitz continuity

We study proximal random reshuffling (PRR) for minimizing the sum of locally Lipschitz or locally smooth functions and a proper lower semicontinuous convex function without assuming coercivity or the existence of limit points. The algorithmic guarantees pertaining to near approximate stationarity rely on a new tracking lemma linking the iterates to trajectories of conservative fields. One of the novelties in the analysis consists in handling set-valued mappings with unbounded values. In the locally smooth case, it improves the known convergence rate from nearly O(k-1/4) to nearly o(k-1/2).