Neither (i) nor (ii) are known for general RMABs. Therefore, to capture the scheduling problems addressed inthiswork,weintroduce anewsubclass ofRMABs,Collapsing Bandits, distinguished by the following feature: when an arm is played, the agent fully observes its state, "collapsing" any uncertainty, but when an arm is passive, no observation is made and uncertainty evolves.

We note that the interchange of the integral and infinite summation is justified by Section 3.7 in [5], since the coefficients Z Now,define action sequence (a)n such thata1 = a and an = a1 for alln > 1. Then we can use subadditivity of measure to bound the maximum difference across all entries of [kZ]. Therefore, the induced infinity norm error ofbZ isless thanεifthe element wise error isless than ε/k. Therefore,bα>Fφ(s) is ρ-Lipschitz if the absolute value of its derivativeisboundedbyρ,i.e. SincebF has all zeros beyond thek-th column and row, each infinite-matrix bF can be treated as ak k matrix.