Let (X,Σ,μ,T) be a measure-preserving dynamical system, and {I_n} a sequence of intervals of nonnegative integers moving to infinity with increasing cardinality. Rosenblatt and Wierdl constructed optimal weights w_n for the averages of the form (1/w_n)∑_{k∈ In}f∘ T^k to converge a.e. in L₁. In this paper, we provide modified versions of those weights to address the question of optimality for more general weighted averages and their differentiation analogues.