We provide a general framework to improve trade-offs between the number of full batch and sample queries used to solve structured optimization problems. Our results apply to a broad class of randomized optimization algorithms that iteratively solve sub-problems to high accuracy. We show that such algorithms can be modified to reuse the randomness used to query the input across sub-problems. Consequently, we improve the trade-off between the number of gradient (full batch) and individual function (sample) queries for finite sum minimization, the number of matrix-vector multiplies (full batch) and random row (sample) queries for top-eigenvector computation, and the number of matrix-vector multiplies with the transition matrix (full batch) and generative model (sample) queries for optimizing Markov Decision Processes. To facilitate our analysis we introduce the notion of pseudo-independent algorithms, a generalization of pseudo-deterministic algorithms [Gat and Goldwasser 2011], that quantifies how independent the output of a randomized algorithm is from a randomness source.