Dimensionality reduction is a critical step in analytics pipelines for high-volume, high-dimensional time series. Principal Component Analysis (PCA) is frequently the method of choice for many applications, yet is often prohibitively expensive for large datasets. Many theoretical means of accelerating PCA via sampling have recently been proposed, but these techniques typically treat PCA as a reusable statistical operator, independent of downstream analytics workflows. We demonstrate how accounting for downstream analytics operations during dimensionality reduction via PCA allows stochastic methods to efficiently operate over very small (e.g., 1%) subsamples of input data, thus reducing computational overhead and end-to-end runtime. This enables end-to-end optimization over both dimensionality reduction and analytics tasks. By combining techniques spanning progressive sampling, approximate query processing, and cost-based optimization, our proposed dimensionality reduction optimizer enables end-to-end speedups of up to 13.9 times over classic SVD-based PCA techniques, and achieves parity with and can exceed conventional approaches like FFT and PAA by up to 3 times in end-to-end workflows.