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Genetically identical cell populations exposed to the same environment can exhibit considerable cell-to-cell variation in the levels of specific proteins. This variation or expression noise arises from the inherent stochastic nature of biochemical reactions that constitute gene expression. Negative feedback loops are common motifs in gene networks that reduce expression noise and intercellular variability in protein levels. Using stochastic models of gene expression we here compare different feedback architectures in their ability to reduce stochasticity in protein levels. A mathematically controlled comparison shows that in physiologically relevant parameter regimes, feedback regulation through the mRNA provides the best suppression of expression noise. Consistent with our theoretical results we find negative feedback loops though the mRNA in essential eukaryotic genes, where feedback is mediated via intron-derived microRNAs. Finally, we find that contrary to previous results, protein-mediated translational regulation may not always provide significantly better noise suppression than protein-mediated transcriptional regulation.