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A concentration theorem for projections
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X in R^D has mean zero and finite second moments. We show that there is a precise sense in which almost all linear projections of X into R^d (for d < D) look like a scale-mixture of spherical Gaussians -- specifically, a mixture of distributions N(0, sigma^2 I_d) where the weight of the particular sigma component is P (| X |^2 = sigma^2 D). The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X's distribution. We explore this result in a variety of experiments.
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Balancing Utility and Privacy: Dynamically Private SGD with Random Projection
D2P2-SGD combines time-decreasing privacy noise with random projection to improve the accuracy of differentially private SGD, with convergence rates matching ordinary SGD.
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