Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Both the discounted and total reward episodic settings are amenable to function approximation.

Denoising and score estimation have long been known to be linked via the classical Tweedie's formula. In this work, we first extend the latter to a wider range of distributions often called "energy models" and denoted elliptical distributions in this work. Next, we examine an alternative view: we consider the denoising posterior $P(X|Y)$ as the optimizer of the energy score (a scoring rule) and derive a fundamental identity that connects the (path-) derivative of a (possibly) non-Euclidean energy score to the score of the noisy marginal. This identity can be seen as an analog of Tweedie's identity for the energy score, and allows for several interesting applications; for example, score estimation, noise distribution parameter estimation, as well as using energy score models in the context of "traditional" diffusion model samplers with a wider array of noising distributions.

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent probability measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, $C$, are equivalent if the difference of their means lies in the Cameron-Martin space of $C$. In the case of releasing finite-dimensional summaries using elliptical perturbations, we show that the privacy parameter $\ep$ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, $t$, Gaussian, and $K$-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve $\ep$-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give $\ep$-DP; only $(\epsilon,\delta)$-DP is possible.

Such embeddings have been traditionally recovered by seeking isometric embeddings in lower dimensional Euclidean spaces, as studied in [Johnson and Lindenstrauss, 1984, Bourgain, 1985].