  We consider the problem of learning uncertainty regions for parameter
estimation problems. The regions are ellipsoids that minimize the average
volumes subject to a prescribed coverage probability. As expected, under the
assumption of jointly Gaussian data, we prove that the optimal ellipsoid is
centered around the conditional mean and shaped as the conditional covariance
matrix. In more practical cases, we propose a differentiable optimization
approach for approximately computing the optimal ellipsoids using a neural
network with proper calibration. Compared to existing methods, our network
requires less storage and less computations in inference time, leading to
accurate yet smaller ellipsoids. We demonstrate these advantages on four
real-world localization datasets.
