Sound Randomized Smoothing in Floating-Point Arithmetics

Randomized smoothing is sound when using infinite precision. However, we show that randomized smoothing is no longer sound for limited floating-point precision. We present a simple example where randomized smoothing certifies a radius of $1.26$ around a point, even though there is an adversarial example in the distance $0.8$ and extend this example further to provide false certificates for CIFAR10. We discuss the implicit assumptions of randomized smoothing and show that they do not apply to generic image classification models whose smoothed versions are commonly certified. In order to overcome this problem, we propose a sound approach to randomized smoothing when using floating-point precision with essentially equal speed and matching the certificates of the standard, unsound practice for standard classifiers tested so far. Our only assumption is that we have access to a fair coin.Comment: Submitted NeurIPS 202

Reducing roundoff errors in numerical integration of planetary ephemeris

Modern lunar-planetary ephemerides are numerically integrated on the observational timespan of more than 100 years (with the last 20 years having very precise astrometrical data). On such long timespans, not only finite difference approximation errors, but also the accumulating arithmetic roundoff errors become important because they exceed random errors of high-precision range observables of Moon, Mars, and Mercury. One way to tackle this problem is using extended-precision arithmetics available on x86 processors. Noting the drawbacks of this approach, we propose an alternative: using double-double arithmetics where appropriate. This will allow to use only double precision floating-point primitives which have ubiquitous support