The Geometry of Randomized Smoothing on Feasible Sets

Abstract

Randomized smoothing certifies the probability of a fixed output event as the center of Gaussian noise moves. Feasibility or confidence filtering reports label probabilities only among retained proposals, producing a ratio. Its numerator is a fixed Gaussian event mass, while its denominator is the probability of retention and can change with the center. Substituting this ratio into the ordinary smoothing formula can therefore certify a ball that contains a decision boundary. We separate the problem into a geometric question and a certification question. Geometry determines when conditioning preserves Gaussian comparisons. Convex retained sets preserve the full comparison, while general sets require geometric control of the retained law as the center moves. Without such control, conditional probabilities imply no positive universal radius. Joint retention-and-label probabilities always yield a valid certificate for the same filtered predictor. A uniform covariance bound transfers divergence certificates to the retained law and can yield larger radii even when the Gaussian event comparison fails. Both methods admit finite-sample bounds. For a learned image classifier with a training-selected nonconvex filter, conditional R\'enyi bounds certify more images than joint-mass bounds without additional model evaluations. A released confidence filter exhibits verified label changes inside radii obtained by conditional substitution. An application of adaptive Gaussian composition covers causal finite-horizon executions with history-dependent center shifts under a pathwise energy bound.