Towards Optimal Randomized Smoothing: A Semi-Infinite Linear Programming Approach

Brendon G. Anderson, Samuel Pfrommer, Somayeh Sojoudi

Abstract:

Randomized smoothing is a leading approach
to producing certifiably robust classifiers. The
goal of optimal randomized smoothing is to maximize the average certified radius over the space
of smoothing distributions. We theoretically
study this problem through the lens of infinitedimensional optimization over measure spaces,
and prove that the nonconvex infinite program is
lower-bounded by a conic linear program wherein
the classifier’s confidence acts as a surrogate objective to optimize. A semi-infinite linear programming approximation to the problem is presented, whose sub-problems are proven to attain
nontrivial strong duality. A proof-of-concept experiment demonstrates the effectiveness of the
proposed approach.