Unbiased First-Order Randomized Smoothing for Differentiable Simulation

Computing informative gradients through nonsmooth physical models—such as those involving hard contact or bouncing—is a cornerstone challenge in robotics and machine learning. Randomized smoothing offers a principled way to obtain well-behaved surrogate gradients, yet standard first-order estimators of these smoothed gradients are biased whenever the underlying function is discontinuous. In this work, we show that this bias arises from neglected discontinuity contributions and derive a corrected first-order formula that eliminates it. When the discontinuity structure is known, we exploit it to build discontinuity-aware gradient estimators, which we instantiate for differentiable collision detection on 3D meshes. When it is unknown, we propose a complementary estimator that leverages classical gradients to reduce variance. We validate this estimator on nonsmooth trajectory optimization instances, demonstrating variance reductions of several orders of magnitude over zeroth-order baselines while remaining unbiased under discontinuities where standard first-order estimators fail.