Generalizing Stochastic Smoothing for Differentiation and Gradient Estimation

Felix Petersen, Christian Borgelt, Aashwin Ananda Mishra, Stefano Ermon

Keywords: differentiable, sorting, rendering, ranking, variance reduction, continuous, cryo-et, differentiable simulator, shortest-path, stochastic relaxation

TL;DR: Reducing assumptions for stochastic smoothing; reducing variance of the gradient estimator; empirical evaluation in 4 domains.

Abstract: We deal with the problem of gradient estimation for stochastic differentiable relaxations of algorithms, operators, simulators, and other non-differentiable functions. Stochastic smoothing conventionally perturbs the input of a non-differentiable function with a differentiable density distribution with full support, smoothing it and enabling gradient estimation. Our theory starts at first principles to derive stochastic smoothing with reduced assumptions, without requiring a differentiable density nor full support, and presenting a general framework for relaxation and gradient estimation of non-differentiable black-box functions $f:\mathbb{R}^n\to\mathbb{R}^m$. We develop variance reduction for gradient estimation from 3 orthogonal perspectives. Empirically, we benchmark 6 distributions and up to 24 variance reduction strategies for differentiable sorting and ranking, differentiable shortest-paths on graphs, differentiable rendering for pose estimation, as well as differentiable cryo-electron tomography simulations.

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

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Submission Number: 3968