# Generalizing Stochastic Smoothing for Differentiation and Gradient Estimation

arXiv: https://arxiv.org/abs/2410.08125 · HTML: https://mlsanity.com/paper/2410.08125/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 57 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Neural 3D Mesh Renderer](https://mlsanity.com/paper/1711.07566/) (2017)
- [Soft Rasterizer: A Differentiable Renderer for Image-based 3D Reasoning](https://mlsanity.com/paper/1904.01786/) (2019)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [Do Differentiable Simulators Give Better Policy Gradients?](https://mlsanity.com/paper/2202.00817/) (2022)
- [Learning with Differentiable Perturbed Optimizers](https://mlsanity.com/paper/2002.08676/) (2020)
- [A Theoretical and Empirical Comparison of Gradient Approximations in Derivative-Free Optimization](https://mlsanity.com/paper/1905.01332/) (2019)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Differentiable Sorting Networks for Scalable Sorting and Ranking Supervision](https://mlsanity.com/paper/2105.04019/) (2021)
- [Automatic Differentiation of Programs with Discrete Randomness](https://mlsanity.com/paper/2210.08572/) (2022)
- [Fast, Differentiable and Sparse Top-k: a Convex Analysis Perspective](https://mlsanity.com/paper/2302.01425/) (2023)
- [Monotonic Differentiable Sorting Networks](https://mlsanity.com/paper/2203.09630/) (2022)
- [Learning with Algorithmic Supervision via Continuous Relaxations](https://mlsanity.com/paper/2110.05651/) (2021)
- [Pix2Vex: Image-to-Geometry Reconstruction using a Smooth Differentiable Renderer](https://mlsanity.com/paper/1903.11149/) (2019)
- [Differentiable Clustering with Perturbed Spanning Forests](https://mlsanity.com/paper/2305.16358/) (2023)
- [Differentiable Sampling of Categorical Distributions Using the CatLog-Derivative Trick](https://mlsanity.com/paper/2311.12569/) (2023)
- [Learning by Sorting: Self-supervised Learning with Group Ordering Constraints](https://mlsanity.com/paper/2301.02009/) (2023)
- [Storchastic: A Framework for General Stochastic Automatic Differentiation](https://mlsanity.com/paper/2104.00428/) (2021)
- [Learning with Differentiable Algorithms](https://mlsanity.com/paper/2209.00616/) (2022)
- [GeoPhy: Differentiable Phylogenetic Inference via Geometric Gradients of Tree Topologies](https://mlsanity.com/paper/2307.03675/) (2023)
- [Uncertainty Quantification via Stable Distribution Propagation](https://mlsanity.com/paper/2402.08324/) (2024)
- [Differentiable Rendering with Perturbed Optimizers](https://mlsanity.com/paper/2110.09107/) (2021)
- [Branches of a Tree: Taking Derivatives of Programs with Discrete and Branching Randomness in High Energy Physics](https://mlsanity.com/paper/2308.16680/) (2023)
- [Style Agnostic 3D Reconstruction via Adversarial Style Transfer](https://mlsanity.com/paper/2110.10784/) (2021)

## Built on by (most cited first, 4 of 4)

- [Newton Losses: Using Curvature Information for Learning with Differentiable Algorithms](https://mlsanity.com/paper/2410.19055/) (2024)
- [A Start To End Machine Learning Approach To Maximize Scientific Throughput From The LCLS-II-HE](https://mlsanity.com/paper/2505.23858/) (2025)
- [Dimensional Peeking for Low-Variance Gradients in Zeroth-Order Discrete Optimization via Simulation](https://mlsanity.com/paper/2602.00075/) (2026)
- [Learning to Approximate Uniform Facility Location via Graph Neural Networks](https://mlsanity.com/paper/2602.13155/) (2026)

## Abstract (arXiv)

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 we present a general framework for relaxation and gradient estimation of non-differentiable black-box functions f:ℝn→ℝm. 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-ET simulations.