# MuProp: Unbiased Backpropagation for Stochastic Neural Networks

arXiv: https://arxiv.org/abs/1511.05176 · HTML: https://mlsanity.com/paper/1511.05176/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2015.

## Built on by (most cited first, 30 of 37)

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- [Towards More Human-like AI Communication: A Review of Emergent Communication Research](https://mlsanity.com/paper/2308.02541/) (2023)
- [A unified view of likelihood ratio and reparameterization gradients](https://mlsanity.com/paper/2105.14900/) (2021)
- [Learning to Inference Adaptively for Multimodal Large Language Models](https://mlsanity.com/paper/2503.10905/) (2025)
- [ARMS: Antithetic-REINFORCE-Multi-Sample Gradient for Binary Variables](https://mlsanity.com/paper/2105.14141/) (2021)
- [Coupled Gradient Estimators for Discrete Latent Variables](https://mlsanity.com/paper/2106.08056/) (2021)
- [Double Control Variates for Gradient Estimation in Discrete Latent Variable Models](https://mlsanity.com/paper/2111.05300/) (2021)
- [Training Discrete Deep Generative Models via Gapped Straight-Through Estimator](https://mlsanity.com/paper/2206.07235/) (2022)
- [Gradient Estimation for Binary Latent Variables via Gradient Variance Clipping](https://mlsanity.com/paper/2208.06124/) (2022)
- [LARO: Learned Acquisition and Reconstruction Optimization to accelerate Quantitative Susceptibility Mapping](https://mlsanity.com/paper/2211.00725/) (2022)
- [Bias-Variance Tradeoffs in Single-Sample Binary Gradient Estimators](https://mlsanity.com/paper/2110.03549/) (2021)
- [CARMS: Categorical-Antithetic-REINFORCE Multi-Sample Gradient Estimator](https://mlsanity.com/paper/2110.14002/) (2021)
- [Debiasing a First-order Heuristic for Approximate Bi-level Optimization](https://mlsanity.com/paper/2106.02487/) (2021)
- [Revisiting the Gumbel-Softmax in MADDPG](https://mlsanity.com/paper/2302.11793/) (2023)
- [Symmetric Equilibrium Learning of VAEs](https://mlsanity.com/paper/2307.09883/) (2023)
- [AskewSGD : An Annealed interval-constrained Optimisation method to train Quantized Neural Networks](https://mlsanity.com/paper/2211.03741/) (2022)
- [On Advantage Estimates for Max@K Policy Gradients](https://mlsanity.com/paper/2606.06080/) (2026)
- [Curious Explorer: a provable exploration strategy in Policy Learning](https://mlsanity.com/paper/2106.15503/) (2021)
- [Anomaly Detection in Aeronautics Data with Quantum-compatible Discrete Deep Generative Model](https://mlsanity.com/paper/2303.12302/) (2023)
- [Revisiting Score Function Estimators for k-Subset Sampling](https://mlsanity.com/paper/2407.16058/) (2024)
- [Hindsight Network Credit Assignment: Efficient Credit Assignment in Networks of Discrete Stochastic Units](https://mlsanity.com/paper/2110.07700/) (2021)
- [Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks](https://mlsanity.com/paper/2201.07934/) (2022)
- [Machine learning using magnetic stochastic synapses](https://mlsanity.com/paper/2303.01886/) (2023)

## Abstract (arXiv)

Deep neural networks are powerful parametric models that can be trained efficiently using the backpropagation algorithm. Stochastic neural networks combine the power of large parametric functions with that of graphical models, which makes it possible to learn very complex distributions. However, as backpropagation is not directly applicable to stochastic networks that include discrete sampling operations within their computational graph, training such networks remains difficult. We present MuProp, an unbiased gradient estimator for stochastic networks, designed to make this task easier. MuProp improves on the likelihood-ratio estimator by reducing its variance using a control variate based on the first-order Taylor expansion of a mean-field network. Crucially, unlike prior attempts at using backpropagation for training stochastic networks, the resulting estimator is unbiased and well behaved. Our experiments on structured output prediction and discrete latent variable modeling demonstrate that MuProp yields consistently good performance across a range of difficult tasks.