# Gradient Estimators for Implicit Models

arXiv: https://arxiv.org/abs/1705.07107 · HTML: https://mlsanity.com/paper/1705.07107/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 May 2017.

## Built on by (most cited first, 30 of 58)

- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [Score-based Causal Representation Learning with Interventions](https://mlsanity.com/paper/2301.08230/) (2023)
- [Posterior Meta-Replay for Continual Learning](https://mlsanity.com/paper/2103.01133/) (2021)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [LiBRe: A Practical Bayesian Approach to Adversarial Detection](https://mlsanity.com/paper/2103.14835/) (2021)
- [A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods](https://mlsanity.com/paper/2305.15027/) (2023)
- [Assumption violations in causal discovery and the robustness of score matching](https://mlsanity.com/paper/2310.13387/) (2023)
- [Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning](https://mlsanity.com/paper/2205.06342/) (2022)
- [Training Neural Samplers with Reverse Diffusive KL Divergence](https://mlsanity.com/paper/2410.12456/) (2024)
- [Semi-Implicit Variational Inference via Score Matching](https://mlsanity.com/paper/2308.10014/) (2023)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [FOOGD: Federated Collaboration for Both Out-of-distribution Generalization and Detection](https://mlsanity.com/paper/2410.11397/) (2024)
- [Bounding the Invertibility of Privacy-preserving Instance Encoding using Fisher Information](https://mlsanity.com/paper/2305.04146/) (2023)
- [Ordering-Based Causal Discovery for Linear and Nonlinear Relations](https://mlsanity.com/paper/2410.05890/) (2024)
- [Bounds all around: training energy-based models with bidirectional bounds](https://mlsanity.com/paper/2111.00929/) (2021)
- [Score matching through the roof: linear, nonlinear, and latent variables causal discovery](https://mlsanity.com/paper/2407.18755/) (2024)
- [Efficient Semi-Implicit Variational Inference](https://mlsanity.com/paper/2101.06070/) (2021)
- [Feature Space Particle Inference for Neural Network Ensembles](https://mlsanity.com/paper/2206.00944/) (2022)
- [Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression](https://mlsanity.com/paper/2305.00608/) (2023)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Distinguishing Cause from Effect with Causal Velocity Models](https://mlsanity.com/paper/2502.05122/) (2025)
- [iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models](https://mlsanity.com/paper/2306.17361/) (2023)
- [On Estimating the Gradient of the Expected Information Gain in Bayesian Experimental Design](https://mlsanity.com/paper/2308.09888/) (2023)
- [Stein Variational Gradient Descent with Multiple Kernel](https://mlsanity.com/paper/2107.09338/) (2021)
- [Shortcuts for causal discovery of nonlinear models by score matching](https://mlsanity.com/paper/2310.14246/) (2023)
- [MVEB: Self-Supervised Learning with Multi-View Entropy Bottleneck](https://mlsanity.com/paper/2403.19078/) (2024)
- [Learning to Explore for Stochastic Gradient MCMC](https://mlsanity.com/paper/2408.09140/) (2024)
- [Active Slices for Sliced Stein Discrepancy](https://mlsanity.com/paper/2102.03159/) (2021)

## Abstract (arXiv)

Implicit models, which allow for the generation of samples but not for point-wise evaluation of probabilities, are omnipresent in real-world problems tackled by machine learning and a hot topic of current research. Some examples include data simulators that are widely used in engineering and scientific research, generative adversarial networks (GANs) for image synthesis, and hot-off-the-press approximate inference techniques relying on implicit distributions. The majority of existing approaches to learning implicit models rely on approximating the intractable distribution or optimisation objective for gradient-based optimisation, which is liable to produce inaccurate updates and thus poor models. This paper alleviates the need for such approximations by proposing the Stein gradient estimator, which directly estimates the score function of the implicitly defined distribution. The efficacy of the proposed estimator is empirically demonstrated by examples that include meta-learning for approximate inference, and entropy regularised GANs that provide improved sample diversity.