# Probability Functional Descent: A Unifying Perspective on GANs, Variational Inference, and Reinforcement Learning

arXiv: https://arxiv.org/abs/1901.10691 · HTML: https://mlsanity.com/paper/1901.10691/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

## Built on by (most cited first, 11 of 11)

- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Mirror Learning: A Unifying Framework of Policy Optimisation](https://mlsanity.com/paper/2201.02373/) (2022)
- [Inferring Smooth Control: Monte Carlo Posterior Policy Iteration with Gaussian Processes](https://mlsanity.com/paper/2210.03512/) (2022)
- [Toward a `Standard Model' of Machine Learning](https://mlsanity.com/paper/2108.07783/) (2021)
- [Adaptive Annealed Importance Sampling with Constant Rate Progress](https://mlsanity.com/paper/2306.15283/) (2023)
- [Dual Approximation Policy Optimization](https://mlsanity.com/paper/2410.01249/) (2024)
- [Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points](https://mlsanity.com/paper/2509.16974/) (2025)
- [Proximal Mean Field Learning in Shallow Neural Networks](https://mlsanity.com/paper/2210.13879/) (2022)
- [Message Passing Descent for Efficient Machine Learning](https://mlsanity.com/paper/2102.08110/) (2021)
- [A Distributed Algorithm for Measure-valued Optimization with Additive Objective](https://mlsanity.com/paper/2202.08930/) (2022)
- [Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Training](https://mlsanity.com/paper/2312.00991/) (2023)

## Abstract (arXiv)

This paper provides a unifying view of a wide range of problems of interest in machine learning by framing them as the minimization of functionals defined on the space of probability measures. In particular, we show that generative adversarial networks, variational inference, and actor-critic methods in reinforcement learning can all be seen through the lens of our framework. We then discuss a generic optimization algorithm for our formulation, called probability functional descent (PFD), and show how this algorithm recovers existing methods developed independently in the settings mentioned earlier.