# Fast and Correct Gradient-Based Optimisation for Probabilistic Programming via Smoothing

arXiv: https://arxiv.org/abs/2301.03415 · HTML: https://mlsanity.com/paper/2301.03415/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 9 Jan 2023.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Neural Variational Inference and Learning in Belief Networks](https://mlsanity.com/paper/1402.0030/) (2014)
- [Automated Variational Inference in Probabilistic Programming](https://mlsanity.com/paper/1301.1299/) (2013)
- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [A Lambda-Calculus Foundation for Universal Probabilistic Programming](https://mlsanity.com/paper/1512.08990/) (2015)
- [Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints](https://mlsanity.com/paper/1601.04943/) (2016)
- [Reparameterization Gradient for Non-differentiable Models](https://mlsanity.com/paper/1806.00176/) (2018)
- [A Domain Theory for Statistical Probabilistic Programming](https://mlsanity.com/paper/1811.04196/) (2018)
- [Towards Verified Stochastic Variational Inference for Probabilistic Programs](https://mlsanity.com/paper/1907.08827/) (2019)
- [Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere](https://mlsanity.com/paper/2004.03924/) (2020)
- [Semantics of higher-order probabilistic programs with conditioning](https://mlsanity.com/paper/1902.11189/) (2019)
- [Multilevel Monte Carlo Method for Statistical Model Checking of Hybrid Systems](https://mlsanity.com/paper/1706.08270/) (2017)

## Built on by (most cited first, 2 of 2)

- [Diagonalisation SGD: Fast & Convergent SGD for Non-Differentiable Models via Reparameterisation and Smoothing](https://mlsanity.com/paper/2402.11752/) (2024)
- [GradInf: Gradient Estimation as Probabilistic Inference](https://mlsanity.com/paper/2607.07840/) (2026)

## Abstract (arXiv)

We study the foundations of variational inference, which frames posterior inference as an optimisation problem, for probabilistic programming. The dominant approach for optimisation in practice is stochastic gradient descent. In particular, a variant using the so-called reparameterisation gradient estimator exhibits fast convergence in a traditional statistics setting. Unfortunately, discontinuities, which are readily expressible in programming languages, can compromise the correctness of this approach. We consider a simple (higher-order, probabilistic) programming language with conditionals, and we endow our language with both a measurable and a smoothed (approximate) value semantics. We present type systems which establish technical pre-conditions. Thus we can prove stochastic gradient descent with the reparameterisation gradient estimator to be correct when applied to the smoothed problem. Besides, we can solve the original problem up to any error tolerance by choosing an accuracy coefficient suitably. Empirically we demonstrate that our approach has a similar convergence as a key competitor, but is simpler, faster, and attains orders of magnitude reduction in work-normalised variance.