# Computation of conditional expectations with guarantees

arXiv: https://arxiv.org/abs/2112.01804 · HTML: https://mlsanity.com/paper/2112.01804/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 38 references that are in the map.
- First posted on arXiv 3 Dec 2021.

## Builds on (most cited first, 4 of 4)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Deep learning based numerical approximation algorithms for stochastic partial differential equations](https://mlsanity.com/paper/2012.01194/) (2020)

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

- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Transformers Can Solve Non-Linear and Non-Markovian Filtering Problems in Continuous Time For Conditionally Gaussian Signals](https://mlsanity.com/paper/2310.19603/) (2023)

## Abstract (arXiv)

Theoretically, the conditional expectation of a square-integrable random variable Y given a d-dimensional random vector X can be obtained by minimizing the mean squared distance between Y and f(X) over all Borel measurable functions f colon ℝd → ℝ. However, in many applications this minimization problem cannot be solved exactly, and instead, a numerical method which computes an approximate minimum over a suitable subfamily of Borel functions has to be used. The quality of the result depends on the adequacy of the subfamily and the performance of the numerical method. In this paper, we derive an expected value representation of the minimal mean squared distance which in many applications can efficiently be approximated with a standard Monte Carlo average. This enables us to provide guarantees for the accuracy of any numerical approximation of a given conditional expectation. We illustrate the method by assessing the quality of approximate conditional expectations obtained by linear, polynomial and neural network regression in different concrete examples.