# Epi-Convergence of Expectation Functions under Varying Measures and Integrands

arXiv: https://arxiv.org/abs/2208.03805 · HTML: https://mlsanity.com/paper/2208.03805/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2022.

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

- [Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints](https://mlsanity.com/paper/2411.19193/) (2024)
- [Open Problem: Learning with Variational Objectives on Measures](https://mlsanity.com/paper/2306.11928/) (2023)

## Abstract (arXiv)

For expectation functions on metric spaces, we provide sufficient conditions for epi-convergence under varying probability measures and integrands, and examine applications in the area of sieve estimators, mollifier smoothing, PDE-constrained optimization, and stochastic optimization with expectation constraints. As a stepping stone to epi-convergence of independent interest, we develop parametric Fatou's lemmas under mild integrability assumptions. In the setting of Suslin metric spaces, the assumptions are expressed in terms of Pasch-Hausdorff envelopes. For general metric spaces, the assumptions shift to semicontinuity of integrands also on the sample space, which then is assumed to be a metric space.