# Regularizing with Bregman-Moreau envelopes

arXiv: https://arxiv.org/abs/1705.06019 · HTML: https://mlsanity.com/paper/1705.06019/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2017.

## Built on by (most cited first, 12 of 12)

- [Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions](https://mlsanity.com/paper/2106.03791/) (2021)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [LPGD: A General Framework for Backpropagation through Embedded Optimization Layers](https://mlsanity.com/paper/2407.05920/) (2024)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Bregman Graph Neural Network](https://mlsanity.com/paper/2309.06645/) (2023)
- [Spurious Stationarity and Hardness Results for Bregman Proximal-Type Algorithms](https://mlsanity.com/paper/2404.08073/) (2024)
- [Compressive Recovery of Sparse Precision Matrices](https://mlsanity.com/paper/2311.04673/) (2023)
- [A Bregman firmly nonexpansive proximal operator for baryconvex optimization](https://mlsanity.com/paper/2411.00928/) (2024)
- [Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems](https://mlsanity.com/paper/2511.12257/) (2025)
- [On the Tension Between Optimality and Adversarial Robustness in Policy Optimization](https://mlsanity.com/paper/2512.01228/) (2025)
- [Entropy-Smooth Convex Optimization Cannot Be Accelerated](https://mlsanity.com/paper/2607.27476/) (2026)

## Abstract (arXiv)

Moreau's seminal paper, introducing what is now called the Moreau envelope and the proximity operator (also known as the proximal mapping), appeared in 1965. The Moreau envelope of a given convex function provides a regularized version which has additional desirable properties such as differentiability and full domain. Fifty years ago, Attouch proposed using the Moreau envelope for regularization. Since then, this branch of convex analysis has developed in many fruitful directions. In 1967, Bregman introduced what is nowadays known as the Bregman distance as a measure of discrepancy between two points generalizing the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in the algorithmic solution of optimization problems. More recently, in 2012, Kan and Song studied regularization aspects of the left Bregman-Moreau envelope even for nonconvex functions. In this paper, we complement previous works by analyzing the left and right Bregman-Moreau envelopes and by providing additional asymptotic results. Several examples are provided.