# Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators

arXiv: https://arxiv.org/abs/2505.22594 · HTML: https://mlsanity.com/paper/2505.22594/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 62 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 6 of 6)

- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression](https://mlsanity.com/paper/1804.09753/) (2018)
- [Generalization error of min-norm interpolators in transfer learning](https://mlsanity.com/paper/2406.13944/) (2024)
- [Abstracting Fairness: Oracles, Metrics, and Interpretability](https://mlsanity.com/paper/2004.01840/) (2020)
- [Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling](https://mlsanity.com/paper/2502.15131/) (2025)
- [Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models](https://mlsanity.com/paper/1309.0579/) (2013)

## Abstract (arXiv)

Approximate Message Passing (AMP) algorithms enable precise characterization of certain classes of random objects in the high-dimensional limit, and have found widespread applications in fields such as signal processing, statistics, and communications. In this work, we introduce Multi-Environment Generalized Long AMP, a novel AMP framework that applies to transfer learning problems with multiple data sources and distribution shifts. We rigorously establish state evolution for multi-environment GLAMP. We demonstrate the utility of this framework by precisely characterizing the risk of three Lasso-based transfer learning estimators for the first time: the Stacked Lasso, the Model Averaging Estimator, and the Second Step Estimator. We also demonstrate the remarkable finite sample accuracy of our theory via extensive simulations.