# AL-CoLe: Augmented Lagrangian for Constrained Learning

arXiv: https://arxiv.org/abs/2510.20995 · HTML: https://mlsanity.com/paper/2510.20995/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 24 references that are in the map.
- First posted on arXiv 23 Oct 2025.

## Builds on (most cited first, 14 of 14)

- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Achieving Zero Constraint Violation for Constrained Reinforcement Learning via Primal-Dual Approach](https://mlsanity.com/paper/2109.06332/) (2021)
- [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Simultaneous Multi-Robot Motion Planning with Projected Diffusion Models](https://mlsanity.com/paper/2502.03607/) (2025)
- [Controlled Sparsity via Constrained Optimization or: How I Learned to Stop Tuning Penalties and Love Constraints](https://mlsanity.com/paper/2208.04425/) (2022)
- [A Lagrangian Duality Approach to Active Learning](https://mlsanity.com/paper/2202.04108/) (2022)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [Learning (Approximately) Equivariant Networks via Constrained Optimization](https://mlsanity.com/paper/2505.13631/) (2025)
- [Near-Optimal Solutions of Constrained Learning Problems](https://mlsanity.com/paper/2403.11844/) (2024)
- [SAFE: Finding Sparse and Flat Minima to Improve Pruning](https://mlsanity.com/paper/2506.06866/) (2025)
- [Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems](https://mlsanity.com/paper/2508.15695/) (2025)
- [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025)
- [Strong Duality Relations in Nonconvex Risk-Constrained Learning](https://mlsanity.com/paper/2312.01110/) (2023)

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

- [Everywhere Learning: Artificial Intelligence with Pointwise Constraints](https://mlsanity.com/paper/2606.01557/) (2026)
- [Constrained Learning with Universally Learnable Concept Classes](https://mlsanity.com/paper/2608.08414/) (2026)

## Abstract (arXiv)

Despite the non-convexity of most modern machine learning parameterizations, Lagrangian duality has become a popular tool for addressing constrained learning problems. We revisit Augmented Lagrangian methods, which aim to mitigate the duality gap in non-convex settings while requiring only minimal modifications, and have remained comparably unexplored in constrained learning settings. We establish strong duality results under mild conditions, prove convergence of dual ascent algorithms to feasible and optimal primal solutions, and provide PAC-style generalization guarantees. Finally, we demonstrate its effectiveness on fairness constrained classification tasks.