# Cooper: A Library for Constrained Optimization in Deep Learning

arXiv: https://arxiv.org/abs/2504.01212 · HTML: https://mlsanity.com/paper/2504.01212/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 29 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Safe RLHF: Safe Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.12773/) (2023)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Responsive Safety in Reinforcement Learning by PID Lagrangian Methods](https://mlsanity.com/paper/2007.03964/) (2020)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals](https://mlsanity.com/paper/1809.04198/) (2018)
- [Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies](https://mlsanity.com/paper/2401.04890/) (2024)
- [Generalized Logit Adjustment: Calibrating Fine-tuned Models by Removing Label Bias in Foundation Models](https://mlsanity.com/paper/2310.08106/) (2023)
- [Partial Disentanglement via Mechanism Sparsity](https://mlsanity.com/paper/2207.07732/) (2022)
- [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)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [Balancing Act: Constraining Disparate Impact in Sparse Models](https://mlsanity.com/paper/2310.20673/) (2023)
- [L0onie: Compressing COINs with L0-constraints](https://mlsanity.com/paper/2207.04144/) (2022)
- [Neural Networks with Quantization Constraints](https://mlsanity.com/paper/2210.15623/) (2022)
- [Active Reinforcement Learning for Robust Building Control](https://mlsanity.com/paper/2312.10289/) (2023)
- [Novel Node Category Detection Under Subpopulation Shift](https://mlsanity.com/paper/2404.01216/) (2024)

## Built on by (most cited first, 11 of 11)

- [Audit Me If You Can: Query-Efficient Active Fairness Auditing of Black-Box LLMs](https://mlsanity.com/paper/2601.03087/) (2026)
- [Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation](https://mlsanity.com/paper/2603.28744/) (2026)
- [Position: Adopt Constraints Over Fixed Penalties in Deep Learning](https://mlsanity.com/paper/2505.20628/) (2025)
- [Provably Safe Model Updates](https://mlsanity.com/paper/2512.01899/) (2025)
- [humancompatible.train: Implementing Optimization Algorithms for Stochastically-Constrained Stochastic Optimization Problems](https://mlsanity.com/paper/2509.21254/) (2025)
- [Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise](https://mlsanity.com/paper/2509.22500/) (2025)
- [Unsupervised Causal Abstractions Discovery](https://mlsanity.com/paper/2606.19594/) (2026)
- [Not All Objectives Are Born Equal: Priority-Constrained Descent for Hierarchical Multi-Objective Optimization](https://mlsanity.com/paper/2606.29521/) (2026)
- [Benchmarking Stochastic Approximation Algorithms for Fairness-Constrained Training of Deep Neural Networks](https://mlsanity.com/paper/2507.04033/) (2025)
- [Stochastic Penalty-Barrier Method for Constrained Machine Learning](https://mlsanity.com/paper/2605.18618/) (2026)
- [Neural Slack Variables for Shape Constraints](https://mlsanity.com/paper/2606.13803/) (2026)

## Abstract (arXiv)

Cooper is an open-source package for solving constrained optimization problems involving deep learning models. Cooper implements several Lagrangian-based first-order update schemes, making it easy to combine constrained optimization algorithms with high-level features of PyTorch such as automatic differentiation, and specialized deep learning architectures and optimizers. Although Cooper is specifically designed for deep learning applications where gradients are estimated based on mini-batches, it is suitable for general non-convex continuous constrained optimization. Cooper's source code is available at https://github.com/cooper-org/cooper.