# Position: Adopt Constraints Over Fixed Penalties in Deep Learning

arXiv: https://arxiv.org/abs/2505.20628 · HTML: https://mlsanity.com/paper/2505.20628/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 48 references that are in the map.
- First posted on arXiv 27 May 2025.

## Builds on (most cited first, 23 of 23)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Safe RLHF: Safe Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.12773/) (2023)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [Responsive Safety in Reinforcement Learning by PID Lagrangian Methods](https://mlsanity.com/paper/2007.03964/) (2020)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals](https://mlsanity.com/paper/1809.04198/) (2018)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Training Well-Generalizing Classifiers for Fairness Metrics and Other Data-Dependent Constraints](https://mlsanity.com/paper/1807.00028/) (2018)
- [Adversarial Robustness with Semi-Infinite Constrained Learning](https://mlsanity.com/paper/2110.15767/) (2021)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [Direct Behavior Specification via Constrained Reinforcement Learning](https://mlsanity.com/paper/2112.12228/) (2021)
- [Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2007.10525/) (2020)
- [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)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [Cooper: A Library for Constrained Optimization in Deep Learning](https://mlsanity.com/paper/2504.01212/) (2025)
- [Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints](https://mlsanity.com/paper/2302.14790/) (2023)
- [Near-Optimal Solutions of Constrained Learning Problems](https://mlsanity.com/paper/2403.11844/) (2024)
- [Balancing Act: Constraining Disparate Impact in Sparse Models](https://mlsanity.com/paper/2310.20673/) (2023)
- [Feasible Learning](https://mlsanity.com/paper/2501.14912/) (2025)

## Built on by (most cited first, 5 of 5)

- [Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise](https://mlsanity.com/paper/2509.22500/) (2025)
- [Benchmarking Stochastic Approximation Algorithms for Fairness-Constrained Training of Deep Neural Networks](https://mlsanity.com/paper/2507.04033/) (2025)
- [CLASP: An online learning algorithm for Convex Losses And Squared Penalties](https://mlsanity.com/paper/2601.16072/) (2026)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)
- [Stochastic Penalty-Barrier Method for Constrained Machine Learning](https://mlsanity.com/paper/2605.18618/) (2026)

## Abstract (arXiv)

Recent efforts to develop trustworthy AI systems have increased interest in learning problems with explicit requirements, or constraints. In deep learning, however, such problems are often handled through fixed weighted-sum penalization: the constraints are added to the task loss with fixed coefficients, and the resulting scalarized objective is minimized. This position paper argues that fixed penalization is often ill-suited for deep learning problems with non-negotiable requirements for several reasons. First, in non-convex settings, the penalized and constrained problems are generally not equivalent, so solving the former need not solve the latter. Second, fixed penalization weakens hard requirements into soft penalties to be traded off against task performance. Third, choosing penalty coefficients to indirectly solve the constrained problem often involves costly trial and error, because changing them alters the penalized objective itself, and hence can mean solving the wrong problem altogether. We therefore argue that, when a deep learning problem specifies non-negotiable requirements, the constrained formulation itself should be the starting point, not the surrogate problem defined by fixed penalization. The appropriate solution strategy should then be chosen based on the problem's structure and scale.