# Learning with Differentiable Perturbed Optimizers

arXiv: https://arxiv.org/abs/2002.08676 · HTML: https://mlsanity.com/paper/2002.08676/

## Key facts

- Cited by 81 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2020.

## Built on by (most cited first, 30 of 81)

- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities](https://mlsanity.com/paper/2307.13565/) (2023)
- [A Review of the Gumbel-max Trick and its Extensions for Discrete Stochasticity in Machine Learning](https://mlsanity.com/paper/2110.01515/) (2021)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Differentiable Patch Selection for Image Recognition](https://mlsanity.com/paper/2104.03059/) (2021)
- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [GrooMeD-NMS: Grouped Mathematically Differentiable NMS for Monocular 3D Object Detection](https://mlsanity.com/paper/2103.17202/) (2021)
- [Strategic Classification Made Practical](https://mlsanity.com/paper/2103.01826/) (2021)
- [Efficient Video Transformers with Spatial-Temporal Token Selection](https://mlsanity.com/paper/2111.11591/) (2021)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [PyEPO: A PyTorch-based End-to-End Predict-then-Optimize Library for Linear and Integer Programming](https://mlsanity.com/paper/2206.14234/) (2022)
- [Backpropagation through Combinatorial Algorithms: Identity with Projection Works](https://mlsanity.com/paper/2205.15213/) (2022)
- [Learning with Combinatorial Optimization Layers: a Probabilistic Approach](https://mlsanity.com/paper/2207.13513/) (2022)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [Learning with Algorithmic Supervision via Continuous Relaxations](https://mlsanity.com/paper/2110.05651/) (2021)
- [ScoreNet: Learning Non-Uniform Attention and Augmentation for Transformer-Based Histopathological Image Classification](https://mlsanity.com/paper/2202.07570/) (2022)
- [Differentiable Top-k Classification Learning](https://mlsanity.com/paper/2206.07290/) (2022)
- [Sparse MoEs meet Efficient Ensembles](https://mlsanity.com/paper/2110.03360/) (2021)
- [Self-Supervised Learning of Audio Representations from Permutations with Differentiable Ranking](https://mlsanity.com/paper/2103.09879/) (2021)
- [GenDR: A Generalized Differentiable Renderer](https://mlsanity.com/paper/2204.13845/) (2022)
- [Neuro-algorithmic Policies enable Fast Combinatorial Generalization](https://mlsanity.com/paper/2102.07456/) (2021)
- [Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting](https://mlsanity.com/paper/2301.13395/) (2023)
- [Combinatorial Optimization for Panoptic Segmentation: A Fully Differentiable Approach](https://mlsanity.com/paper/2106.03188/) (2021)
- [Stochastic smoothing of the top-K calibrated hinge loss for deep imbalanced classification](https://mlsanity.com/paper/2202.02193/) (2022)
- [DynFocus: Dynamic Cooperative Network Empowers LLMs with Video Understanding](https://mlsanity.com/paper/2411.12355/) (2024)
- [Benchmarking PtO and PnO Methods in the Predictive Combinatorial Optimization Regime](https://mlsanity.com/paper/2311.07633/) (2023)
- [CaVE: A Cone-Aligned Approach for Fast Predict-then-optimize with Binary Linear Programs](https://mlsanity.com/paper/2312.07718/) (2023)
- [Unsupervised Resource Allocation with Graph Neural Networks](https://mlsanity.com/paper/2106.09761/) (2021)
- [Differentiable Rendering with Perturbed Optimizers](https://mlsanity.com/paper/2110.09107/) (2021)
- [Structured Reinforcement Learning for Combinatorial Decision-Making](https://mlsanity.com/paper/2505.19053/) (2025)

## Abstract (arXiv)

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed, they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform optimizers into operations that are differentiable and never locally constant. Our approach relies on stochastically perturbed optimizers, and can be used readily together with existing solvers. Their derivatives can be evaluated efficiently, and smoothness tuned via the chosen noise amplitude. We also show how this framework can be connected to a family of losses developed in structured prediction, and give theoretical guarantees for their use in learning tasks. We demonstrate experimentally the performance of our approach on various tasks.