# A Lagrangian Duality Approach to Active Learning

arXiv: https://arxiv.org/abs/2202.04108 · HTML: https://mlsanity.com/paper/2202.04108/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 86 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 30 of 41)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [Contrastive Multiview Coding](https://mlsanity.com/paper/1906.05849/) (2019)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [An Empirical Study of Example Forgetting during Deep Neural Network Learning](https://mlsanity.com/paper/1812.05159/) (2018)
- [Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds](https://mlsanity.com/paper/1906.03671/) (2019)
- [Learning Loss for Active Learning](https://mlsanity.com/paper/1905.03677/) (2019)
- [Not All Samples Are Created Equal: Deep Learning with Importance Sampling](https://mlsanity.com/paper/1803.00942/) (2018)
- [Variational Adversarial Active Learning](https://mlsanity.com/paper/1904.00370/) (2019)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Batch Active Learning at Scale](https://mlsanity.com/paper/2107.14263/) (2021)
- [Discriminative Active Learning](https://mlsanity.com/paper/1907.06347/) (2019)
- [Deep Active Learning: Unified and Principled Method for Query and Training](https://mlsanity.com/paper/1911.09162/) (2019)
- [Diverse mini-batch Active Learning](https://mlsanity.com/paper/1901.05954/) (2019)
- [Gone Fishing: Neural Active Learning with Fisher Embeddings](https://mlsanity.com/paper/2106.09675/) (2021)
- [Deep Active Learning over the Long Tail](https://mlsanity.com/paper/1711.00941/) (2017)
- [On Statistical Bias In Active Learning: How and When To Fix It](https://mlsanity.com/paper/2101.11665/) (2021)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Satisfying Real-world Goals with Dataset Constraints](https://mlsanity.com/paper/1606.07558/) (2016)
- [Mind Your Outliers! Investigating the Negative Impact of Outliers on Active Learning for Visual Question Answering](https://mlsanity.com/paper/2107.02331/) (2021)

## Built on by (most cited first, 24 of 24)

- [A Survey on Deep Active Learning: Recent Advances and New Frontiers](https://mlsanity.com/paper/2405.00334/) (2024)
- [Algorithm Selection for Deep Active Learning with Imbalanced Datasets](https://mlsanity.com/paper/2302.07317/) (2023)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [Resilient Constrained Learning](https://mlsanity.com/paper/2306.02426/) (2023)
- [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)
- [Near-Optimal Solutions of Constrained Learning Problems](https://mlsanity.com/paper/2403.11844/) (2024)
- [Improved Algorithm for Deep Active Learning under Imbalance via Optimal Separation](https://mlsanity.com/paper/2312.09196/) (2023)
- [Balancing Act: Constraining Disparate Impact in Sparse Models](https://mlsanity.com/paper/2310.20673/) (2023)
- [Querying Easily Flip-flopped Samples for Deep Active Learning](https://mlsanity.com/paper/2401.09787/) (2024)
- [Out-of-Distribution Learning with Human Feedback](https://mlsanity.com/paper/2408.07772/) (2024)
- [Deep Active Learning in the Open World](https://mlsanity.com/paper/2411.06353/) (2024)
- [Position: Adopt Constraints Over Fixed Penalties in Deep Learning](https://mlsanity.com/paper/2505.20628/) (2025)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Primal Dual Continual Learning: Balancing Stability and Plasticity through Adaptive Memory Allocation](https://mlsanity.com/paper/2310.00154/) (2023)
- [Loss Shaping Constraints for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2402.09373/) (2024)
- [How To Overcome Confirmation Bias in Semi-Supervised Image Classification By Active Learning](https://mlsanity.com/paper/2308.08224/) (2023)
- [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025)
- [AL-CoLe: Augmented Lagrangian for Constrained Learning](https://mlsanity.com/paper/2510.20995/) (2025)
- [Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise](https://mlsanity.com/paper/2509.22500/) (2025)
- [A State-Augmented Approach for Learning Optimal Resource Management Decisions in Wireless Networks](https://mlsanity.com/paper/2210.16412/) (2022)
- [Omega: Optimistic EMA Gradients](https://mlsanity.com/paper/2306.07905/) (2023)
- [Optimal and Efficient Binary Questioning for Human-in-the-Loop Annotation](https://mlsanity.com/paper/2307.01578/) (2023)
- [Active Learning with Foundation Model Priors: Efficient Learning under Class Imbalance](https://mlsanity.com/paper/2606.07630/) (2026)

## Abstract (arXiv)

We consider the pool-based active learning problem, where only a subset of the training data is labeled, and the goal is to query a batch of unlabeled samples to be labeled so as to maximally improve model performance. We formulate the problem using constrained learning, where a set of constraints bounds the performance of the model on labeled samples. Considering a primal-dual approach, we optimize the primal variables, corresponding to the model parameters, as well as the dual variables, corresponding to the constraints. As each dual variable indicates how significantly the perturbation of the respective constraint affects the optimal value of the objective function, we use it as a proxy of the informativeness of the corresponding training sample. Our approach, which we refer to as Active Learning via Lagrangian dualitY, or ALLY, leverages this fact to select a diverse set of unlabeled samples with the highest estimated dual variables as our query set. We demonstrate the benefits of our approach in a variety of classification and regression tasks and discuss its limitations depending on the capacity of the model used and the degree of redundancy in the dataset. We also examine the impact of the distribution shift induced by active sampling and show that ALLY can be used in a generative mode to create novel, maximally-informative samples.