---
title: "Deep Batch Active Learning by Diverse, Uncertain Gradient"
description: "The authors propose the BADGE method. 51 references, cited by 90 papers here. Among them: 272."
authors: ["Jordan T. Ash", "Chicheng Zhang", "Akshay Krishnamurthy", "John Langford", "Alekh Agarwal"]
year: 2019
venue: "arXiv:1906.03671 · 2020 International Conference on Learning Representations"
source: "https://arxiv.org/abs/1906.03671"
doi: "10.48550/arxiv.1906.03671"
arxiv: "1906.03671"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/ash-2019-deep-batch-active-learning-diverse/"
---
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# Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds

**Authors:** Jordan T. Ash, Chicheng Zhang, Akshay Krishnamurthy, [John Langford](https://md.research.elamur.ai/free/en/authors/john-c-langford/) and [Alekh Agarwal](https://md.research.elamur.ai/free/en/authors/alekh-agarwal/) · **Year:** 2019 · **Venue:** [arXiv:1906.03671 · 2020 International Conference on Learning Representations](https://md.research.elamur.ai/free/en/venues/international-conference-on-learning-representations-iclr/) ([ICLR 2019](https://md.research.elamur.ai/free/en/venues/international-conference-on-learning-representations-iclr/2019/)) · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/1906.03671) · [PDF](https://arxiv.org/pdf/1906.03671) · [DOI 10.48550/arxiv.1906.03671](https://doi.org/10.48550/arxiv.1906.03671) · [arXiv 1906.03671](https://arxiv.org/abs/1906.03671) · [OpenAlex](https://openalex.org/W2948367246)

**Citations:** 272 citations (OpenAlex, 2026-10-14) · [90](https://md.research.elamur.ai/free/en/papers/ash-2019-deep-batch-active-learning-diverse/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
The paper addresses batch active learning for deep neural networks. The task is to select a whole group of unlabeled examples for labeling at once while spending as little annotator effort as possible. Methods that consider only model uncertainty often fill a batch with nearly identical examples. Methods that seek only diversity may select points that give the model little benefit. Moreover, the success of both approaches depends on the architecture, the batch size, and the data.

The authors propose the BADGE method. Each example is represented as the gradient of the loss function with respect to the output-layer weights, computed for the label the model considers most likely. The length of this vector shows how uncertain the model is, and its direction shows what the model update would be. The batch is formed using k-MEANS++ initialization. This favors long vectors and vectors far from one another at the same time, and it needs no hyperparameters to balance these criteria. Experiments with MLP and ResNet on SVHN, CIFAR10, MNIST, and OpenML datasets show that BADGE consistently performs no worse than the best baseline, and often better. This matters because in active learning, hyperparameter tuning itself costs additional labels.

## Key points
- Theoretical basis (Proposition 1): if the network output is a softmax, the gradient norm for the predicted label is a lower bound on the gradient norm for any other label. Therefore, the vector length serves as a measure of uncertainty.
- k-MEANS++ is used as a cheap substitute for k-DPP. The accuracy of the two selection methods is nearly the same, but k-DPP (MCMC sampling following Kang, 2013) is much slower for batches of size 1000. For batches of size 10,000, it did not fit in memory at all.
- Experimental settings: batch sizes 100, 1000, and 10,000; MLP and ResNet architectures; datasets OpenML #6, #155, #156, #184, SVHN, MNIST, CIFAR10. Each learning curve is the average of five independent runs.
- Batch diversity analysis: CONF has the largest average gradient norm. However, it sometimes selects linearly dependent examples: the log-determinant of the Gram matrix for such a batch equals −∞. These cases coincide with drops in accuracy. k-DPP and FF-k-CENTER form less diverse batches than BADGE.
- The paper was published at ICLR 2020 (arXiv:1906.03671). The authors include researchers from Princeton, University of Arizona, and Microsoft Research.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Jordan T. Ash, Chicheng Zhang, Akshay Krishnamurthy, John Langford and Alekh Agarwal. 2019. Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds. arXiv:1906.03671 · 2020 International Conference on Learning Representations.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{ash2019deep,
  title = {{Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds}},
  author = {Jordan T. Ash, Chicheng Zhang, Akshay Krishnamurthy, John Langford and Alekh Agarwal},
  year = {2019},
  doi = {10.48550/arxiv.1906.03671},
  eprint = {1906.03671},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.1906.03671}
}
```

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50. Stephen Mussmann and Percy S Liang. Uncertainty sampling is preconditioned stochastic gradient descent on zero-one loss. In Neural Information Processing Systems, 2018. [find](https://scholar.google.com/scholar?q=Stephen+Mussmann+and+Percy+S+Liang%2E+Uncertainty+sampling+is+preconditioned+stochastic+gradient+descent+on+zero%2Done+loss%2E+In+Neural+Information+Processing+Systems%2C+2018%2E)
51. T HE k- MEANS ++ SEEDING ALGORITHM A [find](https://scholar.google.com/scholar?q=T+HE+k%2D+MEANS+%2B%2B+SEEDING+ALGORITHM+A)

*Updated 2 November 2026*