{ "id": "epmr5kkaw5", "slug": "multiple-instance-learning", "title": "Multiple-instance learning", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_methods" }, { "id": "technology.computing.ai.supervised_unsupervised", "label": "Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.supervised_unsupervised" }, { "id": "technology.computing.ai.supervised_unsupervised.semi_supervised_learning", "label": "Semi-supervised and weakly supervised learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.supervised_unsupervised.semi_supervised_learning" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.supervised_unsupervised", "label": "Non-geographic · 1946 to 2000: Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.supervised_unsupervised", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t1946.technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.ml_methods" }, { "id": "geo.nongeo.t1946.technology.computing.ai.supervised_unsupervised", "label": "Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.supervised_unsupervised" } ] } ], "excerpt": "Multiple-instance learning (MIL) is a weakly supervised machine learning method in which labels apply to whole bags of instances, introduced in 1997 for drug-activity prediction.", "snippet": "Multiple-instance learning (MIL) is a weakly supervised machine learning method in which labels apply to whole bags of instances, introduced in 1997 for drug-activity prediction.", "node": "technology.computing.ai.supervised_unsupervised.semi_supervised_learning", "markdown": "# Multiple-instance learning\n\nMultiple-instance learning (MIL) is a weakly supervised machine learning paradigm in which training examples are bags of instances and a class label is provided only for the entire bag, so the algorithm must learn to classify individual instances from bag-level supervision alone. It was introduced for drug-activity prediction, where a molecule offers many alternative conformations but only some may bind a target protein, and it is now widely used in whole-slide histopathology, where a slide carries one diagnosis label but contains thousands of patches.[1](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf) • [2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009)\n\n| Key fact | Detail |\n|---|---|\n| Bag | A set of feature vectors (instances) sharing one label; the individual instance labels are hidden.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) |\n| Standard MI assumption | A bag is positive if and only if at least one of its instances is positive; those instances are called witnesses.[3](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A) |\n| Origin | Dietterich, Lathrop, and Lozano-Pérez, *Artificial Intelligence*, 1997; their inside-out APR algorithm gave 89% correct predictions on the musk task.[1](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf) |\n| Method taxonomy | Instance-space, bag-space, and embedded-space paradigms, per Amores' review.[4](https://www.sciencedirect.com/science/article/pii/S0004370213000581) |\n| Evaluation asymmetry | Misclassifying 1% of instances in each negative bag gives 0% negative-bag accuracy but 99% instance accuracy.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) |\n| Theory | MIL sample complexity depends only poly-logarithmically on bag size, for any hypothesis class.[5](https://www.jmlr.org/papers/volume13/sabato12a/sabato12a.pdf) |\n| Deep MIL milestone | Attention-based MIL pooling with a Bernoulli bag-label formulation, Ilse, Tomczak and Welling, 2018.[6](https://proceedings.mlr.press/v80/ilse18a/ilse18a.pdf) |\n\n## How it works\n\nA bag \\( X = (X_{1}, \\ldots, X_{n}) \\) is a collection of instance feature vectors with a single observed label \\( \\nu_{S}(X) \\). Under the standard MI assumption each instance has a hidden label \\( c \\in \\{+,-\\} \\), and\n\n\\[ \\nu_{S}(X) \\Leftrightarrow g(X_{1}) \\vee g(X_{2}) \\vee \\ldots \\vee g(X_{n}) \\]\n\nfor some instance concept \\( g \\); a bag is negative only if all of its instances are negative.[3](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A) This assumption is asymmetric and traces to the musk drug-discovery problem, where one binding conformation makes a molecule active.[4](https://www.sciencedirect.com/science/article/pii/S0004370213000581) It is not guaranteed to hold in other domains, and much later work relaxes it, for example for text categorization, robot localization, and drugs binding at multiple sites.[3](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A)\n\nAlternative assumptions change what the bag label means. The collective assumption treats a bag as a sample of an underlying population, with all instances contributing equally to the label.[3](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A) Generalized assumptions, used by DD-SVM and MILES, determine the bag label from distances to a set of target points.[3](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A) The MI-GEN model treats each bag as a latent probability distribution \\( P(x \\mid B) \\) rather than a finite set, motivated by molecules in conformational equilibrium governed by [Gibbs free energy](https://www.edgepedia.org/gibbs-free-energy); a bag is negative if and only if \\( P_{x \\sim B}[f(x)=1] = 0 \\).[7](https://www.jmlr.org/papers/volume17/15-171/15-171.pdf)\n\nMethods are grouped by where they extract discriminative information: instance-space methods learn at the instance level, bag-space methods compare whole bags, and embedded-space methods map each bag to a single feature vector, reducing MIL to standard supervised learning.[4](https://www.sciencedirect.com/science/article/pii/S0004370213000581)\n\n## How it is done\n\nSetting up a MIL problem involves several choices. First, define bags: group instances that share a label, such as the conformations of one molecule or the patches of one slide. Second, represent instances; in computational pathology a feature extractor embeds each patch, an aggregator pools the patch embeddings into a slide representation, and a predictor maps that representation to the slide label.[8](https://arxiv.org/html/2408.09476) Third, choose the learning paradigm and loss. In deep MIL, Ilse, Tomczak and Welling state the problem as learning the [Bernoulli distribution](https://www.edgepedia.org/bernoulli-distribution) of the bag label, with the bag probability fully parameterized by neural networks and trained by maximizing the log-likelihood; their general pipeline transforms instances with a function \\( f \\), combines them with a permutation-invariant function \\( \\sigma \\) (the MIL pooling), and maps the result to a bag probability with \\( g \\).[6](https://proceedings.mlr.press/v80/ilse18a/ilse18a.pdf) Fourth, decide what to evaluate. Bag-level accuracy, instance-level accuracy, and AU-ROC answer different questions, and the gap between them can be extreme: a classifier that errs on 1% of instances in each negative bag scores 0% on negative bags but 99% on negative instances.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009)\n\n## Origin\n\nThe multiple instance problem was introduced by Thomas G. Dietterich, Richard H. Lathrop and [Tomás Lozano-Pérez](https://www.edgepedia.org/tomas-lozano-perez) in \"Solving the multiple instance problem with axis-parallel rectangles\" (*Artificial Intelligence*, 1997), motivated by predicting whether a drug molecule binds strongly to a target protein.[1](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf) Because molecules rotate internal bonds, any shape representation yields multiple feature vectors per molecule; the musk data used 166-dimensional vectors describing low-energy conformations of roughly 100 molecules, about half of them musky.[1](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf) Their three axis-parallel rectangle (APR) algorithms included an inside-out variant that directly confronts the ambiguity, reaching 89% correct predictions; Crippen's distance-geometry approach had earlier confronted the same problem but was limited by combinatorial explosion to constraints on four or five key atoms.[1](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf)\n\nMaron and Lozano-Pérez's Diverse Density framework followed in 1997 (some secondary sources cite it as 1998).[9](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf) Early theory developed quickly around the formalization: a PAC bound, a more efficient algorithm, and a reduction showing that a hypothesis class PAC-learnable from one-sided random classification noise is PAC-learnable in MIL under the independence assumption.[9](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf) • [5](https://www.jmlr.org/papers/volume13/sabato12a/sabato12a.pdf) • [10](https://doi.org/10.1023/a:1007402410823) Early applications beyond drug activity included natural scene classification by Maron and Ratan (1998).[9](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf)\n\n## Variants\n\nClassic algorithms differ mainly in how they treat the hidden instance labels. Diverse Density searches for the intersection of the positive bags minus the union of the negative bags, using a noisy-or probabilistic model with instance causal probability \\( \\Pr(x = t \\mid B_{i,j}) = \\exp(-\\| B_{i,j} - x \\|^{2}) \\).[9](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf) EM-DD, by Qi Zhang and Sally A. Goldman (2001), combines an EM-style estimate of the single instance determining each bag's label with Diverse Density, and on the drug-activity problem outperformed DD, Citation-kNN, and APR.[11](https://www.cs.cmu.edu/~juny/MILL/mil_review.pdf) The SVM pair of Stuart Andrews, Ioannis Tsochantaridis, and Thomas Hofmann (2002) treats instance labels as unobserved variables: mi-SVM uses all instances, iteratively adjusting labels of instances in positive bags by their distance to the current hyperplane, while MI-SVM uses only the most positive instance (the witness) of each bag; both were implemented with mixed [integer quadratic programming](https://www.edgepedia.org/integer-quadratic-programming).[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) • [11](https://www.cs.cmu.edu/~juny/MILL/mil_review.pdf) • [12](https://www.sciencedirect.com/science/article/abs/pii/S003132031830061X) Other classic lines include Citation-kNN with minimum Hausdorff distances between bags, the MI-Kernel of Thomas Gärtner, Peter Flach, Adam Kowalczyk and Alex Smola (2002), MILES, which embeds each bag as a vector of maximal similarities to prototypes selected by a 1-norm SVM (Yixin Chen, Jinbo Bi and J.Z. Wang, *IEEE TPAMI*, 2006), an SVM for sparse positive bags that directly enforces at least one positive instance per positive bag, and MIGraph and miGraph, which treat instances as non-i.i.d. samples (Zhi-Hua Zhou, Yu-Yin Sun, and Yu-Feng Li, 2008).[11](https://www.cs.cmu.edu/~juny/MILL/mil_review.pdf) • [13](https://doi.org/10.1109/tpami.2006.248) • [14](https://dl.acm.org/doi/10.1145/1273496.1273510) • [15](https://icml.cc/Conferences/2009/papers/422.pdf)\n\nDeep MIL replaced fixed pooling with learned aggregation. Attention-based MIL (AMIL) uses a trainable weighted average in place of max or mean pooling, with a gated variant combining tanh and sigmoid gating.[6](https://proceedings.mlr.press/v80/ilse18a/ilse18a.pdf) TransMIL (Shao and colleagues, NeurIPS 2021) adds a Transformer-based correlated MIL architecture with two [Transformer](https://www.edgepedia.org/transformer) layers and a Pyramid Position Encoding Generator, converging with roughly two to three times fewer training epochs than ABMIL, DSMIL, and CLAM.[16](https://papers.nips.cc/paper_files/paper/2021/file/10c272d06794d3e5785d5e7c5356e9ff-Paper.pdf)\n\n## Applications\n\nDrug-activity prediction is the founding application, and the Musk datasets remain the classic benchmarks: Musk1 has 47 positive and 45 negative bags, Musk2 has 39 positive and 63 negative bags, and Elephant, Fox, and Tiger each have 100 positive and 100 negative bags.[15](https://icml.cc/Conferences/2009/papers/422.pdf) Whole-slide histopathology is a prominent application: CLAM demonstrated data-efficient, weakly supervised computational pathology on whole-slide images in *Nature Biomedical Engineering* in 2021, and a slide's patches form a bag with a single binary label under the original MIL assumptions.[17](https://doi.org/10.1038/s41551-020-00682-w) • [8](https://arxiv.org/html/2408.09476) A comparative study applied 16 MIL methods to cancer detection from [T-cell receptor](https://www.edgepedia.org/t-cell-receptor) sequences and to TCGA sequencing data from ten cancer types.[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC8192570/) Other documented uses include natural scene classification, image region classification, and stock selection.[9](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf) • [14](https://dl.acm.org/doi/10.1145/1273496.1273510)\n\n## Limitations and alternatives\n\nThe main failure modes follow from the supervision. Positive bags with a low witness rate can perform poorly because positive and negative bags become similar.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) The survey distinguishes polymorphism ambiguity, where each instance is a distinct version of an entity such as a molecule conformation, from part-whole ambiguity, where instances are parts of one object such as image segments; methods that optimize bag accuracy (APR, MI-SVM, MIL-Boost, EM-DD, MILD) can learn suboptimal instance-level boundaries.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) In pathology, positive instances are a small proportion of the data, inviting overfitting.[8](https://arxiv.org/html/2408.09476) A practical trade-off: bag-space and embedded-space methods are more robust for bag classification but cannot identify the key instances that trigger the bag label.[19](https://link.springer.com/article/10.1007/s00521-024-09417-3)\n\nOn theory, published analyses note that efficient PAC learning of MIL with APRs would imply efficient learning of DNF formulas, and that many practical algorithms, including mi-SVM, MI-SVM, and Multi-Instance Kernels, had no generalization guarantees before Sabato and colleagues showed the sample complexity is only poly-logarithmic in bag size, with a PAC algorithm using a supervised learner as an oracle.[5](https://www.jmlr.org/papers/volume13/sabato12a/sabato12a.pdf) MIL sits inside [weakly supervised learning](https://www.edgepedia.org/weakly-supervised-learning), and the Blum–Kalai theorem links it formally to learning from one-sided classification noise.[2](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009) • [5](https://www.jmlr.org/papers/volume13/sabato12a/sabato12a.pdf)\n\nRecent work has addressed efficiency and foundation models. nnMIL connects pathology foundation models to slide-level prediction through random sampling at patch and feature levels with sliding-window ensemble inference, and across 40,000 whole-slide images, 35 clinical tasks, and 4 pathology foundation models it consistently outperformed existing MIL methods for diagnosis, subtyping, biomarker detection, and pan-cancer prognosis.[20](https://www.nature.com/articles/s41551-026-01767-8)\n\n## References\n\n1. [Solving the Multiple Instance Problem with Axis-Parallel Rectangles (Dietterich, Lathrop, Lozano-Pérez, Artificial Intelligence 89, 1997)](https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1997%20-%20DietterichLathropLozano-Perez%20-%20AI.pdf)\n2. [Multiple instance learning (Carbonneau et al., Pattern Recognition 2018 survey; excerpts merged from arXiv 1612.03365 copy)](https://dl.acm.org/doi/10.1016/j.patcog.2017.10.009)\n3. [A review of multi-instance learning assumptions (Foulds & Frank, Knowledge Engineering Review, 2010)](https://www.cambridge.org/core/journals/knowledge-engineering-review/article/abs/review-of-multiinstance-learning-assumptions/0915098C83BF119A377015A45952247A)\n4. [Multiple instance classification: Review, taxonomy and comparative study (Amores, Artificial Intelligence, 2013)](https://www.sciencedirect.com/science/article/pii/S0004370213000581)\n5. [Multi-Instance Learning with Any Hypothesis Class (Sabato et al., JMLR 2012)](https://www.jmlr.org/papers/volume13/sabato12a/sabato12a.pdf)\n6. [Attention-based Deep Multiple Instance Learning (Ilse, Tomczak, Welling, ICML 2018)](https://proceedings.mlr.press/v80/ilse18a/ilse18a.pdf)\n7. [Multiple-Instance Learning from Distributions (Doran & Ray, JMLR 2016)](https://www.jmlr.org/papers/volume17/15-171/15-171.pdf)\n8. [Advances in Multiple Instance Learning for Whole Slide Image Analysis: Techniques, Challenges, and Future Directions (2024 survey)](https://arxiv.org/html/2408.09476)\n9. [A Framework for Multiple-Instance Learning (Maron & Lozano-Pérez, NIPS 1997)](https://proceedings.neurips.cc/paper_files/paper/1997/file/82965d4ed8150294d4330ace00821d77-Paper.pdf)\n10. [Avrim Blum, Adam Kalai (1998). A Note on Learning from Multiple-Instance Examples. Machine Learning.](https://doi.org/10.1023/a:1007402410823)\n11. [Review of Multi-Instance Learning and Its Applications (CMU course review; excerpts merged from review.htm copy)](https://www.cs.cmu.edu/~juny/MILL/mil_review.pdf)\n12. [MIRSVM: Multi-instance support vector machine with bag representatives (Pattern Recognition)](https://www.sciencedirect.com/science/article/abs/pii/S003132031830061X)\n13. [Yixin Chen, Jinbo Bi, J.Z. Wang (2006). MILES: Multiple-Instance Learning via Embedded Instance Selection. IEEE Transactions on Pattern Analysis and Machine Intelligence.](https://doi.org/10.1109/tpami.2006.248)\n14. [Multiple instance learning for sparse positive bags (Bunescu & Mooney, ICML 2007)](https://dl.acm.org/doi/10.1145/1273496.1273510)\n15. [Multi-Instance Learning by Treating Instances As Non-I.I.D. Samples (Zhou, Sun, Li, ICML 2009)](https://icml.cc/Conferences/2009/papers/422.pdf)\n16. [TransMIL: Transformer based Correlated Multiple Instance Learning for Whole Slide Image Classification (Shao et al., NeurIPS 2021)](https://papers.nips.cc/paper_files/paper/2021/file/10c272d06794d3e5785d5e7c5356e9ff-Paper.pdf)\n17. [Ming Y. Lu and colleagues (2021). Data-efficient and weakly supervised computational pathology on whole-slide images. Nature Biomedical Engineering.](https://doi.org/10.1038/s41551-020-00682-w)\n18. [A comparative study of multiple instance learning methods for cancer detection using T-cell receptor sequences](https://pmc.ncbi.nlm.nih.gov/articles/PMC8192570/)\n19. [Simultaneous instance pooling and bag representation selection approach for MIL using vision transformer (ViT-IWRS, Neural Computing and Applications, 2024)](https://link.springer.com/article/10.1007/s00521-024-09417-3)\n20. [nnMIL: a generalizable multiple instance learning framework for computational pathology (Luo et al., Nature Biomedical Engineering)](https://www.nature.com/articles/s41551-026-01767-8)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Machine learning methods › Supervised, unsupervised, and semi-supervised learning › Semi-supervised and weakly supervised learning*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.cs.cmu.edu/~juny/MILL/mil_review.pdf" ], "url": "https://www.edgepedia.org/multiple-instance-learning", "markdown_url": "https://www.edgepedia.org/multiple-instance-learning.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Multiple-instance learning\", Edgepedia (EdgeChat), https://www.edgepedia.org/multiple-instance-learning. Edgepedia Community License 1.0.", "credit_md": "\"[Multiple-instance learning](https://www.edgepedia.org/multiple-instance-learning)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/multiple-instance-learning](https://www.edgepedia.org/multiple-instance-learning). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Multiple-instance learning\", Edgepedia (EdgeChat), https://www.edgepedia.org/multiple-instance-learning. Edgepedia Community License 1.0.", "speakable": "Multiple-instance learning is a weakly supervised machine learning method in which labels apply to whole bags of instances, introduced in 1997 for drug-activity prediction." }