mlsanityData to 7 Oct 2026

Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions

arXiv 2106.01798 ↗First posted 3 Jun 2021cs.LG cs.AIcited by 97

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Abstract (arXiv)

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.

Key facts. Cited by 97 papers in the map (data through 7 Oct 2026). Builds on 34 of its 55 references that are in the map. First posted on arXiv 3 Jun 2021.

Builds on

34 of 55 references.

References · 34

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22 more
  1. 2019Learning Discrete Structures for Graph Neural Networkscited by 228
  2. 2019Differentiation of Blackbox Combinatorial Solverscited by 224
  3. 2018Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimizationcited by 220
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 18; 2023: 30; 2024: 19; 2025: 16; 2026: 12.

97 papers.

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  5. 2022Theseus: A Library for Differentiable Nonlinear Optimizationcited by 64
  6. 2021Decision-Focused Learning: Through the Lens of Learning to Rankcited by 61
  7. 2023LVM-Med: Learning Large-Scale Self-Supervised Vision Models for Medical Imaging via Second-order Graph Matchingcited by 60
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  9. 2022A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inferencecited by 50
  10. 2023Diffusion Models With Learned Adaptive Noisecited by 50
  11. 2022SIMPLE: A Gradient Estimator for k-Subset Samplingcited by 34
  12. 2022Backpropagation through Combinatorial Algorithms: Identity with Projection Workscited by 29
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  4. 2023Graph Deep Learning for Time Series Forecastingcited by 26
  5. 2023Neural Probabilistic Logic Programming in Discrete-Continuous Domainscited by 24
  6. 2022Gradient Estimation with Discrete Stein Operatorscited by 22
  7. 2022Sparse Graph Learning from Spatiotemporal Time Seriescited by 22
  8. 2022SurCo: Learning Linear Surrogates For Combinatorial Nonlinear Optimization Problemscited by 21
  9. 2023From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Modulecited by 19
  10. 2023From Relational Pooling to Subgraph GNNs: A Universal Framework for More Expressive Graph Neural Networkscited by 17
  11. 2023Landscape Surrogate: Learning Decision Losses for Mathematical Optimization Under Partial Informationcited by 17
  12. 2023Efficient Subgraph GNNs by Learning Effective Selection Policiescited by 17
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  17. 2022Adaptive Perturbation-Based Gradient Estimation for Discrete Latent Variable Modelscited by 12
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  19. 2023Score Function Gradient Estimation to Widen the Applicability of Decision-Focused Learningcited by 10
  20. 2024BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learningcited by 10
  21. 2023DAG Learning on the Permutahedroncited by 9
  22. 2024On the Hardness of Probabilistic Neurosymbolic Learningcited by 9
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  24. 2023Benchmarking PtO and PnO Methods in the Predictive Combinatorial Optimization Regimecited by 8
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  28. 2022L2XGNN: Learning to Explain Graph Neural Networkscited by 7
  29. 2023Adaptive Computation Modules: Granular Conditional Computation For Efficient Inferencecited by 6
  30. 2024LPGD: A General Framework for Backpropagation through Embedded Optimization Layerscited by 6
  31. 2024ExGra-Med: Extended Context Graph Alignment for Medical Vision-Language Modelscited by 6
  32. 2022A Solver-Free Framework for Scalable Learning in Neural ILP Architecturescited by 5
  33. 2024Mixtures of In-Context Learnerscited by 5
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  35. 2022On Quantum Circuits for Discrete Graphical Modelscited by 4
  36. 2023Scalable Coupling of Deep Learning with Logical Reasoningcited by 4
  37. 2023REFER: An End-to-end Rationale Extraction Framework for Explanation Regularizationcited by 4
  38. 2024Learning Latent Graph Structures and their Uncertaintycited by 4

50 most cited of 97 All 97 (JSON)