mlsanityData to 14 Oct 2026

On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning

arXiv 1704.00805 ↗First posted 3 Apr 2017math.OC cs.LGcited by 142

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Abstract (arXiv)

In this paper, we utilize results from convex analysis and monotone operator theory to derive additional properties of the softmax function that have not yet been covered in the existing literature. In particular, we show that the softmax function is the monotone gradient map of the log-sum-exp function. By exploiting this connection, we show that the inverse temperature parameter determines the Lipschitz and co-coercivity properties of the softmax function. We then demonstrate the usefulness of these properties through an application in game-theoretic reinforcement learning.

Key facts. Cited by 142 papers in the map (data through 14 Oct 2026). First posted on arXiv 3 Apr 2017.

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Citing papers per year: 2021: 12; 2022: 19; 2023: 25; 2024: 26; 2025: 29; 2026: 31.

142 papers.

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  10. 2024Do LLM Agents Have Regret? A Case Study in Online Learning and Gamescited by 35
  11. 2023Towards Consistent Video Editing with Text-to-Image Diffusion Modelscited by 34
  12. 2023Quality-Agnostic Deepfake Detection with Intra-model Collaborative Learningcited by 32
38 more
  1. 2025Attention Is All You Need for KV Cache in Diffusion LLMscited by 32
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  5. 2025Softmax is 1/2-Lipschitz: A tight bound across all ℓp normscited by 22
  6. 2021Transformer Uncertainty Estimation with Hierarchical Stochastic Attentioncited by 21
  7. 2023Replicability in Reinforcement Learningcited by 21
  8. 2022Oracle-free Reinforcement Learning in Mean-Field Games along a Single Sample Pathcited by 20
  9. 2023Distill n' Explain: explaining graph neural networks using simple surrogatescited by 18
  10. 2023With a Little Help from your own Past: Prototypical Memory Networks for Image Captioningcited by 18
  11. 2022Neural Architecture Search on Efficient Transformers and Beyondcited by 17
  12. 2023Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimationcited by 17
  13. 2022Beyond Adult and COMPAS: Fairness in Multi-Class Predictioncited by 16
  14. 2024Statistical Advantages of Perturbing Cosine Router in Mixture of Expertscited by 16
  15. 2022Lambda-DARTS: Mitigating Performance Collapse by Harmonizing Operation Selection among Cellscited by 15
  16. 2023A Finite-Sample Analysis of Payoff-Based Independent Learning in Zero-Sum Stochastic Gamescited by 15
  17. 2022Explainable AI via Learning to Optimizecited by 14
  18. 2024The Effect of Intrinsic Dataset Properties on Generalization: Unraveling Learning Differences Between Natural and Medical Imagescited by 13
  19. 2024Towards Bridging Generalization and Expressivity of Graph Neural Networkscited by 12
  20. 2025How Transformers Learn Regular Language Recognition: A Theoretical Study on Training Dynamics and Implicit Biascited by 11
  21. 2025Pay Attention to Attention Distribution: A New Local Lipschitz Bound for Transformerscited by 11
  22. 2023Offline Minimax Soft-Q-learning Under Realizability and Partial Coveragecited by 10
  23. 2025Biologically Plausible Brain Graph Transformercited by 10
  24. 2021MixACM: Mixup-Based Robustness Transfer via Distillation of Activated Channel Mapscited by 9
  25. 2023Convergence Rates for Localized Actor-Critic in Networked Markov Potential Gamescited by 9
  26. 2023Adapt On-the-Go: Behavior Modulation for Single-Life Robot Deploymentcited by 8
  27. 2024Proximal Interacting Particle Langevin Algorithmscited by 8
  28. 2025Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiencycited by 8
  29. 2021Theoretical Guarantees of Fictitious Discount Algorithms for Episodic Reinforcement Learning and Global Convergence of Policy Gradient Methodscited by 7
  30. 2023Synaptic Plasticity Models and Bio-Inspired Unsupervised Deep Learning: A Surveycited by 7
  31. 2026Path-conditioned training: a principled way to rescale ReLU neural networkscited by 6
  32. 2022What is Flagged in Uncertainty Quantification? Latent Density Models for Uncertainty Categorizationcited by 5
  33. 2023SmoothHess: ReLU Network Feature Interactions via Stein's Lemmacited by 5
  34. 2024Graphon Mean Field Games with a Representative Player: Analysis and Learning Algorithmcited by 5
  35. 2024MultiMax: Sparse and Multi-Modal Attention Learningcited by 5
  36. 2024Regression under demographic parity constraints via unlabeled post-processingcited by 5
  37. 2024Stochastic Semi-Gradient Descent for Learning Mean Field Games with Population-Aware Function Approximationcited by 5
  38. 2025Inverse Q-Learning Done Right: Offline Imitation Learning in Qπ-Realizable MDPscited by 5

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