ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts

Heng Zhao, Zilei Shao, Guy Van Den Broeck, Zhe Zeng
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:162093-162113, 2026.

Abstract

Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert’s exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert allocation per token. Across benchmarks and model backbones, ProbMoE Exact-$k$ achieves strong performance compared to competitive baselines, with improved expert utilization and routing diversity; ProbMoE Dynamic-$k$ achieves comparable performance with fewer activated experts. Code is available at: https://github.com/HengHugoZhao/ProbMoE.git

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-zhao26ap, title = {{P}rob{M}o{E}: Differentiable Probabilistic Routing for Mixture-of-Experts}, author = {Zhao, Heng and Shao, Zilei and Van Den Broeck, Guy and Zeng, Zhe}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {162093--162113}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/zhao26ap/zhao26ap.pdf}, url = {https://proceedings.mlr.press/v306/zhao26ap.html}, abstract = {Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert’s exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert allocation per token. Across benchmarks and model backbones, ProbMoE Exact-$k$ achieves strong performance compared to competitive baselines, with improved expert utilization and routing diversity; ProbMoE Dynamic-$k$ achieves comparable performance with fewer activated experts. Code is available at: https://github.com/HengHugoZhao/ProbMoE.git} }
Endnote
%0 Conference Paper %T ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts %A Heng Zhao %A Zilei Shao %A Guy Van Den Broeck %A Zhe Zeng %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-zhao26ap %I PMLR %P 162093--162113 %U https://proceedings.mlr.press/v306/zhao26ap.html %V 306 %X Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert’s exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert allocation per token. Across benchmarks and model backbones, ProbMoE Exact-$k$ achieves strong performance compared to competitive baselines, with improved expert utilization and routing diversity; ProbMoE Dynamic-$k$ achieves comparable performance with fewer activated experts. Code is available at: https://github.com/HengHugoZhao/ProbMoE.git
APA
Zhao, H., Shao, Z., Van Den Broeck, G. & Zeng, Z.. (2026). ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:162093-162113 Available from https://proceedings.mlr.press/v306/zhao26ap.html.

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