# Rethinking Multinomial Logistic Mixture of Experts with Sigmoid Gating Function

arXiv: https://arxiv.org/abs/2602.01466 · HTML: https://mlsanity.com/paper/2602.01466/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 30 of its 39 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 30 of 30)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [M3ViT: Mixture-of-Experts Vision Transformer for Efficient Multi-task Learning with Model-Accelerator Co-design](https://mlsanity.com/paper/2210.14793/) (2022)
- [Sparse Mixture-of-Experts are Domain Generalizable Learners](https://mlsanity.com/paper/2206.04046/) (2022)
- [Flex-MoE: Modeling Arbitrary Modality Combination via the Flexible Mixture-of-Experts](https://mlsanity.com/paper/2410.08245/) (2024)
- [FuseMoE: Mixture-of-Experts Transformers for Fleximodal Fusion](https://mlsanity.com/paper/2402.03226/) (2024)
- [EM Converges for a Mixture of Many Linear Regressions](https://mlsanity.com/paper/1905.12106/) (2019)
- [Approximating Two-Layer Feedforward Networks for Efficient Transformers](https://mlsanity.com/paper/2310.10837/) (2023)
- [Sigmoid Gating is More Sample Efficient than Softmax Gating in Mixture of Experts](https://mlsanity.com/paper/2405.13997/) (2024)
- [A General Theory for Softmax Gating Multinomial Logistic Mixture of Experts](https://mlsanity.com/paper/2310.14188/) (2023)
- [Refined Convergence Rates for Maximum Likelihood Estimation under Finite Mixture Models](https://mlsanity.com/paper/2202.08786/) (2022)
- [Statistical Advantages of Perturbing Cosine Router in Mixture of Experts](https://mlsanity.com/paper/2405.14131/) (2024)
- [Revisiting Prefix-tuning: Statistical Benefits of Reparameterization among Prompts](https://mlsanity.com/paper/2410.02200/) (2024)
- [The power of fine-grained experts: Granularity boosts expressivity in Mixture of Experts](https://mlsanity.com/paper/2505.06839/) (2025)
- [Time series modeling by a regression approach based on a latent process](https://mlsanity.com/paper/1312.6969/) (2013)
- [RepLoRA: Reparameterizing Low-Rank Adaptation via the Perspective of Mixture of Experts](https://mlsanity.com/paper/2502.03044/) (2025)
- [On DeepSeekMoE: Statistical Benefits of Shared Experts and Normalized Sigmoid Gating](https://mlsanity.com/paper/2505.10860/) (2025)
- [On Zero-Initialized Attention: Optimal Prompt and Gating Factor Estimation](https://mlsanity.com/paper/2502.03029/) (2025)
- [On the Expressive Power of Mixture-of-Experts for Structured Complex Tasks](https://mlsanity.com/paper/2505.24205/) (2025)
- [Functional mixture-of-experts for classification](https://mlsanity.com/paper/2202.13934/) (2022)
- [The EM-algorithm and the Method of Moments in Softmax Mixture Models](https://mlsanity.com/paper/2409.09903/) (2024)

## Abstract (arXiv)

The sigmoid gate in mixture-of-experts (MoE) models has been empirically shown to outperform the softmax gate across several tasks, ranging from approximating feed-forward networks to language modeling. Additionally, recent efforts have demonstrated that the sigmoid gate is provably more sample-efficient than its softmax counterpart under regression settings. Nevertheless, there are three notable concerns that have not been addressed in the literature, namely (i) the benefits of the sigmoid gate have not been established under classification settings; (ii) existing sigmoid-gated MoE models may not converge to their ground-truth; and (iii) the effects of a temperature parameter in the sigmoid gate remain theoretically underexplored. To tackle these open problems, we perform a comprehensive analysis of multinomial logistic MoE equipped with a modified sigmoid gate to ensure model convergence. Our results indicate that the sigmoid gate exhibits a lower sample complexity than the softmax gate for both parameter and expert estimation. Furthermore, we find that incorporating a temperature into the sigmoid gate leads to a sample complexity of exponential order due to an intrinsic interaction between the temperature and gating parameters. To overcome this issue, we propose replacing the vanilla inner product score in the gating function with a Euclidean score that effectively removes that interaction, thereby substantially improving the sample complexity to a polynomial order.