# Sigmoid Self-Attention has Lower Sample Complexity than Softmax Self-Attention: A Mixture-of-Experts Perspective

arXiv: https://arxiv.org/abs/2502.00281 · HTML: https://mlsanity.com/paper/2502.00281/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Expert routing strategies, one of 12 directions in Mixture-of-experts.
- Builds on 38 of its 51 references that are in the map.
- First posted on arXiv 1 Feb 2025.

## Placed in

- Mixture-of-experts / [Expert routing strategies](https://mlsanity.com/moe/expert-routing-strategies/)

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [OLMoE: Open Mixture-of-Experts Language Models](https://mlsanity.com/paper/2409.02060/) (2024)
- [From Sparse to Soft Mixtures of Experts](https://mlsanity.com/paper/2308.00951/) (2023)
- [When Attention Sink Emerges in Language Models: An Empirical View](https://mlsanity.com/paper/2410.10781/) (2024)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [DSelect-k: Differentiable Selection in the Mixture of Experts with Applications to Multi-Task Learning](https://mlsanity.com/paper/2106.03760/) (2021)
- [Preference Transformer: Modeling Human Preferences using Transformers for RL](https://mlsanity.com/paper/2303.00957/) (2023)
- [Theory, Analysis, and Best Practices for Sigmoid Self-Attention](https://mlsanity.com/paper/2409.04431/) (2024)
- [SpeechMoE: Scaling to Large Acoustic Models with Dynamic Routing Mixture of Experts](https://mlsanity.com/paper/2105.03036/) (2021)
- [MoH: Multi-Head Attention as Mixture-of-Head Attention](https://mlsanity.com/paper/2410.11842/) (2024)
- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [SwitchHead: Accelerating Transformers with Mixture-of-Experts Attention](https://mlsanity.com/paper/2312.07987/) (2023)
- [Demystifying Softmax Gating Function in Gaussian Mixture of Experts](https://mlsanity.com/paper/2305.03288/) (2023)
- [MoLE : Mixture of Language Experts for Multi-Lingual Automatic Speech Recognition](https://mlsanity.com/paper/2302.13750/) (2023)
- [Brainformers: Trading Simplicity for Efficiency](https://mlsanity.com/paper/2306.00008/) (2023)

## Built on by (most cited first, 4 of 4)

- [Weaver: Kronecker Product Approximations of Spatiotemporal Attention for Traffic Network Forecasting](https://mlsanity.com/paper/2511.08888/) (2025)
- [Integrated electro-optic attention nonlinearities for transformers](https://mlsanity.com/paper/2604.09512/) (2026)
- [Boltzmann Attention: Learnable Ising Couplings for Cooperative Attention](https://mlsanity.com/paper/2606.12478/) (2026)
- [PANDA: Prior-guided Attentional Dual-path Architecture](https://mlsanity.com/paper/2610.13635/) (2026)

## Abstract (arXiv)

At the core of the popular Transformer architecture is the self-attention mechanism, which dynamically assigns softmax weights to each input token so that the model can focus on the most salient information. However, the softmax structure slows down the attention computation due to its row-wise nature, and it inherently introduces competition among tokens: as the weight assigned to one token increases, the weights of others decrease. This competitive dynamic may narrow the focus of self-attention to a limited set of features, potentially overlooking other informative characteristics. Recent experimental studies have shown that using the element-wise sigmoid function helps eliminate token competition and reduce the computational overhead. Despite these promising empirical results, a rigorous comparison between sigmoid and softmax self-attention mechanisms remains absent in the literature. This paper closes this gap by theoretically demonstrating that sigmoid self-attention is more sample-efficient than its softmax counterpart. Toward that goal, we represent the self-attention matrix as a mixture of experts and show that ``experts'' in sigmoid self-attention require significantly less data to achieve the same approximation error as those in softmax self-attention.