# Statistical Advantages of Perturbing Cosine Router in Mixture of Experts

arXiv: https://arxiv.org/abs/2405.14131 · HTML: https://mlsanity.com/paper/2405.14131/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Placed in Expert routing strategies, one of 12 directions in Mixture-of-experts.
- Builds on 34 of its 52 references that are in the map.
- First posted on arXiv 23 May 2024.

## Placed in

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

## Builds on (most cited first, 30 of 34)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Conformer: Convolution-augmented Transformer for Speech Recognition](https://mlsanity.com/paper/2005.08100/) (2020)
- [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)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [Domain Generalization: A Survey](https://mlsanity.com/paper/2103.02503/) (2021)
- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [From Sparse to Soft Mixtures of Experts](https://mlsanity.com/paper/2308.00951/) (2023)
- [A Fine-Grained Analysis on Distribution Shift](https://mlsanity.com/paper/2110.11328/) (2021)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [DSelect-k: Differentiable Selection in the Mixture of Experts with Applications to Multi-Task Learning](https://mlsanity.com/paper/2106.03760/) (2021)
- [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)
- [FuseMoE: Mixture-of-Experts Transformers for Fleximodal Fusion](https://mlsanity.com/paper/2402.03226/) (2024)
- [SpeechMoE: Scaling to Large Acoustic Models with Dynamic Routing Mixture of Experts](https://mlsanity.com/paper/2105.03036/) (2021)
- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [Demystifying Softmax Gating Function in Gaussian Mixture of Experts](https://mlsanity.com/paper/2305.03288/) (2023)
- [Brainformers: Trading Simplicity for Efficiency](https://mlsanity.com/paper/2306.00008/) (2023)
- [Statistical Perspective of Top-K Sparse Softmax Gating Mixture of Experts](https://mlsanity.com/paper/2309.13850/) (2023)
- [CompeteSMoE -- Effective Training of Sparse Mixture of Experts via Competition](https://mlsanity.com/paper/2402.02526/) (2024)
- [On Least Square Estimation in Softmax Gating Mixture of Experts](https://mlsanity.com/paper/2402.02952/) (2024)
- [Refined Convergence Rates for Maximum Likelihood Estimation under Finite Mixture Models](https://mlsanity.com/paper/2202.08786/) (2022)
- [Improving Expert Specialization in Mixture of Experts](https://mlsanity.com/paper/2302.14703/) (2023)

## Built on by (most cited first, 16 of 16)

- [A Comprehensive Survey of Mixture-of-Experts: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2503.07137/) (2025)
- [LIBMoE: A Library for comprehensive benchmarking Mixture of Experts in Large Language Models](https://mlsanity.com/paper/2411.00918/) (2024)
- [RepLoRA: Reparameterizing Low-Rank Adaptation via the Perspective of Mixture of Experts](https://mlsanity.com/paper/2502.03044/) (2025)
- [On Zero-Initialized Attention: Optimal Prompt and Gating Factor Estimation](https://mlsanity.com/paper/2502.03029/) (2025)
- [Convergence Rates for Softmax Gating Mixture of Experts](https://mlsanity.com/paper/2503.03213/) (2025)
- [Sparsity is Combinatorial Depth: Quantifying MoE Expressivity via Tropical Geometry](https://mlsanity.com/paper/2602.03204/) (2026)
- [One-Prompt Strikes Back: Sparse Mixture of Experts for Prompt-based Continual Learning](https://mlsanity.com/paper/2509.24483/) (2025)
- [On Minimax Estimation of Parameters in Softmax-Contaminated Mixture of Experts](https://mlsanity.com/paper/2505.18455/) (2025)
- [L2R: Low-Rank and Lipschitz-Controlled Routing for Mixture-of-Experts](https://mlsanity.com/paper/2601.21349/) (2026)
- [Equifinality in Mixture of Experts: Routing Topology Does Not Determine Language Modeling Quality](https://mlsanity.com/paper/2604.14419/) (2026)
- [Soft-to-Hard Routing in Sparse Mixture-of-Experts Models](https://mlsanity.com/paper/2605.02124/) (2026)
- [LiMoDE: Rethinking Lifelong Robot Manipulation from a Mixture-of-Dynamic-Experts Perspective](https://mlsanity.com/paper/2606.26183/) (2026)
- [Cosine-Similarity Routing with Semantic Anchors for Interpretable Mixture-of-Experts Language Models](https://mlsanity.com/paper/2509.14255/) (2025)
- [Improving Minimax Estimation Rates for Contaminated Mixture of Multinomial Logistic Experts via Expert Heterogeneity](https://mlsanity.com/paper/2602.00939/) (2026)
- [Rethinking Multinomial Logistic Mixture of Experts with Sigmoid Gating Function](https://mlsanity.com/paper/2602.01466/) (2026)
- [Share First, Route What Remains: A Unified Framework for Token-Adaptive MoE Computation](https://mlsanity.com/paper/2608.10392/) (2026)

## Abstract (arXiv)

The cosine router in Mixture of Experts (MoE) has recently emerged as an attractive alternative to the conventional linear router. Indeed, the cosine router demonstrates favorable performance in image and language tasks and exhibits better ability to mitigate the representation collapse issue, which often leads to parameter redundancy and limited representation potentials. Despite its empirical success, a comprehensive analysis of the cosine router in MoE has been lacking. Considering the least square estimation of the cosine routing MoE, we demonstrate that due to the intrinsic interaction of the model parameters in the cosine router via some partial differential equations, regardless of the structures of the experts, the estimation rates of experts and model parameters can be as slow as O(1/logτ(n)) where τ > 0 is some constant and n is the sample size. Surprisingly, these pessimistic non-polynomial convergence rates can be circumvented by the widely used technique in practice to stabilize the cosine router -- simply adding noises to the ℓ2-norms in the cosine router, which we refer to as textitperturbed cosine router. Under the strongly identifiable settings of the expert functions, we prove that the estimation rates for both the experts and model parameters under the perturbed cosine routing MoE are significantly improved to polynomial rates. Finally, we conduct extensive simulation studies in both synthetic and real data settings to empirically validate our theoretical results.