# On Minimax Estimation of Parameters in Softmax-Contaminated Mixture of Experts

arXiv: https://arxiv.org/abs/2505.18455 · HTML: https://mlsanity.com/paper/2505.18455/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 31 of its 46 references that are in the map.
- First posted on arXiv 24 May 2025.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 30 of 31)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [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)
- [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)
- [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)
- [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)
- [Mixtures of Experts Unlock Parameter Scaling for Deep RL](https://mlsanity.com/paper/2402.08609/) (2024)
- [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)
- [Theory on Mixture-of-Experts in Continual Learning](https://mlsanity.com/paper/2406.16437/) (2024)
- [SpeechMoE: Scaling to Large Acoustic Models with Dynamic Routing Mixture of Experts](https://mlsanity.com/paper/2105.03036/) (2021)
- [Mixture of Experts Meets Prompt-Based Continual Learning](https://mlsanity.com/paper/2405.14124/) (2024)
- [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)
- [EM Converges for a Mixture of Many Linear Regressions](https://mlsanity.com/paper/1905.12106/) (2019)
- [SpeechMoE2: Mixture-of-Experts Model with Improved Routing](https://mlsanity.com/paper/2111.11831/) (2021)
- [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)
- [A Mixture-of-Expert Approach to RL-based Dialogue Management](https://mlsanity.com/paper/2206.00059/) (2022)
- [Statistical Advantages of Perturbing Cosine Router in Mixture of Experts](https://mlsanity.com/paper/2405.14131/) (2024)
- [On Expert Estimation in Hierarchical Mixture of Experts: Beyond Softmax Gating Functions](https://mlsanity.com/paper/2410.02935/) (2024)
- [Is Temperature Sample Efficient for Softmax Gaussian Mixture of Experts?](https://mlsanity.com/paper/2401.13875/) (2024)
- [Revisiting Prefix-tuning: Statistical Benefits of Reparameterization among Prompts](https://mlsanity.com/paper/2410.02200/) (2024)
- [Understanding Expert Structures on Minimax Parameter Estimation in Contaminated Mixture of Experts](https://mlsanity.com/paper/2410.12258/) (2024)
- [Revisit Visual Prompt Tuning: The Expressiveness of Prompt Experts](https://mlsanity.com/paper/2501.18936/) (2025)
- [On Parameter Estimation in Deviated Gaussian Mixture of Experts](https://mlsanity.com/paper/2402.05220/) (2024)

## Built on by (most cited first, 1 of 1)

- [Improving Minimax Estimation Rates for Contaminated Mixture of Multinomial Logistic Experts via Expert Heterogeneity](https://mlsanity.com/paper/2602.00939/) (2026)

## Abstract (arXiv)

The softmax-contaminated mixture of experts (MoE) model is deployed when a large-scale pre-trained model, which plays the role of a fixed expert, is fine-tuned for learning downstream tasks by including a new contamination part, or prompt, functioning as a new, trainable expert. Despite its popularity and relevance, the theoretical properties of the softmax-contaminated MoE have remained unexplored in the literature. In the paper, we study the convergence rates of the maximum likelihood estimator of gating and prompt parameters in order to gain insights into the statistical properties and potential challenges of fine-tuning with a new prompt. We find that the estimability of these parameters is compromised when the prompt acquires overlapping knowledge with the pre-trained model, in the sense that we make precise by formulating a novel analytic notion of distinguishability. Under distinguishability of the pre-trained and prompt models, we derive minimax optimal estimation rates for all the gating and prompt parameters. By contrast, when the distinguishability condition is violated, these estimation rates become significantly slower due to their dependence on the prompt convergence rate to the pre-trained model. Finally, we empirically corroborate our theoretical findings through several numerical experiments.