# Theory on Mixture-of-Experts in Continual Learning

arXiv: https://arxiv.org/abs/2406.16437 · HTML: https://mlsanity.com/paper/2406.16437/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Placed in Mixture of LoRA experts, one of 12 directions in Mixture-of-experts.
- Builds on 44 of its 58 references that are in the map.
- First posted on arXiv 24 Jun 2024.

## Placed in

- Mixture-of-experts / [Mixture of LoRA experts](https://mlsanity.com/moe/mixture-of-lora-experts/)

## Builds on (most cited first, 30 of 44)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [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)
- [Knowledge Distillation: A Survey](https://mlsanity.com/paper/2006.05525/) (2020)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Scaling Vision with Sparse Mixture of Experts](https://mlsanity.com/paper/2106.05974/) (2021)
- [Overcoming catastrophic forgetting with hard attention to the task](https://mlsanity.com/paper/1801.01423/) (2018)
- [Mixture-of-Experts with Expert Choice Routing](https://mlsanity.com/paper/2202.09368/) (2022)
- [Orthogonal Gradient Descent for Continual Learning](https://mlsanity.com/paper/1910.07104/) (2019)
- [Gradient Projection Memory for Continual Learning](https://mlsanity.com/paper/2103.09762/) (2021)
- [MoE-LLaVA: Mixture of Experts for Large Vision-Language Models](https://mlsanity.com/paper/2401.15947/) (2024)
- [Learning Factored Representations in a Deep Mixture of Experts](https://mlsanity.com/paper/1312.4314/) (2013)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Online Structured Laplace Approximations For Overcoming Catastrophic Forgetting](https://mlsanity.com/paper/1805.07810/) (2018)
- [Boosting Continual Learning of Vision-Language Models via Mixture-of-Experts Adapters](https://mlsanity.com/paper/2403.11549/) (2024)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [OpenMoE: An Early Effort on Open Mixture-of-Experts Language Models](https://mlsanity.com/paper/2402.01739/) (2024)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [Pushing Mixture of Experts to the Limit: Extremely Parameter Efficient MoE for Instruction Tuning](https://mlsanity.com/paper/2309.05444/) (2023)
- [A Neural Dirichlet Process Mixture Model for Task-Free Continual Learning](https://mlsanity.com/paper/2001.00689/) (2020)
- [Scalable and Order-robust Continual Learning with Additive Parameter Decomposition](https://mlsanity.com/paper/1902.09432/) (2019)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [A Theoretical Analysis of Catastrophic Forgetting through the NTK Overlap Matrix](https://mlsanity.com/paper/2010.04003/) (2020)
- [Continual Learning in the Teacher-Student Setup: Impact of Task Similarity](https://mlsanity.com/paper/2107.04384/) (2021)
- [TRGP: Trust Region Gradient Projection for Continual Learning](https://mlsanity.com/paper/2202.02931/) (2022)
- [How catastrophic can catastrophic forgetting be in linear regression?](https://mlsanity.com/paper/2205.09588/) (2022)
- [Theory on Forgetting and Generalization of Continual Learning](https://mlsanity.com/paper/2302.05836/) (2023)

## Built on by (most cited first, 30 of 54)

- [A Comprehensive Survey of Mixture-of-Experts: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2503.07137/) (2025)
- [Mixture of A Million Experts](https://mlsanity.com/paper/2407.04153/) (2024)
- [A Survey on Model MoErging: Recycling and Routing Among Specialized Experts for Collaborative Learning](https://mlsanity.com/paper/2408.07057/) (2024)
- [Drift-aware Collaborative Assistance Mixture of Experts for Heterogeneous Multistream Learning](https://mlsanity.com/paper/2508.01598/) (2025)
- [Monet: Mixture of Monosemantic Experts for Transformers](https://mlsanity.com/paper/2412.04139/) (2024)
- [Unlocking the Power of Rehearsal in Continual Learning: A Theoretical Perspective](https://mlsanity.com/paper/2506.00205/) (2025)
- [KIF: Knowledge Identification and Fusion for Language Model Continual Learning](https://mlsanity.com/paper/2408.05200/) (2024)
- [Hierarchical-Task-Aware Multi-modal Mixture of Incremental LoRA Experts for Embodied Continual Learning](https://mlsanity.com/paper/2506.04595/) (2025)
- [Nile-Chat: Egyptian Language Models for Arabic and Latin Scripts](https://mlsanity.com/paper/2507.04569/) (2025)
- [Theory of Mixture-of-Experts for Mobile Edge Computing](https://mlsanity.com/paper/2412.15690/) (2024)
- [Little by Little: Continual Learning via Incremental Mixture of Rank-1 Associative Memory Experts](https://mlsanity.com/paper/2506.21035/) (2025)
- [On Zero-Initialized Attention: Optimal Prompt and Gating Factor Estimation](https://mlsanity.com/paper/2502.03029/) (2025)
- [Continual Adaptation for Autonomous Driving with the Mixture of Progressive Experts Network](https://mlsanity.com/paper/2502.05943/) (2025)
- [Plasticity-Aware Mixture of Experts for Learning Under QoE Shifts in Adaptive Video Streaming](https://mlsanity.com/paper/2504.09906/) (2025)
- [On the Expressive Power of Mixture-of-Experts for Structured Complex Tasks](https://mlsanity.com/paper/2505.24205/) (2025)
- [Mixture of Experts Provably Detect and Learn the Latent Cluster Structure in Gradient-Based Learning](https://mlsanity.com/paper/2506.01656/) (2025)
- [Putting a Face to Forgetting: Continual Learning meets Mechanistic Interpretability](https://mlsanity.com/paper/2601.22012/) (2026)
- [Seeing Clearly, Forgetting Deeply: Revisiting Fine-Tuned Video Generators for Driving Simulation](https://mlsanity.com/paper/2508.16512/) (2025)
- [On the Theory of Continual Learning with Gradient Descent for Neural Networks](https://mlsanity.com/paper/2510.05573/) (2025)
- [Continual Policy Consolidation for Lifelong Robot Learning](https://mlsanity.com/paper/2601.22475/) (2026)
- [HiCL: Hippocampal-Inspired Continual Learning](https://mlsanity.com/paper/2508.16651/) (2025)
- [Guided by the Experts: Provable Feature Learning Dynamic of Soft-Routed Mixture-of-Experts](https://mlsanity.com/paper/2510.07205/) (2025)
- [CP-MoE: Consistency-Preserving Mixture-of-Experts for Continual Learning](https://mlsanity.com/paper/2605.20247/) (2026)
- [Algorithm Design for Continual Learning in IoT Networks](https://mlsanity.com/paper/2412.16830/) (2024)
- [On Minimax Estimation of Parameters in Softmax-Contaminated Mixture of Experts](https://mlsanity.com/paper/2505.18455/) (2025)
- [DriftMoE: A Mixture of Experts Approach to Handle Concept Drifts](https://mlsanity.com/paper/2507.18464/) (2025)
- [STM3: Mixture of Multiscale Mamba for Long-Term Spatio-Temporal Time-Series Prediction](https://mlsanity.com/paper/2508.12247/) (2025)
- [RobIA: Robust Instance-aware Continual Test-time Adaptation for Deep Stereo](https://mlsanity.com/paper/2511.10107/) (2025)
- [MECAD: A multi-expert architecture for continual anomaly detection](https://mlsanity.com/paper/2512.15323/) (2025)
- [Robustness of Mixtures of Experts to Feature Noise](https://mlsanity.com/paper/2601.14792/) (2026)

## Abstract (arXiv)

Continual learning (CL) has garnered significant attention because of its ability to adapt to new tasks that arrive over time. Catastrophic forgetting (of old tasks) has been identified as a major issue in CL, as the model adapts to new tasks. The Mixture-of-Experts (MoE) model has recently been shown to effectively mitigate catastrophic forgetting in CL, by employing a gating network to sparsify and distribute diverse tasks among multiple experts. However, there is a lack of theoretical analysis of MoE and its impact on the learning performance in CL. This paper provides the first theoretical results to characterize the impact of MoE in CL via the lens of overparameterized linear regression tasks. We establish the benefit of MoE over a single expert by proving that the MoE model can diversify its experts to specialize in different tasks, while its router learns to select the right expert for each task and balance the loads across all experts. Our study further suggests an intriguing fact that the MoE in CL needs to terminate the update of the gating network after sufficient training rounds to attain system convergence, which is not needed in the existing MoE studies that do not consider the continual task arrival. Furthermore, we provide explicit expressions for the expected forgetting and overall generalization error to characterize the benefit of MoE in the learning performance in CL. Interestingly, adding more experts requires additional rounds before convergence, which may not enhance the learning performance. Finally, we conduct experiments on both synthetic and real datasets to extend these insights from linear models to deep neural networks (DNNs), which also shed light on the practical algorithm design for MoE in CL.