Mixture of experts

Short description: Machine learning technique

"MoE" redirects here. For other uses, see Moe.

Machine learning and
data mining

Problems

* Classification
* Clustering
* Regression
* Anomaly detection
* AutoML
* Association rules
* Reinforcement learning
* Structured prediction
* Feature engineering
* Feature learning
* Online learning
* Semi-supervised learning
* Unsupervised learning
* Learning to rank
* Grammar induction

Supervised learning
(classification • regression)

* Decision trees
* Ensembles
* Bagging
* Boosting
* Random forest
* k-NN
* Linear regression
* Naive Bayes
* Artificial neural networks
* Logistic regression
* Perceptron
* Relevance vector machine (RVM)
* Support vector machine (SVM)

Clustering

* BIRCH
* CURE
* Hierarchical
* k-means
* Expectation–maximization (EM)

DBSCAN
* OPTICS
* Mean-shift

Dimensionality reduction

* Factor analysis
* CCA
* ICA
* LDA
* NMF
* PCA
* t-SNE

Structured prediction

* Graphical models
* Bayes net
* Conditional random field
* Hidden Markov

Anomaly detection

* k-NN
* Local outlier factor

Artificial neural networks

* Autoencoder
* Deep learning
* DeepDream
* Multilayer perceptron
* RNN
* LSTM
* GRU
* Restricted Boltzmann machine
* GAN
* SOM
* Convolutional neural network
* U-Net

Reinforcement learning

* Q-learning
* SARSA
* Temporal difference (TD)

Theory

* Bias–variance dilemma
* Computational learning theory
* Empirical risk minimization
* Occam learning
* PAC learning
* Statistical learning
* VC theory

Machine-learning venues

* NeurIPS
* ICML
* ML
* JMLR
* ArXiv:cs.LG

Glossary of artificial intelligence

* Glossary of artificial intelligence

Related articles

* List of datasets for machine-learning research
* Outline of machine learning

* v
* t
* e

Mixture of experts (MoE) is a machine learning technique where multiple expert networks (learners) are used to divide a problem space into homogeneous regions.[1] MoE represents a form of ensemble learning.[2] They were also called committee machines.[3]

## Contents

* 1 Basic theory
* 1.1 Meta-pi network
* 1.2 Adaptive mixtures of local experts
* 1.3 Hierarchical MoE
* 1.4 Variants
* 2 Deep learning
* 2.1 Sparsely-gated MoE layer
* 2.2 Load balancing
* 2.3 Capacity factor
* 2.4 Routing
* 2.5 Applications to transformer models
* 3 See also
* 4 References
* 5 Further reading

## Basic theory

MoE always has the following components, but they are implemented and combined differently according to the problem being solved:

* Experts f1,...,fn, each taking the same input x, and producing outputs f1(x),...,fn(x).
* A weighting function (also known as a gating function) w, which takes input x and produces a vector of outputs (w(x)1,...,w(x)n). This may or may not be a probability distribution, but in both cases, its entries are non-negative.
* θ=(θ0,θ1,...,θn) is the set of parameters. The parameter θ0 is for the weighting function. The parameters θ1,…,θn are for the experts.
* Given an input x, the mixture of experts produces a single output by combining f1(x),...,fn(x) according to the weights w(x)1,...,w(x)n in some way, usually by f(x)=∑iw(x)ifi(x).

Both the experts and the weighting function are trained by minimizing some loss function, generally via gradient descent. There is much freedom in choosing the precise form of experts, the weighting function, and the loss function.

### Meta-pi network

The meta-pi network, reported by Hampshire and Waibel,[4] uses f(x)=∑iw(x)ifi(x) as the output. The model is trained by performing gradient descent on the mean-squared error loss L:=1N∑k‖yk−f(xk)‖2. The experts may be arbitrary functions.

In their original publication, they were solving the problem of classifying phonemes in speech signal from 6 different Japanese speakers, 2 females and 4 males. They trained 6 experts, each being a "time-delayed neural network"[5] (essentially a multilayered convolution network over the mel spectrogram). They found that the resulting mixture of experts dedicated 5 experts for 5 of the speakers, but the 6th (male) speaker does not have a dedicated expert, instead his voice was classified by a linear combination of the experts for the other 3 male speakers.

### Adaptive mixtures of local experts

The adaptive mixtures of local experts [6][7] uses a Gaussian mixture model. Each expert simply predicts a Gaussian distribution, and totally ignores the input. Specifically, the i-th expert predicts that the output is y∼N(μi,I), where μi is a learnable parameter. The weighting function is a linear-softmax function:w(x)i=ekiTx+bi∑jekjTx+bjThe mixture of experts predict that the output is distributed according to the log-probability density function:ln⁡fθ(y|x)=ln⁡[∑iekiTx+bi∑jekjTx+bjN(y|μi,I)]=ln⁡[(2π)−d/2∑iekiTx+bi∑jekjTx+bje−12‖y−μi‖2]It is trained by maximal likelihood estimation, that is, gradient ascent on f(y|x). The gradient for the i-th expert is

∇μiln⁡fθ(y|x)=w(x)iN(y|μi,I)∑jw(x)jN(y|μj,I)(y−μi)

and the gradient for the weighting function is∇[ki,bi]ln⁡fθ(y|x)=[x1]w(x)i∑jw(x)jN(y|μj,I)(fi(x)−fθ(y|x))

For each input-output pair (x,y), the weighting function is changed to increase the weight on all experts that performed above average, and decrease the weight on all experts that performed below average. This encourages the weighting function to learn to select only the experts that make the right predictions for each input.

The i-th expert is changed to make its prediction closer to y, but the amount of change is proportional to w(x)iN(y|μi,I). This has a Bayesian interpretation. Given input x, the prior probability that expert i is the right one is w(x)i, and N(y|μi,I) is the likelihood of evidence y. So, w(x)iN(y|μi,I)∑jw(x)jN(y|μj,I) is the posterior probability for expert i, and so the rate of change for the i-th expert is proportional to its posterior probability.

In words, the experts that, in hindsight, seemed like the good experts to consult, are asked to learn on the example. The experts that, in hindsight, were not, are left alone.

The combined effect is that the experts become specialized: Suppose two experts are both good at predicting a certain kind of input, but one is slightly better, then the weighting function would eventually learn to favor the better one. After that happens, the lesser expert is unable to obtain a high gradient signal, and becomes even worse at predicting such kind of input. Conversely, the lesser expert can become better at predicting other kinds of input, and increasingly pulled away into another region. This has a positive feedback effect, causing each expert to move apart from the rest and take care of a local region alone (thus the name "local experts").

### Hierarchical MoE

Hierarchical mixtures of experts[8][9] uses multiple levels of gating in a tree. Each gating is a probability distribution over the next level of gatings, and the experts are on the leaf nodes of the tree. They are similar to decision trees.

For example, a 2-level hierarchical MoE would have a first order gating function wi, and second order gating functions wj|i and experts fj|i. The total prediction is then ∑iwi(x)∑jwj|i(x)fj|i(x).

### Variants

The mixture of experts, being similar to the gaussian mixture model, can also be trained by the expectation-maximization algorithm, just like gaussian mixture models. Specifically, during the expectation step, the "burden" for explaining each data point is assigned over the experts, and during the maximization step, the experts are trained to improve the explanations they got a high burden for, while the gate is trained to improve its burden assignment. This can converge faster than gradient ascent on the log-likelihood.[9][10]

The choice of gating function is often softmax. Other than that, gating may use gaussian distributions[11] and exponential families.[10]

Instead of performing a weighted sum of all the experts, in hard MoE,[12] only the highest ranked expert is chosen. That is, f(x)=farg⁡maxiwi(x)(x). This can accelerate training and inference time.[13]

The experts can use more general forms of multivariant gaussian distributions. For example,[8] proposed fi(y|x)=N(y|Aix+bi,Σi), where Ai,bi,Σi are learnable parameters. In words, each expert learns to do linear regression, with a learnable uncertainty estimate.

One can use different experts than gaussian distributions. For example, one can use Laplace distribution,[14] or Student's t-distribution.[15] For binary classification, it also proposed logistic regression experts, withfi(y|x)={11+eβiTx+βi,0,y=01−11+eβiTx+βi,0,y=1where βi,βi,0 are learnable parameters. This is later generalized for multi-class classification, with multinomial logistic regression experts.[16]

One paper proposed mixture of softmaxes for autoregressive language modelling.[17] Specifically, consider a language model that given a previous text c, predicts the next word x. The network encodes the text into a vector vc, and predicts the probability distribution of the next word as Softmax(vcW) for an embedding matrix W. In mixture of softmaxes, the model outputs multiple vectors vc,1,…,vc,n, and predict the next word as ∑i=1npiSoftmax(vc,iWi), where pi is a probability distribution by a linear-softmax operation on the activations of the hidden neurons within the model. The original paper demonstrated its effectiveness for recurrent neural networks. This was later found to work for Transformers as well.[18]

## Deep learning

The previous section described MoE as it was used before the era of deep learning. After deep learning, MoE found applications in running the largest models, as a simple way to perform conditional computation: only parts of the model are used, the parts chosen according to what the input is.[19]

The earliest paper that applies MoE to deep learning dates back to 2013,[20] which proposed to use a different gating network at each layer in a deep neural network. Specifically, each gating is a linear-ReLU-linear-softmax network, and each expert is a linear-ReLU network. Since the output from the gating is not sparse, all expert outputs are needed, and no conditional computation is performed.

The key goal when using MoE in deep learning is to reduce computing cost. Consequently, for each query, only a small subset of the experts should be queried. This makes MoE in deep learning different from classical MoE. In classical MoE, the output for each query is a weighted sum of all experts' outputs. In deep learning MoE, the output for each query can only involve a few experts' outputs. Consequently, the key design choice in MoE becomes routing: given a batch of queries, how to route the queries to the best experts.

### Sparsely-gated MoE layer

The sparsely-gated MoE layer,[21] published by researchers from Google Brain, uses feedforward networks as experts, and linear-softmax gating. Similar to the previously proposed hard MoE, they achieve sparsity by a weighted sum of only the top-k experts, instead of the weighted sum of all of them. Specifically, in a MoE layer, there are feedforward networks f1,...,fn, and a gating network w. The gating network is defined by w(x)=softmax(topk(Wx+noise)), where topk is a function that keeps the top-k entries of a vector the same, but sets all other entries to −∞. The addition of noise helps with load balancing.

The choice of k is a hyperparameter that is chosen according to application. Typical values are k=1,2. The k=1 version is also called the Switch Transformer. The original Switch Transformer was applied to a T5 language model.[22]

As demonstration, they trained a series of models for machine translation with alternating layers of MoE and LSTM, and compared with deep LSTM models.[23] Table 3 shows that the MoE models used less inference time compute, despite having 30x more parameters.

This architectural module was published in 2017-01, within a few months of the publication of the Transformer architecture (2017-06-12), and they were combined into a multimodal architecture called MultiModel published 4 days later (2017-06-16).[24]

### Load balancing

Vanilla MoE tend to have issues of load balancing: some experts are consulted often, while other experts rarely or not at all. To encourage the gate to select each expert with equal frequency (proper load balancing) within each batch, each MoE layer has two auxiliary loss functions. This is improved by Switch Transformer[22] into a single auxiliary loss function. Specifically, let

be the number of experts, then for a given batch of queries

{x1,x2,...,xT}

, the auxiliary loss for the batch is

n∑i=1nfiPi

Here,

fi=1T#(queries sent to expert i)

is the fraction of tokens that chose expert

, and

Pi=1T∑j=1Twi(xj)∑i′∈expertswi′(xj)

is the fraction of weight on expert

. This loss is minimized at

, precisely when every expert has equal weight

1/n

in all situations.

The DeepSeek MoE architecture. Also shown is MLA, a variant of attention mechanism in Transformer.[25]Template:Pg

Researchers at DeepSeek designed a variant of MoE, with "shared experts" that are always queried, and "routed experts" that might not be. They found that standard load balancing encourages the experts to be equally consulted, but this then causes experts to replicate the same core capacity, such as English grammar. They proposed the shared experts to learn core capacities that are often used, and let the routed experts to learn the peripheral capacities that are rarely used.[26]

They also proposed "auxiliary-loss-free load balancing strategy", which does not use auxiliary loss. Instead, each expert i has an extra "expert bias" bi. If an expert is being neglected, then their bias increases, and vice versa. During token assignment, each token picks the top-k experts, but with the bias added in. That is:[27]f(x)=∑i is in the top-k of {w(x)j+bj}jw(x)ifi(x)Note that the expert bias matters for picking the experts, but not in adding up the responses from the experts.

### Capacity factor

Suppose there are n experts in a layer. For a given batch of queries {x1,x2,...,xT}, each query is routed to one or more experts. For example, if each query is routed to one expert as in Switch Transformers, and if the experts are load-balanced, then each expert should expect on average T/n queries in a batch. In practice, the experts cannot expect perfect load balancing: in some batches, one expert might be underworked, while in other batches, it would be overworked.

Since the inputs cannot move through the layer until every expert in the layer has finished the queries it is assigned, load balancing is important. The capacity factor is sometimes used to enforce a hard constraint on load balancing. Each expert is only allowed to process up to c⋅T/n queries in a batch. The ST-MoE report found c∈[1.25,2] to work well in practice.[28]

### Routing

In the original sparsely-gated MoE, only the top-k experts are queried, and their outputs are weighted-summed. There are other methods.[28] Generally speaking, routing is an assignment problem: How to assign tokens to experts, such that a variety of constraints are followed (such as throughput, load balancing, etc.)? There are typically three classes of routing algorithm: the experts choose the tokens ("expert choice"),[29] the tokens choose the experts (the original sparsely-gated MoE), and a global assigner matching experts and tokens.[30]

During inference, the MoE works over a large batch of tokens at any time. If the tokens were to choose the experts, then some experts might get few tokens, while a few experts get so many tokens that it exceeds their maximum batch size, so they would have to ignore some of the tokens. Similarly, if the experts were to choose the tokens, then some tokens might not be picked by any expert. This is the "token drop" problem. Dropping a token is not necessarily a serious problem, since in Transformers, due to residual connections, if a token is "dropped", it does not disappear. Instead, its vector representation simply passes through the feedforward layer without change.[30]

Other approaches include solving it as a constrained linear programming problem,[31] using reinforcement learning to train the routing algorithm (since picking an expert is a discrete action, like in RL).[32] The token-expert match may involve no learning ("static routing"): It can be done by a deterministic hash function[33] or a random number generator.[34]

### Applications to transformer models

MoE layers are used in the largest transformer models, for which learning and inferring over the full model is too costly. They are typically sparsely-gated, with sparsity 1 or 2. In Transformer models, the MoE layers are often used to select the feedforward layers (typically a linear-ReLU-linear network), appearing in each Transformer block after the multiheaded attention. This is because the feedforward layers take up an increasing portion of the computing cost as models grow larger. For example, in the Palm-540B model, 90% of parameters are in its feedforward layers.[35]

A trained Transformer can be converted to a MoE by duplicating its feedforward layers, with randomly initialized gating, then trained further. This is a technique called "sparse upcycling".[36]

There are a large number of design choices involved in Transformer MoE that affect the training stability and final performance. The OLMoE report describes these in some detail.[37]

As of 2023[update], models large enough to use MoE tend to be large language models, where each expert has on the order of 10 billion parameters. Other than language models, Vision MoE[38] is a Transformer model with MoE layers. They demonstrated it by training a model with 15 billion parameters. MoE Transformer has also been applied for diffusion models.[39]

A series of large language models from Google used MoE. GShard[40] uses MoE with up to top-2 experts per layer. Specifically, the top-1 expert is always selected, and the top-2th expert is selected with probability proportional to that experts' weight according to the gating function. Later, GLaM[41] demonstrated a language model with 1.2 trillion parameters, each MoE layer using top-2 out of 64 experts. Switch Transformers[22] use top-1 in all MoE layers.

The NLLB-200 by Meta AI is a machine translation model for 200 languages.[42] Each MoE layer uses a hierarchical MoE with two levels. On the first level, the gating function chooses to use either a "shared" feedforward layer, or to use the experts. If using the experts, then another gating function computes the weights and chooses the top-2 experts.[43]

MoE large language models can be adapted for downstream tasks by instruction tuning.[44]

In December 2023, Mistral AI released Mixtral 8x7B under Apache 2.0 license. It is a MoE language model with 46.7B parameters, 8 experts, and sparsity 2. They also released a version finetuned for instruction following.[45][46]

In March 2024, Databricks released DBRX. It is a MoE language model with 132B parameters, 16 experts, and sparsity 4. They also released a version finetuned for instruction following.[47][48]

## See also

* Product of experts
* Mixture models
* Mixture of gaussians
* Ensemble learning

## References

1. ↑ Baldacchino, Tara; Cross, Elizabeth J.; Worden, Keith; Rowson, Jennifer (2016). "Variational Bayesian mixture of experts models and sensitivity analysis for nonlinear dynamical systems". Mechanical Systems and Signal Processing 66–67: 178–200. doi:10.1016/j.ymssp.2015.05.009. Bibcode: 2016MSSP...66..178B.
2. ↑ Rokach, Lior (November 2009) (in en). Pattern Classification Using Ensemble Methods. Series in Machine Perception and Artificial Intelligence. 75. WORLD SCIENTIFIC. p. 142. doi:10.1142/7238. ISBN 978-981-4271-06-6.
3. ↑ TRESP, V. (2001). "Committee Machines". Handbook of Neural Network Signal Processing. Electrical Engineering & Applied Signal Processing Series. 5. doi:10.1201/9781420038613.ch5. ISBN 978-0-8493-2359-1. https://cir.nii.ac.jp/crid/1573387450424676096.
4. ↑ Hampshire, J.B.; Waibel, A. (July 1992). "The Meta-Pi network: building distributed knowledge representations for robust multisource pattern recognition". IEEE Transactions on Pattern Analysis and Machine Intelligence 14 (7): 751–769. doi:10.1109/34.142911. https://isl.anthropomatik.kit.edu/downloads/The_Meta-Pi_Network-_Building_Distributed_Knowledge_Representations_for_Robust_Multi-Source_Pattern_Recognition.pdf.
5. ↑ Alexander Waibel; Toshiyuki Hanazawa; Geoffrey Hinton; Kiyohiro Shikano; Kevin J. Lang (1995). "Phoneme Recognition Using Time-Delay Neural Networks*". Backpropagation. Psychology Press. doi:10.4324/9780203763247. ISBN 978-0-203-76324-7. https://www.taylorfrancis.com/chapters/edit/10.4324/9780203763247-2/phoneme-recognition-using-time-delay-neural-networks-alexander-waibel-toshiyuki-hanazawa-geoffrey-hinton-kiyohiro-shikano-kevin-lang.
6. ↑ Nowlan, Steven; Hinton, Geoffrey E (1990). "Evaluation of Adaptive Mixtures of Competing Experts". Advances in Neural Information Processing Systems (Morgan-Kaufmann) 3. https://proceedings.neurips.cc/paper/1990/hash/432aca3a1e345e339f35a30c8f65edce-Abstract.html.
7. ↑ Jacobs, Robert A.; Jordan, Michael I.; Nowlan, Steven J.; Hinton, Geoffrey E. (February 1991). "Adaptive Mixtures of Local Experts" (in en). Neural Computation 3 (1): 79–87. doi:10.1162/neco.1991.3.1.79. ISSN 0899-7667. PMID 31141872. https://direct.mit.edu/neco/article/3/1/79-87/5560.
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9. ↑ 9.0 9.1 Jordan, Michael I.; Jacobs, Robert A. (March 1994). "Hierarchical Mixtures of Experts and the EM Algorithm" (in en). Neural Computation 6 (2): 181–214. doi:10.1162/neco.1994.6.2.181. ISSN 0899-7667. https://direct.mit.edu/neco/article/6/2/181-214/5779.
10. ↑ 10.0 10.1 Jordan, Michael I.; Xu, Lei (1995-01-01). "Convergence results for the EM approach to mixtures of experts architectures (95)00014-3". Neural Networks 8 (9): 1409–1431. doi:10.1016/0893-6080(95)00014-3. ISSN 0893-6080.
11. ↑ Xu, Lei; Jordan, Michael; Hinton, Geoffrey E (1994). "An Alternative Model for Mixtures of Experts". Advances in Neural Information Processing Systems (MIT Press) 7. https://proceedings.neurips.cc/paper/1994/hash/c8fbbc86abe8bd6a5eb6a3b4d0411301-Abstract.html.
12. ↑ Collobert, Ronan; Bengio, Samy; Bengio, Yoshua (2001). "A Parallel Mixture of SVMs for Very Large Scale Problems". Advances in Neural Information Processing Systems (MIT Press) 14. https://proceedings.neurips.cc/paper_files/paper/2001/hash/36ac8e558ac7690b6f44e2cb5ef93322-Abstract.html.
13. ↑ Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). "12: Applications". Deep learning. Adaptive computation and machine learning. Cambridge, Mass: The MIT press. ISBN 978-0-262-03561-3.
14. ↑ Nguyen, Hien D.; McLachlan, Geoffrey J. (2016-01-01). "Laplace mixture of linear experts". Computational Statistics & Data Analysis 93: 177–191. doi:10.1016/j.csda.2014.10.016. ISSN 0167-9473. https://www.sciencedirect.com/science/article/pii/S0167947314003089.
15. ↑ Chamroukhi, F. (2016-07-01). "Robust mixture of experts modeling using the t distribution". Neural Networks 79: 20–36. doi:10.1016/j.neunet.2016.03.002. ISSN 0893-6080. PMID 27093693. https://www.sciencedirect.com/science/article/pii/S0893608016000435.
16. ↑ Chen, K.; Xu, L.; Chi, H. (1999-11-01). "Improved learning algorithms for mixture of experts in multiclass classification". Neural Networks 12 (9): 1229–1252. doi:10.1016/S0893-6080(99)00043-X. ISSN 0893-6080. PMID 12662629. https://www.sciencedirect.com/science/article/pii/S089360809900043X.
17. ↑ Yang, Zhilin; Dai, Zihang; Salakhutdinov, Ruslan; Cohen, William W. (2017-11-10). "Breaking the Softmax Bottleneck: A High-Rank RNN Language Model". arXiv:1711.03953 [cs.CL].
18. ↑ Narang, Sharan; Chung, Hyung Won; Tay, Yi; Fedus, William; Fevry, Thibault; Matena, Michael; Malkan, Karishma; Fiedel, Noah; Shazeer, Noam (2021-02-23). "Do Transformer Modifications Transfer Across Implementations and Applications?". arXiv:2102.11972 [cs.LG].
19. ↑ Bengio, Yoshua; Léonard, Nicholas; Courville, Aaron (2013). "Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation". arXiv:1308.3432 [cs.LG].
20. ↑ Eigen, David; Ranzato, Marc'Aurelio; Sutskever, Ilya (2013). "Learning Factored Representations in a Deep Mixture of Experts". arXiv:1312.4314 [cs.LG].
21. ↑ Shazeer, Noam; Mirhoseini, Azalia; Maziarz, Krzysztof; Davis, Andy; Le, Quoc; Hinton, Geoffrey; Dean, Jeff (2017). "Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer". arXiv:1701.06538 [cs.LG].
22. ↑ 22.0 22.1 22.2 Fedus, William; Zoph, Barret; Shazeer, Noam (2022-01-01). "Switch transformers: scaling to trillion parameter models with simple and efficient sparsity". The Journal of Machine Learning Research 23 (1): 5232–5270. ISSN 1532-4435. https://dl.acm.org/doi/abs/10.5555/3586589.3586709.
23. ↑ Wu, Yonghui; Schuster, Mike; Chen, Zhifeng; Le, Quoc V.; Norouzi, Mohammad; Macherey, Wolfgang; Krikun, Maxim; Cao, Yuan; Gao, Qin; Macherey, Klaus; Klingner, Jeff; Shah, Apurva; Johnson, Melvin; Liu, Xiaobing; Kaiser, Łukasz (2016). "Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation". arXiv:1609.08144 [cs.CL].
24. ↑ Kaiser, Lukasz; Gomez, Aidan N.; Shazeer, Noam; Vaswani, Ashish; Parmar, Niki; Jones, Llion; Uszkoreit, Jakob (2017-06-16). "One Model To Learn Them All" (in en). https://arxiv.org/abs/1706.05137v1.
25. ↑ DeepSeek-AI; Liu, Aixin; Feng, Bei; Wang, Bin; Wang, Bingxuan; Liu, Bo; Zhao, Chenggang; Dengr, Chengqi; Ruan, Chong (19 June 2024). "DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model". arXiv:2405.04434 [cs.CL]..
26. ↑ Dai, Damai; Deng, Chengqi; Zhao, Chenggang; Xu, R. X.; Gao, Huazuo; Chen, Deli; Li, Jiashi; Zeng, Wangding; Yu, Xingkai (11 January 2024). "DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models". arXiv:2401.06066 [cs.CL].
27. ↑ DeepSeek-AI; Liu, Aixin; Feng, Bei; Xue, Bing; Wang, Bingxuan; Wu, Bochao; Lu, Chengda; Zhao, Chenggang; Deng, Chengqi (2024-12-27). "DeepSeek-V3 Technical Report". arXiv:2412.19437 [cs.CL].
28. ↑ 28.0 28.1 Zoph, Barret; Bello, Irwan; Kumar, Sameer; Du, Nan; Huang, Yanping; Dean, Jeff; Shazeer, Noam; Fedus, William (2022). "ST-MoE: Designing Stable and Transferable Sparse Expert Models". arXiv:2202.08906 [cs.CL].
29. ↑ Zhou, Yanqi; Lei, Tao; Liu, Hanxiao; Du, Nan; Huang, Yanping; Zhao, Vincent; Dai, Andrew M.; Chen, Zhifeng et al. (2022-12-06). "Mixture-of-Experts with Expert Choice Routing" (in en). Advances in Neural Information Processing Systems 35: 7103–7114. https://proceedings.neurips.cc/paper_files/paper/2022/hash/2f00ecd787b432c1d36f3de9800728eb-Abstract-Conference.html.
30. ↑ 30.0 30.1 Fedus, William; Dean, Jeff; Zoph, Barret (2022-09-04). "A Review of Sparse Expert Models in Deep Learning". arXiv:2209.01667 [cs.LG].
31. ↑ Lewis, Mike; Bhosale, Shruti; Dettmers, Tim; Goyal, Naman; Zettlemoyer, Luke (2021-07-01). "BASE Layers: Simplifying Training of Large, Sparse Models" (in en). Proceedings of the 38th International Conference on Machine Learning (PMLR): 6265–6274. https://proceedings.mlr.press/v139/lewis21a.html.
32. ↑ Bengio, Emmanuel; Bacon, Pierre-Luc; Pineau, Joelle; Precup, Doina (2015). "Conditional Computation in Neural Networks for faster models". arXiv:1511.06297 [cs.LG].
33. ↑ Roller, Stephen; Sukhbaatar, Sainbayar; szlam, arthur; Weston, Jason (2021). "Hash Layers For Large Sparse Models". Advances in Neural Information Processing Systems (Curran Associates, Inc.) 34: 17555–17566. https://proceedings.neurips.cc/paper_files/paper/2021/hash/92bf5e6240737e0326ea59846a83e076-Abstract.html.
34. ↑ Zuo, Simiao; Liu, Xiaodong; Jiao, Jian; Kim, Young Jin; Hassan, Hany; Zhang, Ruofei; Zhao, Tuo; Gao, Jianfeng (2022-02-03). "Taming Sparsely Activated Transformer with Stochastic Experts". arXiv:2110.04260 [cs.CL].
35. ↑ "Transformer Deep Dive: Parameter Counting" (in en). https://orenleung.com/transformer-parameter-counting.
36. ↑ Komatsuzaki, Aran; Puigcerver, Joan; Lee-Thorp, James; Ruiz, Carlos Riquelme; Mustafa, Basil; Ainslie, Joshua; Tay, Yi; Dehghani, Mostafa; Houlsby, Neil (2023-02-17). "Sparse Upcycling: Training Mixture-of-Experts from Dense Checkpoints". arXiv:2212.05055 [cs.LG].
37. ↑ Muennighoff, Niklas; Soldaini, Luca; Groeneveld, Dirk; Lo, Kyle; Morrison, Jacob; Min, Sewon; Shi, Weijia; Walsh, Pete; Tafjord, Oyvind (2024-09-03). "OLMoE: Open Mixture-of-Experts Language Models". arXiv:2409.02060 [cs.CL].
38. ↑ Riquelme, Carlos; Puigcerver, Joan; Mustafa, Basil; Neumann, Maxim; Jenatton, Rodolphe; Susano Pinto, André; Keysers, Daniel; Houlsby, Neil (2021). "Scaling Vision with Sparse Mixture of Experts". Advances in Neural Information Processing Systems 34: 8583–8595. https://proceedings.neurips.cc/paper/2021/hash/48237d9f2dea8c74c2a72126cf63d933-Abstract.html.
39. ↑ Fei, Zhengcong; Fan, Mingyuan; Yu, Changqian; Li, Debang; Huang, Junshi (2024-07-16). "Scaling Diffusion Transformers to 16 Billion Parameters". arXiv:2407.11633 [cs.CV].
40. ↑ Lepikhin, Dmitry; Lee, HyoukJoong; Xu, Yuanzhong; Chen, Dehao; Firat, Orhan; Huang, Yanping; Krikun, Maxim; Shazeer, Noam; Chen, Zhifeng (2020). "GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding". arXiv:2006.16668 [cs.CL].
41. ↑ Du, Nan; Huang, Yanping; Dai, Andrew M.; Tong, Simon; Lepikhin, Dmitry; Xu, Yuanzhong; Krikun, Maxim; Zhou, Yanqi; Yu, Adams Wei; Firat, Orhan; Zoph, Barret; Fedus, Liam; Bosma, Maarten; Zhou, Zongwei; Wang, Tao (2021). "GLaM: Efficient Scaling of Language Models with Mixture-of-Experts". arXiv:2112.06905 [cs.CL].
42. ↑ "200 languages within a single AI model: A breakthrough in high-quality machine translation" (in en). 2022-06-19. https://ai.facebook.com/blog/nllb-200-high-quality-machine-translation/.
43. ↑ NLLB Team; Costa-jussà, Marta R.; Cross, James; Çelebi, Onur; Elbayad, Maha; Heafield, Kenneth; Heffernan, Kevin; Kalbassi, Elahe; Lam, Janice; Licht, Daniel; Maillard, Jean; Sun, Anna; Wang, Skyler; Wenzek, Guillaume; Youngblood, Al (2022). "No Language Left Behind: Scaling Human-Centered Machine Translation". arXiv:2207.04672 [cs.CL].
44. ↑ Shen, Sheng; Hou, Le; Zhou, Yanqi; Du, Nan; Longpre, Shayne; Wei, Jason; Chung, Hyung Won; Zoph, Barret; Fedus, William; Chen, Xinyun; Vu, Tu; Wu, Yuexin; Chen, Wuyang; Webson, Albert; Li, Yunxuan (2023). "Mixture-of-Experts Meets Instruction Tuning:A Winning Combination for Large Language Models". arXiv:2305.14705 [cs.CL].
45. ↑ AI, Mistral (2023-12-11). "Mixtral of experts" (in en-us). https://mistral.ai/news/mixtral-of-experts/.
46. ↑ Jiang, Albert Q.; Sablayrolles, Alexandre; Roux, Antoine; Mensch, Arthur; Savary, Blanche; Bamford, Chris; Chaplot, Devendra Singh; Casas, Diego de las; Hanna, Emma Bou (2024-01-08). "Mixtral of Experts". arXiv:2401.04088 [cs.LG].
47. ↑ "Introducing DBRX: A New State-of-the-Art Open LLM" (in en-US). 2024-03-27. https://www.databricks.com/blog/introducing-dbrx-new-state-art-open-llm.
48. ↑ Knight, Will. "Inside the Creation of the World's Most Powerful Open Source AI Model" (in en-US). Wired. ISSN 1059-1028. https://www.wired.com/story/dbrx-inside-the-creation-of-the-worlds-most-powerful-open-source-ai-model/. Retrieved 2024-03-28.

## Further reading

* Before deep learning era
* McLachlan, Geoffrey J.; Peel, David (2000). Finite mixture models. Wiley series in probability and statistics applied probability and statistics section. New York Chichester Weinheim Brisbane Singapore Toronto: John Wiley & Sons, Inc. ISBN 978-0-471-00626-8.
* Yuksel, S. E.; Wilson, J. N.; Gader, P. D. (August 2012). "Twenty Years of Mixture of Experts". IEEE Transactions on Neural Networks and Learning Systems 23 (8): 1177–1193. doi:10.1109/TNNLS.2012.2200299. ISSN 2162-237X. PMID 24807516. Bibcode: 2012ITNNL..23.1177Y.
* Masoudnia, Saeed; Ebrahimpour, Reza (12 May 2012). "Mixture of experts: a literature survey". Artificial Intelligence Review 42 (2): 275–293. doi:10.1007/s10462-012-9338-y.
* Nguyen, Hien D.; Chamroukhi, Faicel (July 2018). "Practical and theoretical aspects of mixture-of-experts modeling: An overview" (in en). WIREs Data Mining and Knowledge Discovery 8 (4). doi:10.1002/widm.1246. ISSN 1942-4787. https://wires.onlinelibrary.wiley.com/doi/10.1002/widm.1246.
* Practical techniques for training MoE Transformer models
* Zoph, Barret; Bello, Irwan; Kumar, Sameer; Du, Nan; Huang, Yanping; Dean, Jeff; Shazeer, Noam; Fedus, William (2022). "ST-MoE: Designing Stable and Transferable Sparse Expert Models". arXiv:2202.08906 [cs.CL].
* Muennighoff, Niklas; Soldaini, Luca; Groeneveld, Dirk; Lo, Kyle; Morrison, Jacob; Min, Sewon; Shi, Weijia; Walsh, Pete; Tafjord, Oyvind; Lambert, Nathan; Gu, Yuling; Arora, Shane; Bhagia, Akshita; Schwenk, Dustin; Wadden, David; Wettig, Alexander; Hui, Binyuan; Dettmers, Tim; Kiela, Douwe; Farhadi, Ali; Smith, Noah A.; Pang Wei Koh; Singh, Amanpreet; Hajishirzi, Hannaneh (2024). "OLMoE: Open Mixture-of-Experts Language Models". arXiv:2409.02060 [cs.CL]., with associated data release at "allenai/OLMoE". Ai2. 2024-10-17. https://github.com/allenai/OLMoE.
* Rajbhandari, Samyam; Li, Conglong; Yao, Zhewei; Zhang, Minjia; Reza Yazdani Aminabadi; Ammar Ahmad Awan; Rasley, Jeff; He, Yuxiong (2022). "DeepSpeed-MoE: Advancing Mixture-of-Experts Inference and Training to Power Next-Generation AI Scale". arXiv:2201.05596 [cs.LG].
* DeepSeek-AI; et al. (2024). "DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model". arXiv:2405.04434 [cs.CL].
* DeepSeek-AI; et al. (2024). "DeepSeek-V3 Technical Report". arXiv:2412.19437 [cs.CL].
* Jin, Chao; Jiang, Ziheng; Bai, Zhihao; Zhong, Zheng; Liu, Juncai; Li, Xiang; Zheng, Ningxin; Wang, Xi; Xie, Cong; Huang, Qi; Heng, Wen; Ma, Yiyuan; Bao, Wenlei; Zheng, Size; Peng, Yanghua; Lin, Haibin; Liu, Xuanzhe; Jin, Xin; Liu, Xin (2025). "MegaScale-MoE: Large-Scale Communication-Efficient Training of Mixture-of-Experts Models in Production". arXiv:2505.11432 [cs.LG].
* Literature review for deep learning era
* Fedus, William; Dean, Jeff; Zoph, Barret (2022). "A Review of Sparse Expert Models in Deep Learning". arXiv:2209.01667 [cs.LG].
* Fuzhao, Xue (2024-07-21). "XueFuzhao/awesome-mixture-of-experts". https://github.com/XueFuzhao/awesome-mixture-of-experts.
* Vats, Arpita (2024-09-02). "arpita8/Awesome-Mixture-of-Experts-Papers". https://github.com/arpita8/Awesome-Mixture-of-Experts-Papers?tab=readme-ov-file.
* Cai, Weilin; Jiang, Juyong; Wang, Fan; Tang, Jing; Kim, Sunghun; Huang, Jiayi (2025). "A Survey on Mixture of Experts in Large Language Models". IEEE Transactions on Knowledge and Data Engineering 37 (7): 3896. doi:10.1109/TKDE.2025.3554028. Bibcode: 2025IDSO...37.3896C.

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