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69-621: MoE always has the following components, but they are implemented and combined differently according to the problem being solved: 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. The meta-pi network , reported by Hampshire and Waibel, uses f ( x ) = ∑ i w ( x ) i f i ( x ) {\displaystyle f(x)=\sum _{i}w(x)_{i}f_{i}(x)} as

138-400: A loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called

207-439: A reward function , a profit function , a utility function , a fitness function , etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation , and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace ,

276-674: A 2-level hierarchical MoE would have a first order gating function w i {\displaystyle w_{i}} , and second order gating functions w j | i {\displaystyle w_{j|i}} and experts f j | i {\displaystyle f_{j|i}} . The total prediction is then ∑ i w i ( x ) ∑ j w j | i ( x ) f j | i ( x ) {\displaystyle \sum _{i}w_{i}(x)\sum _{j}w_{j|i}(x)f_{j|i}(x)} . The mixture of experts, being similar to

345-598: A MoE layer, there are feedforward networks f 1 , . . . , f n {\displaystyle f_{1},...,f_{n}} , and a gating network w {\displaystyle w} . The gating network is defined by w ( x ) = s o f t m a x ( t o p k ( W x + noise ) ) {\displaystyle w(x)=\mathrm {softmax} (\mathrm {top} _{k}(Wx+{\text{noise}}))} , where t o p k {\displaystyle \mathrm {top} _{k}}

414-426: 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. MoE large language models can be adapted for downstream tasks by instruction tuning . In December 2023, Mistral AI released Mixtral 8x7B under Apache 2.0 license. It

483-1009: A learnable uncertainty estimate. One can use different experts than gaussian distributions. For example, one can use Laplace distribution , or Student's t-distribution . For binary classification, it also proposed logistic regression experts, with f i ( y | x ) = { 1 1 + e β i T x + β i , 0 , y = 0 1 − 1 1 + e β i T x + β i , 0 , y = 1 {\displaystyle f_{i}(y|x)={\begin{cases}{\frac {1}{1+e^{\beta _{i}^{T}x+\beta _{i,0}}}},&y=0\\1-{\frac {1}{1+e^{\beta _{i}^{T}x+\beta _{i,0}}}},&y=1\end{cases}}} where β i , β i , 0 {\displaystyle \beta _{i},\beta _{i,0}} are learnable parameters. This

552-402: A particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in

621-474: A plane gate closure can still make the plane, but a person who arrives after can not, a discontinuity and asymmetry which makes arriving slightly late much more costly than arriving slightly early. In drug dosing, the cost of too little drug may be lack of efficacy, while the cost of too much may be tolerable toxicity, another example of asymmetry. Traffic, pipes, beams, ecologies, climates, etc. may tolerate increased load or stress with little noticeable change up to

690-405: 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 mixtures of experts 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,

759-557: A vector v c {\displaystyle v_{c}} , and predicts the probability distribution of the next word as S o f t m a x ( v c W ) {\displaystyle \mathrm {Softmax} (v_{c}W)} for an embedding matrix W {\displaystyle W} . In mixture of softmaxes, the model outputs multiple vectors v c , 1 , … , v c , n {\displaystyle v_{c,1},\dots ,v_{c,n}} , and predict

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828-468: Is ∑ k w i , j , k x k {\displaystyle \sum _{k}w_{i,j,k}x_{k}} . However, this does not work with autoregressive modelling, since the weights w i , j , k {\displaystyle w_{i,j,k}} over one token depends on all other tokens'. Other approaches include solving it as a constrained linear programming problem, making each expert choose

897-407: Is a MoE language model with 46.7B parameters, 8 experts, and sparsity 2. They also released a version finetuned for instruction following. 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. Loss function In mathematical optimization and decision theory ,

966-399: Is a fixed but possibly unknown state of nature, X is a vector of observations stochastically drawn from a population , E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X , dP θ is a probability measure over the event space of X (parametrized by  θ ) and the integral is evaluated over

1035-501: Is a function that keeps the top-k entries of a vector the same, but sets all other entries to − ∞ {\displaystyle -\infty } . The addition of noise helps with load balancing. The choice of k {\displaystyle k} is a hyperparameter that is chosen according to application. Typical values are k = 1 , 2 {\displaystyle k=1,2} . The k = 1 {\displaystyle k=1} version

1104-434: Is a learnable parameter. The weighting function is a linear-softmax function: w ( x ) i = e k i T x + b i ∑ j e k j T x + b j {\displaystyle w(x)_{i}={\frac {e^{k_{i}^{T}x+b_{i}}}{\sum _{j}e^{k_{j}^{T}x+b_{j}}}}} The mixture of experts predict that

1173-627: Is also called the Switch Transformer. The original Switch Transformer was applied to a T5 language model . As demonstration, they trained a series of models for machine translation with alternating layers of MoE and LSTM , and compared with deep LSTM models. Table 3 shows that the MoE models used less inference time compute, despite having 30x more parameters. Vanilla MoE tend to have issues of load balancing: some experts are consulted often, while other experts rarely or not at all. To encourage

1242-471: Is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems . In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions . In

1311-399: Is desirable to have a loss function that is globally continuous and differentiable . Two very commonly used loss functions are the squared loss , L ( a ) = a 2 {\displaystyle L(a)=a^{2}} , and the absolute loss , L ( a ) = | a | {\displaystyle L(a)=|a|} . However the absolute loss has

1380-401: Is later generalized for multi-class classification, with multinomial logistic regression experts. One paper proposed mixture of softmaxes for autoregressive language modelling. Specifically, consider a language model that given a previous text c {\displaystyle c} , predicts the next word x {\displaystyle x} . The network encodes the text into

1449-463: Is often modelled using the von Neumann–Morgenstern utility function of the uncertain variable of interest, such as end-of-period wealth. Since the value of this variable is uncertain, so is the value of the utility function; it is the expected value of utility that is maximized. A decision rule makes a choice using an optimality criterion. Some commonly used criteria are: Sound statistical practice requires selecting an estimator consistent with

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1518-434: Is performed. The key design desideratum for 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,

1587-553: Is ranked highest, and P i = 1 T ∑ j = 1 T w i ( x j ) {\displaystyle P_{i}={\frac {1}{T}}\sum _{j=1}^{T}w_{i}(x_{j})} is the fraction of weight on expert i {\displaystyle i} . This loss is minimized at 1 {\displaystyle 1} , precisely when every expert has equal weight 1 / n {\displaystyle 1/n} in all situations. In sparsely-gated MoE, only

1656-528: Is referred to as Bayes Risk . In the latter equation, the integrand inside dx is known as the Posterior Risk , and minimising it with respect to decision a also minimizes the overall Bayes Risk. This optimal decision, a is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ, over all possible (probability-weighted) data outcomes. One advantage of

1725-513: 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 {\displaystyle 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

1794-498: 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

1863-419: Is the likelihood of evidence y {\displaystyle y} . So, w ( x ) i N ( y | μ i , I ) ∑ j w ( x ) j N ( y | μ j , I ) {\displaystyle {\frac {w(x)_{i}N(y|\mu _{i},I)}{\sum _{j}w(x)_{j}N(y|\mu _{j},I)}}}

1932-424: Is the posterior probability for expert i {\displaystyle i} , and so the rate of change for the i {\displaystyle 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

2001-798: Is trained by maximal likelihood estimation, that is, gradient ascent on f ( y | x ) {\displaystyle f(y|x)} . The gradient for the i {\displaystyle i} -th expert is ∇ μ i f θ ( y | x ) = w ( x ) i N ( y | μ i , I ) ∑ j w ( x ) j N ( y | μ j , I ) ( y − μ i ) {\displaystyle \nabla _{\mu _{i}}f_{\theta }(y|x)={\frac {w(x)_{i}N(y|\mu _{i},I)}{\sum _{j}w(x)_{j}N(y|\mu _{j},I)}}\;(y-\mu _{i})} and

2070-403: The mean or average is the statistic for estimating location that minimizes the expected loss experienced under the squared-error loss function, while the median is the estimator that minimizes expected loss experienced under the absolute-difference loss function. Still different estimators would be optimal under other, less common circumstances. In economics, when an agent is risk neutral ,

2139-408: The 1920s. In optimal control , the loss is the penalty for failing to achieve a desired value. In financial risk management , the function is mapped to a monetary loss. Leonard J. Savage argued that using non-Bayesian methods such as minimax , the loss function should be based on the idea of regret , i.e., the loss associated with a decision should be the difference between the consequences of

Mixture of experts - Misplaced Pages Continue

2208-564: The Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ. In economics, decision-making under uncertainty

2277-464: The European subsidies for equalizing unemployment rates among 271 German regions. In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X . Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under

2346-425: 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. 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

2415-416: The actual acceptable variation experienced in the context of a particular applied problem. Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the losses that will be experienced from being wrong under the problem's particular circumstances. A common example involves estimating " location ". Under typical statistical assumptions,

2484-596: The amount of change is proportional to w ( x ) i N ( y | μ i , I ) {\displaystyle w(x)_{i}N(y|\mu _{i},I)} . This has a Bayesian interpretation. Given input x {\displaystyle x} , the prior probability that expert i {\displaystyle i} is the right one is w ( x ) i {\displaystyle w(x)_{i}} , and N ( y | μ i , I ) {\displaystyle N(y|\mu _{i},I)}

2553-468: The auxiliary loss for the batch is n ∑ i = 1 n f i P i {\displaystyle n\sum _{i=1}^{n}f_{i}P_{i}} Here, f i = 1 T # ( queries sent to expert  i ) {\displaystyle f_{i}={\frac {1}{T}}\#({\text{queries sent to expert }}i)} is the fraction of time where expert i {\displaystyle i}

2622-407: The best decision that could have been made had the underlying circumstances been known and the decision that was in fact taken before they were known. The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances , as well as being symmetric: an error above

2691-406: The case of i.i.d. observations, the principle of complete information, and some others. W. Edwards Deming and Nassim Nicholas Taleb argue that empirical reality, not nice mathematical properties, should be the sole basis for selecting loss functions, and real losses often are not mathematically nice and are not differentiable, continuous, symmetric, etc. For example, a person who arrives before

2760-526: The computing cost as models grow larger. For example, in the Palm-540B model, 90% of parameters are in its feedforward layers. 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". There are a large number of design choices involved in Transformer MoE that affect

2829-706: The context of stochastic control , the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber , Log-Cosh and SMAE losses are used when the data has many large outliers. In statistics and decision theory , a frequently used loss function is the 0-1 loss function using Iverson bracket notation, i.e. it evaluates to 1 when y ^ ≠ y {\displaystyle {\hat {y}}\neq y} , and 0 otherwise. In many applications, objective functions, including loss functions as

Mixture of experts - Misplaced Pages Continue

2898-424: The disadvantage that it is not differentiable at a = 0 {\displaystyle a=0} . The squared loss has the disadvantage that it has the tendency to be dominated by outliers —when summing over a set of a {\displaystyle a} 's (as in ∑ i = 1 n L ( a i ) {\textstyle \sum _{i=1}^{n}L(a_{i})} ),

2967-414: The entire support of  X . In a Bayesian approach, the expectation is calculated using the prior distribution π of the parameter  θ : where m(x) is known as the predictive likelihood wherein θ has been "integrated out," π (θ | x) is the posterior distribution, and the order of integration has been changed. One then should choose the action a which minimises this expected loss, which

3036-403: The final sum tends to be the result of a few particularly large a -values, rather than an expression of the average a -value. The choice of a loss function is not arbitrary. It is very restrictive and sometimes the loss function may be characterized by its desirable properties. Among the choice principles are, for example, the requirement of completeness of the class of symmetric statistics in

3105-487: 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 into a single auxiliary loss function. Specifically, let n {\displaystyle n} be the number of experts, then for a given batch of queries { x 1 , x 2 , . . . , x T } {\displaystyle \{x_{1},x_{2},...,x_{T}\}} ,

3174-467: 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

3243-798: The gradient for the weighting function is ∇ [ k i , b i ] f θ ( y | x ) = [ x 1 ] w ( x ) i ∑ j w ( x ) j N ( y | μ j , I ) ( f i ( x ) − f θ ( y | x ) ) {\displaystyle \nabla _{[k_{i},b_{i}]}f_{\theta }(y|x)={\begin{bmatrix}x\\1\end{bmatrix}}{\frac {w(x)_{i}}{\sum _{j}w(x)_{j}N(y|\mu _{j},I)}}(f_{i}(x)-f_{\theta }(y|x))} For each input-output pair ( x , y ) {\displaystyle (x,y)} ,

3312-517: The inputs cannot move through the layer until every expert in the layer has finished the queries it is assigned, load balancing is important. As a hard constraint on load balancing, there is the capacity factor : each expert is only allowed to process up to c ⋅ T / n {\displaystyle c\cdot T/n} queries in a batch. found c ∈ [ 1.25 , 2 ] {\displaystyle c\in [1.25,2]} to work in practice. MoE layers are used in

3381-437: The key design choice in MoE becomes routing: given a batch of queries, how to route the queries to the best experts. The sparsely-gated MoE layer , 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

3450-429: 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

3519-1095: The log-likelihood. The choice of gating function is often softmax. Other than that, gating may use gaussian distributions and exponential families . Instead of performing a weighted sum of all the experts, in hard MoE, only the highest ranked expert is chosen. That is, f ( x ) = f arg ⁡ max i w i ( x ) ( x ) {\displaystyle f(x)=f_{\arg \max _{i}w_{i}(x)}(x)} . This can accelerate training and inference time. The experts can use more general forms of multivariant gaussian distributions. For example, proposed f i ( y | x ) = N ( y | A i x + b i , Σ i ) {\displaystyle f_{i}(y|x)=N(y|A_{i}x+b_{i},\Sigma _{i})} , where A i , b i , Σ i {\displaystyle A_{i},b_{i},\Sigma _{i}} are learnable parameters. In words, each expert learns to do linear regression, with

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3588-453: The model are used, the parts chosen according to what the input is. The earliest paper that applies MoE to deep learning dates back to 2013, 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

3657-403: The next word as ∑ i = 1 n p i S o f t m a x ( v c , i W i ) {\displaystyle \sum _{i=1}^{n}p_{i}\;\mathrm {Softmax} (v_{c,i}W_{i})} , where p i {\displaystyle p_{i}} is a probability distribution by a linear-softmax operation on

3726-491: The objective function is simply expressed as the expected value of a monetary quantity, such as profit, income, or end-of-period wealth. For risk-averse or risk-loving agents, loss is measured as the negative of a utility function , and the objective function to be optimized is the expected value of utility. Other measures of cost are possible, for example mortality or morbidity in the field of public health or safety engineering . For most optimization algorithms , it

3795-1170: The output is distributed according to the probability density function: f θ ( y | x ) = ln ⁡ [ ∑ i e k i T x + b i ∑ j e k j T x + b j N ( y | μ i , I ) ] = ln ⁡ [ ( 2 π ) − d / 2 ∑ i e k i T x + b i ∑ j e k j T x + b j e − 1 2 ‖ y − μ i ‖ 2 ] {\displaystyle f_{\theta }(y|x)=\ln \left[\sum _{i}{\frac {e^{k_{i}^{T}x+b_{i}}}{\sum _{j}e^{k_{j}^{T}x+b_{j}}}}N(y|\mu _{i},I)\right]=\ln \left[(2\pi )^{-d/2}\sum _{i}{\frac {e^{k_{i}^{T}x+b_{i}}}{\sum _{j}e^{k_{j}^{T}x+b_{j}}}}e^{-{\frac {1}{2}}\|y-\mu _{i}\|^{2}}\right]} It

3864-432: The output. The model is trained by performing gradient descent on the mean-squared error loss L := 1 N ∑ k ‖ y k − f ( x k ) ‖ 2 {\displaystyle L:={\frac {1}{N}}\sum _{k}\|y_{k}-f(x_{k})\|^{2}} . The experts may be arbitrary functions. In their original publication, they were solving

3933-430: 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" (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

4002-526: The proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and

4071-475: The soft MoE layer computes an array w i , j , k {\displaystyle w_{i,j,k}} , such that ( w i , j , 1 , . . . , w i , j , T ) {\displaystyle (w_{i,j,1},...,w_{i,j,T})} is a probability distribution over queries, and the i {\displaystyle i} -th expert's j {\displaystyle j} -th query

4140-511: The target causes the same loss as the same magnitude of error below the target. If the target is t , then a quadratic loss function is for some constant C ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss ( SEL ). Many common statistics , including t-tests , regression models, design of experiments , and much else, use least squares methods applied using linear regression theory, which

4209-663: The token would be routed to the 1st expert in layer 1, 4th expert in layer 2, etc. Despite its simplicity, it achieves competitive performance as sparsely gated MoE with k = 1 {\displaystyle k=1} . In soft MoE, suppose in each batch, each expert can process p {\displaystyle p} queries, then there are n × p {\displaystyle n\times p} queries that can be assigned per batch. Now for each batch of queries { x 1 , x 2 , . . . , x T } {\displaystyle \{x_{1},x_{2},...,x_{T}\}} ,

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4278-424: 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 demonstrated a language model with 1.2 trillion parameters, each MoE layer using top-2 out of 64 experts. Switch Transformers use top-1 in all MoE layers. The NLLB-200 by Meta AI is a machine translation model for 200 languages. Each MoE layer uses

4347-414: The top-k experts are queried, and their outputs are weighted-summed. There are other methods. In Hash MoE, routing is performed deterministically by a hash function, fixed before learning begins. For example, if the model is a 4-layered Transformer, and input is a token for word "eat", and the hash of "eat" is ( 1 , 4 , 2 , 3 ) {\displaystyle (1,4,2,3)} , then

4416-546: The top-k queries it wants (instead of each query choosing the top-k experts for it), using reinforcement learning to train the routing algorithm (since picking an expert is a discrete action, like in RL), etc. Suppose there are n {\displaystyle n} experts in a layer. For a given batch of queries { x 1 , x 2 , . . . , x T } {\displaystyle \{x_{1},x_{2},...,x_{T}\}} , each query

4485-572: The training stability and final performance. The OLMoE report describes these in some detail. As of 2023, 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 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 . A series of large language models from Google used MoE. GShard uses MoE with up to top-2 experts per layer. Specifically,

4554-409: The two paradigms. We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution , P θ , of the observed data, X . This is also referred to as the risk function of the decision rule δ and the parameter θ . Here the decision rule depends on the outcome of X . The risk function is given by: Here, θ

4623-450: 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 {\displaystyle i} -th expert is changed to make its prediction closer to y {\displaystyle y} , but

4692-548: Was classified by a linear combination of the experts for the other 3 male speakers. The adaptive mixtures of local experts uses a gaussian mixture model . Each expert simply predicts a gaussian distribution, and totally ignores the input. Specifically, the i {\displaystyle i} -th expert predicts that the output is y ∼ N ( μ i , I ) {\displaystyle y\sim N(\mu _{i},I)} , where μ i {\displaystyle \mu _{i}}

4761-409: Was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics , for example, this is usually economic cost or regret . In classification , it is the penalty for an incorrect classification of an example. In actuarial science , it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in

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