# Loss Functions

Flux provides a large number of common loss functions used for training machine learning models. They are grouped together in the Flux.Losses module.

Loss functions for supervised learning typically expect as inputs a target y, and a prediction ŷ. In Flux's convention, the order of the arguments is the following

loss(ŷ, y)

Most loss functions in Flux have an optional argument agg, denoting the type of aggregation performed over the batch:

loss(ŷ, y)                         # defaults to mean
loss(ŷ, y, agg=sum)                # use sum for reduction
loss(ŷ, y, agg=x->sum(x, dims=2))  # partial reduction
loss(ŷ, y, agg=x->mean(w .* x))    # weighted mean
loss(ŷ, y, agg=identity)           # no aggregation.

## Losses Reference

Flux.Losses.maeFunction
mae(ŷ, y; agg=mean)

Return the loss corresponding to mean absolute error:

agg(abs.(ŷ .- y))
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Flux.Losses.mseFunction
mse(ŷ, y; agg=mean)

Return the loss corresponding to mean square error:

agg((ŷ .- y).^2)
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Flux.Losses.msleFunction
msle(ŷ, y; agg=mean, ϵ=eps(ŷ))

The loss corresponding to mean squared logarithmic errors, calculated as

agg((log.(ŷ .+ ϵ) .- log.(y .+ ϵ)).^2)

The ϵ term provides numerical stability. Penalizes an under-estimation more than an over-estimatation.

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Flux.Losses.huber_lossFunction
huber_loss(ŷ, y; δ=1, agg=mean)

Return the mean of the Huber loss given the prediction ŷ and true values y.

             | 0.5 * |ŷ - y|,            for |ŷ - y| <= δ
Huber loss = |
|  δ * (|ŷ - y| - 0.5 * δ), otherwise
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Flux.Losses.crossentropyFunction
crossentropy(ŷ, y; dims=1, ϵ=eps(ŷ), agg=mean)

Return the cross entropy between the given probability distributions; calculated as

agg(-sum(y .* log.(ŷ .+ ϵ); dims=dims))

Cross entropy is tipically used as a loss in multi-class classification, in which case the labels y are given in a one-hot format. dims specifies the dimension (or the dimensions) containing the class probabilities. The prediction ŷ is supposed to sum to one across dims, as would be the case with the output of a softmax operation.

Use of logitcrossentropy is recomended over crossentropy for numerical stability.

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Flux.Losses.binarycrossentropyFunction
binarycrossentropy(ŷ, y; agg=mean, ϵ=eps(ŷ))

Return the binary cross-entropy loss, computed as

agg(@.(-y*log(ŷ + ϵ) - (1-y)*log(1-ŷ + ϵ)))

The ϵ term provides numerical stability.

Typically, the prediction ŷ is given by the output of a sigmoid activation.

Use of logitbinarycrossentropy is recomended over binarycrossentropy for numerical stability.

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Flux.Losses.kldivergenceFunction
kldivergence(ŷ, y; agg=mean)

Return the Kullback-Leibler divergence between the given probability distributions.

KL divergence is a measure of how much one probability distribution is different from the other. It is always non-negative and zero only when both the distributions are equal everywhere.

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Flux.Losses.poisson_lossFunction
poisson_loss(ŷ, y)

Return how much the predicted distribution ŷ diverges from the expected Poisson

distribution y; calculated as sum(ŷ .- y .* log.(ŷ)) / size(y, 2).

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Flux.Losses.dice_coeff_lossFunction
dice_coeff_loss(ŷ, y; smooth=1)

Return a loss based on the dice coefficient. Used in the V-Net image segmentation architecture. Similar to the F1_score. Calculated as:

1 - 2*sum(|ŷ .* y| + smooth) / (sum(ŷ.^2) + sum(y.^2) + smooth)
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Flux.Losses.tversky_lossFunction
tversky_loss(ŷ, y; β=0.7)

Return the Tversky loss. Used with imbalanced data to give more weight to false negatives. Larger β weigh recall more than precision (by placing more emphasis on false negatives) Calculated as: 1 - sum(|y .* ŷ| + 1) / (sum(y .* ŷ + β(1 .- y) . ŷ + (1 - β)y . (1 .- ŷ)) + 1)

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