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.mae — Function

mae(ŷ, y; agg = mean)

Return the loss corresponding to mean absolute error:

agg(abs.(ŷ .- y))

Example

julia> y_model = [1.1, 1.9, 3.1];

julia> Flux.mae(y_model, 1:3)
0.10000000000000009

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Flux.Losses.mse — Function

mse(ŷ, y; agg = mean)

Return the loss corresponding to mean square error:

agg((ŷ .- y) .^ 2)

See also: mae, msle, crossentropy.

Example

julia> y_model = [1.1, 1.9, 3.1];

julia> y_true = 1:3;

julia> Flux.mse(y_model, y_true)
0.010000000000000018

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Flux.Losses.msle — Function

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.

Example

julia> Flux.msle(Float32[1.1, 2.2, 3.3], 1:3)
0.009084041f0

julia> Flux.msle(Float32[0.9, 1.8, 2.7], 1:3)
0.011100831f0

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Flux.Losses.huber_loss — Function

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|^2,            for |ŷ - y| <= δ
Huber loss = |
             |  δ * (|ŷ - y| - 0.5 * δ), otherwise

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Flux.Losses.label_smoothing — Function

label_smoothing(y::Union{Number, AbstractArray}, α; dims::Int=1)

Returns smoothed labels, meaning the confidence on label values are relaxed.

When y is given as one-hot vector or batch of one-hot, its calculated as

y .* (1 - α) .+ α / size(y, dims)

when y is given as a number or batch of numbers for binary classification, its calculated as

y .* (1 - α) .+ α / 2

in which case the labels are squeezed towards 0.5.

α is a number in interval (0, 1) called the smoothing factor. Higher the value of α larger the smoothing of y.

dims denotes the one-hot dimension, unless dims=0 which denotes the application of label smoothing to binary distributions encoded in a single number.

Example

julia> y = Flux.onehotbatch([1, 1, 1, 0, 1, 0], 0:1)
2×6 OneHotMatrix(::Vector{UInt32}) with eltype Bool:
 ⋅  ⋅  ⋅  1  ⋅  1
 1  1  1  ⋅  1  ⋅

julia> y_smoothed = Flux.label_smoothing(y, 0.2f0)
2×6 Matrix{Float32}:
 0.1  0.1  0.1  0.9  0.1  0.9
 0.9  0.9  0.9  0.1  0.9  0.1

julia> y_sim = softmax(y .* log(2f0))
2×6 Matrix{Float32}:
 0.333333  0.333333  0.333333  0.666667  0.333333  0.666667
 0.666667  0.666667  0.666667  0.333333  0.666667  0.333333

julia> y_dis = vcat(y_sim[2,:]', y_sim[1,:]')
2×6 Matrix{Float32}:
 0.666667  0.666667  0.666667  0.333333  0.666667  0.333333
 0.333333  0.333333  0.333333  0.666667  0.333333  0.666667

julia> Flux.crossentropy(y_sim, y) < Flux.crossentropy(y_sim, y_smoothed)
true

julia> Flux.crossentropy(y_dis, y) > Flux.crossentropy(y_dis, y_smoothed)
true

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Flux.Losses.crossentropy — Function

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))

Cross entropy is typically 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.

For numerical stability, it is recommended to use logitcrossentropy rather than softmax followed by crossentropy .

Use label_smoothing to smooth the true labels as preprocessing before computing the loss.

Example

julia> y_label = Flux.onehotbatch([0, 1, 2, 1, 0], 0:2)
3×5 OneHotMatrix(::Vector{UInt32}) with eltype Bool:
 1  ⋅  ⋅  ⋅  1
 ⋅  1  ⋅  1  ⋅
 ⋅  ⋅  1  ⋅  ⋅

julia> y_model = softmax(reshape(-7:7, 3, 5) .* 1f0)
3×5 Matrix{Float32}:
 0.0900306  0.0900306  0.0900306  0.0900306  0.0900306
 0.244728   0.244728   0.244728   0.244728   0.244728
 0.665241   0.665241   0.665241   0.665241   0.665241

julia> sum(y_model; dims=1)
1×5 Matrix{Float32}:
 1.0  1.0  1.0  1.0  1.0

julia> Flux.crossentropy(y_model, y_label)
1.6076053f0

julia> 5 * ans ≈ Flux.crossentropy(y_model, y_label; agg=sum)
true

julia> y_smooth = Flux.label_smoothing(y_label, 0.15f0)
3×5 Matrix{Float32}:
 0.9   0.05  0.05  0.05  0.9
 0.05  0.9   0.05  0.9   0.05
 0.05  0.05  0.9   0.05  0.05

julia> Flux.crossentropy(y_model, y_smooth)
1.5776052f0

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Flux.Losses.logitcrossentropy — Function

logitcrossentropy(ŷ, y; dims = 1, agg = mean)

Return the cross entropy calculated by

agg(-sum(y .* logsoftmax(ŷ; dims); dims))

This is mathematically equivalent to crossentropy(softmax(ŷ), y), but is more numerically stable than using functions crossentropy and softmax separately.

Example

julia> y_label = Flux.onehotbatch(collect("abcabaa"), 'a':'c')
3×7 OneHotMatrix(::Vector{UInt32}) with eltype Bool:
 1  ⋅  ⋅  1  ⋅  1  1
 ⋅  1  ⋅  ⋅  1  ⋅  ⋅
 ⋅  ⋅  1  ⋅  ⋅  ⋅  ⋅

julia> y_model = reshape(vcat(-9:0, 0:9, 7.5f0), 3, 7)
3×7 Matrix{Float32}:
 -9.0  -6.0  -3.0  0.0  2.0  5.0  8.0
 -8.0  -5.0  -2.0  0.0  3.0  6.0  9.0
 -7.0  -4.0  -1.0  1.0  4.0  7.0  7.5

julia> Flux.logitcrossentropy(y_model, y_label)
1.5791205f0

julia> Flux.crossentropy(softmax(y_model), y_label)
1.5791197f0

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Flux.Losses.binarycrossentropy — Function

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)))

Where typically, the prediction ŷ is given by the output of a sigmoid activation. The ϵ term is included to avoid infinity. Using logitbinarycrossentropy is recomended over binarycrossentropy for numerical stability.

Use label_smoothing to smooth the y value as preprocessing before computing the loss.

Examples

julia> y_bin = Bool[1,0,1]
3-element Vector{Bool}:
 1
 0
 1

julia> y_prob = softmax(reshape(vcat(1:3, 3:5), 2, 3) .* 1f0)
2×3 Matrix{Float32}:
 0.268941  0.5  0.268941
 0.731059  0.5  0.731059

julia> Flux.binarycrossentropy(y_prob[2,:], y_bin)
0.43989f0

julia> all(p -> 0 < p < 1, y_prob[2,:])  # else DomainError
true

julia> y_hot = Flux.onehotbatch(y_bin, 0:1)
2×3 OneHotMatrix(::Vector{UInt32}) with eltype Bool:
 ⋅  1  ⋅
 1  ⋅  1

julia> Flux.crossentropy(y_prob, y_hot)
0.43989f0

source

Flux.Losses.logitbinarycrossentropy — Function

logitbinarycrossentropy(ŷ, y; agg = mean)

Mathematically equivalent to binarycrossentropy(σ(ŷ), y) but is more numerically stable.

Examples

julia> y_bin = Bool[1,0,1];

julia> y_model = Float32[2, -1, pi]
3-element Vector{Float32}:
  2.0
 -1.0
  3.1415927

julia> Flux.logitbinarycrossentropy(y_model, y_bin)
0.160832f0

julia> Flux.binarycrossentropy(sigmoid.(y_model), y_bin)
0.16083185f0

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Flux.Losses.kldivergence — Function

kldivergence(ŷ, y; agg = mean, ϵ = eps(ŷ))

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

The 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.

Example

julia> p1 = [1 0; 0 1]
2×2 Matrix{Int64}:
 1  0
 0  1

julia> p2 = fill(0.5, 2, 2)
2×2 Matrix{Float64}:
 0.5  0.5
 0.5  0.5

julia> Flux.kldivergence(p2, p1) ≈ log(2)
true

julia> Flux.kldivergence(p2, p1; agg = sum) ≈ 2log(2)
true

julia> Flux.kldivergence(p2, p2; ϵ = 0)  # about -2e-16 with the regulator
0.0

julia> Flux.kldivergence(p1, p2; ϵ = 0)  # about 17.3 with the regulator
Inf

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Flux.Losses.poisson_loss — Function

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).

More information..

source

Flux.Losses.hinge_loss — Function

hinge_loss(ŷ, y; agg = mean)

Return the hinge_loss loss given the prediction ŷ and true labels y (containing 1 or -1); calculated as sum(max.(0, 1 .- ŷ .* y)) / size(y, 2).