# Regularisation

Applying regularisation to model parameters is straightforward. We just need to apply an appropriate regulariser, such as norm, to each model parameter and add the result to the overall loss.

For example, say we have a simple regression.

using Flux: crossentropy
m = Dense(10, 5)
loss(x, y) = crossentropy(softmax(m(x)), y)

We can regularise this by taking the (L2) norm of the parameters, m.W and m.b.

using LinearAlgebra

penalty() = norm(m.W) + norm(m.b)
loss(x, y) = crossentropy(softmax(m(x)), y) + penalty()

When working with layers, Flux provides the params function to grab all parameters at once. We can easily penalise everything with sum(norm, params).

julia> params(m)
2-element Array{Any,1}:
param([0.355408 0.533092; … 0.430459 0.171498])
param([0.0, 0.0, 0.0, 0.0, 0.0])

julia> sum(norm, params(m))
26.01749952921026

Here's a larger example with a multi-layer perceptron.

m = Chain(
Dense(28^2, 128, relu),
Dense(128, 32, relu),
Dense(32, 10), softmax)

loss(x, y) = crossentropy(m(x), y) + sum(norm, params(m))

loss(rand(28^2), rand(10))

One can also easily add per-layer regularisation via the activations function:

julia> using Flux: activations

julia> c = Chain(Dense(10, 5, σ), Dense(5, 2), softmax)
Chain(Dense(10, 5, σ), Dense(5, 2), softmax)

julia> activations(c, rand(10))
3-element Array{Any,1}:
Float32[0.84682214, 0.6704139, 0.42177814, 0.257832, 0.36255655]
Float32[0.1501253, 0.073269576]
Float32[0.5192045, 0.48079553]

julia> sum(norm, ans)
2.1166067f0
Flux.activationsFunction
activations(c::Chain, input)

Calculate the forward results of each layers in Chain c with input as model input.

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