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Transforming_cost_functions.jl
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Transforming_cost_functions.jl
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### A Pluto.jl notebook ###
# v0.18.1
using Markdown
using InteractiveUtils
# ╔═╡ dae3f41f-87a6-4f4b-90fe-e0379c623b51
using MinimallyDisruptiveCurves
# ╔═╡ ab1dcd16-9e4e-11ec-3d44-d71e0e4f4f39
md"""
# Transforming cost functions using MinimallyDisruptiveCurves.jl
The premise of MinimallyDisruptiveCurves.jl is to
> move as far away as possible from the initial parameters, while keeping model behaviour as similar as possible.
(in other words, while keeping a cost function as low as possible).
The words *as far away as possible* imply some metric on the space of parameters. A lot of key workflows involve manipulating this metric. For instance, by
- fixing/freeing parameters that the minimally disruptive curve can change
- looking at relative, rather than absolute changes in parameters
- biasing parameters to have a larger/smaller influence on the metric, so that minimally disruptive curves are encouraged (not /) to align with them
- something custom.
Each of these corresponds to a **transformation** of parameter space. The easiest way to do all of these things is by **reparameterising** the cost function $C(\theta)$. We take a transformation of parameter space: $T(\theta)$, and a new cost function $D$ satisfying
$$D[T(\theta)] = C(\theta).$$
**The purpose of this notebook is to show you an easy way to make/perform these reparameterisations.**
The (slight) complication is that cost functions compatible with MinimallyDisruptiveCurves.jl must be **differentiable**. That is, they have two methods:
```
## method 1
function cost(𝜃)
...
return cost
end
## method 2
function cost(𝜃, grad_template)
....
grad_template[:] = ∇C # mutate to get gradient wrt parameters
return cost
end
```
So we want an easy way of applying composable transformations to cost functions, which also recompute the gradient automatically. And that is what we shall provide, through the `TransformationStructure` type.
First let's make an arbitrary cost function. We'll keep it very simple for didactic purposes.
"""
# ╔═╡ 45b60fe5-d8b6-45b5-91de-3dce747423c9
θ₀ = [5.,6.,7.]
# ╔═╡ d5e5ba1b-6657-48b6-8167-95b6a3de5479
C(θ) = sum( @. (θ - θ₀)^2)
# ╔═╡ 7caa9840-2434-45f6-b3c8-bc4dee5b9c7e
function C(θ, g)
g[:] = @. 2(θ-θ₀)
return C(θ)
end
# ╔═╡ 64e63817-e342-4b6b-8ac4-e5b56c5658f3
g = deepcopy(θ₀)
# ╔═╡ d50dcff0-8b71-4a60-863f-520613a511f3
C(θ₀, g)
# ╔═╡ 152f8871-6aa0-4e31-8876-1f42db154d03
g
# ╔═╡ 5f320fc7-b73c-4179-8e45-89dbb7ba9fab
md"""
Next let's build a transformation structure:
"""
# ╔═╡ af46717e-b7a8-4aac-ac49-b3b43f9cb9e5
T(p) = @. p + 1
# ╔═╡ 31b5ec9a-7ac1-484b-b490-2f2175702957
Tinv(p) = @. p - 1
# ╔═╡ a920abd7-1e76-4176-9505-4c2ec7bb246d
Trf = TransformationStructure("simple_transform", T, Tinv)
# ╔═╡ f38cd26b-b3a0-480a-b2cb-6523b6096a82
md"""
and reparameterise the cost function with it...
"""
# ╔═╡ d72db3a6-46e4-4e14-93b5-5bbcb4005214
D, θnew = transform_cost(C, θ₀, Trf)
# ╔═╡ 57d8b290-29a3-458b-8086-30f953ed34cb
[θnew, D(θnew, g), g]
# ╔═╡ 352cd9e4-665c-46c6-b4b6-13ec700571ca
md"""
Done! Now we have made some preloaded `TransformationStructure`s to get you started:
"""
# ╔═╡ bb7b6b09-f404-4e25-ad64-7097652456d0
LA = logabs_transform(θ₀);
# ╔═╡ 2ade0e82-80ab-46e7-b072-aa1a84c466c2
md"""
This one transforms parameters to their absolute values, then takes the logs. Useful if you want to quantify relative (instead of absolute) changes to parameters
"""
# ╔═╡ 2ebe244b-9366-408f-a383-2a39f97fcae2
idxs = [1,3]
# ╔═╡ d857639d-4c5f-49e1-b01f-f0fff647cc94
biases = [400.,1/400]
# ╔═╡ 0565c581-4fb5-44a0-817c-760baad4d6ab
BA = bias_transform(θnew, idxs, biases );
# ╔═╡ 216d0253-c8c6-4684-bef6-d6140a6e4295
md"""
This one does the transformation
$$ θ[i] \to b_i θ[i] $$
for $i$ in idxs and $b_i$ in biases.
"""
# ╔═╡ 0b15de14-535b-44c7-8969-f8753a717827
FA = fix_params(θ₀, idxs); #is self explanatory
# ╔═╡ 00b75908-0fc4-4067-984e-9862f97ae98e
OFA = only_free_params(θ₀, idxs); # opposite of above: all except params[idxs] are fixed.
# ╔═╡ ac18b550-bdc9-4e9c-8a75-bd29720a1d0d
md"""
Any of these transformations can be composably applied to any cost function amenable to MinimallyDisruptiveCurves.jl. And as we showed, you can make your own.
"""
# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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