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TP-reinforcement-learning/TP1/notebook.jl

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2023-06-23 18:10:32 +00:00
### A Pluto.jl notebook ###
# v0.19.19
#> [frontmatter]
#> title = " TP1 - Reinforcement learning "
#> date = "2022-12-14"
#> tags = ["RL"]
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 26fdd17e-f03a-4835-93be-85303fe526d8
begin
using Plots # pour le tracé de figures
using PlutoUI # pour les objets Pluto
using LinearAlgebra # pour les matrices identité
using SparseArrays # pour les matrices creuses
using LaTeXStrings # pour les strings en LaTeX (pour les plots)
import Random
TableOfContents(depth=4)
end
# ╔═╡ 56ac3473-24f4-42d7-84e1-cfce6a70d8d5
html"""
<style>
pluto-output, img {
display: block;
margin: auto;
}
.sliders {
position: fixed;
left: 2rem;
top: 40%;
z-index: 1000;
}
</style>
"""
# ╔═╡ ccf4d63e-7ace-11ed-2123-d9dbb62bd308
html"""
<center>
<strong style="font-size: 2rem;">
TP1 - Reinforcement learning <br/>
Laurent Fainsin <br/>
2021 - 2022
</strong>
</center>
"""
# ╔═╡ 9f2879c1-c22b-4067-ad20-4e4c56cc8d00
begin
REWARD_MOVE_slider = @bind REWARD_MOVE Slider(-0.1:0.01:0, default=-0.04, show_value=true)
REWARD_GOAL_slider = @bind REWARD_GOAL Slider(0:1:10, default=1, show_value=true)
REWARD_DEATH_slider = @bind REWARD_DEATH Slider(-10:1:0, default=-2, show_value=true)
DISCOUNT_FACTOR_slider = @bind DISCOUNT_FACTOR Slider(0.9:0.01:0.99, default=0.9, show_value=true)
div = html"""<div class="sliders">"""
div_end = html"""</div>"""
md"""
$(div)
Hyper-paramètres:
REWARD\_MOVE: $(REWARD_MOVE_slider)
REWARD\_GOAL: $(REWARD_GOAL_slider)
REWARD\_DEATH: $(REWARD_DEATH_slider)
DISCOUNT\_FACTOR: $(DISCOUNT_FACTOR_slider)
$(div_end)
"""
end
# ╔═╡ 0a30a68a-068e-41fb-92c4-000869ba7dff
RANDOM_SEED = 420
# ╔═╡ 07b57746-fba0-49aa-ba17-6dcb0bbe44e5
MAX_ITERATIONS = 350
# ╔═╡ 92d6874b-651c-4551-840e-ad5d1e934aeb
MOVEMENTS = [
0.7 0.1 0.1 0.1
0.1 0.7 0.1 0.1
0.1 0.1 0.7 0.1
0.1 0.1 0.1 0.7
]
# ╔═╡ fe44d7f2-155e-42f2-83c3-dd18aadb3810
md"""
On définit notre environnement comme une grille 3x4:
"""
# ╔═╡ 28b769a6-dd3c-43ab-bae0-646d8ebc35d6
begin
ARROW_SYMBOLS = ["←", "↑", "↓", "→"]
DEATH_SYMBOL = "☠"
SUCCESS_SYMBOL = "✓"
WALL_SYMBOL = "■"
EMPTY_SYMBOL = "□"
[
EMPTY_SYMBOL EMPTY_SYMBOL EMPTY_SYMBOL SUCCESS_SYMBOL
EMPTY_SYMBOL WALL_SYMBOL EMPTY_SYMBOL DEATH_SYMBOL
EMPTY_SYMBOL EMPTY_SYMBOL EMPTY_SYMBOL EMPTY_SYMBOL
]
end
# ╔═╡ 3881603c-619b-4976-ac4c-2c7e7f3a6ec7
md"""
On peut définir nos rewards tels que:
"""
# ╔═╡ fb797a9b-6a0a-4a77-a9b6-6804f98639bb
begin
REWARDS = [
REWARD_MOVE, REWARD_MOVE, REWARD_MOVE, REWARD_GOAL,
REWARD_MOVE, REWARD_MOVE, REWARD_DEATH,
REWARD_MOVE, REWARD_MOVE, REWARD_MOVE, REWARD_MOVE,
]
local REWARDS_display = copy(REWARDS)
insert!(REWARDS_display, 6, 0)
REWARDS_display = permutedims(reshape(REWARDS_display, 4, 3))
REWARDS_display = sparse(REWARDS_display)
end
# ╔═╡ 1e3abda8-6645-48ba-874d-28e1011fc3e3
md"""
# Performance Prediction
"""
# ╔═╡ beb410a8-03e2-4f18-8ccd-941cc926ee12
md"""
## Question 1
> Assume the random policy, that is, the policy that takes every possible action with probability 1/4. Compute its value function by solving \
> $V = (I − \gamma P )^{-1} R$. \
> Since there are 11 possible states in the problem, the vectors ``R`` and ``V`` have have length 11, and the matrix ``P`` has dimension 11x11. There are two absorbing states, i.e., they are visited once, and their respective reward (+1 or -1) is only accrued once. To model this, you can simply put all 0’s in all the elements of the respective two lines.
"""
# ╔═╡ 133f291f-6f21-4441-86f7-ba190a7d6b1f
md"""
On définit une politique aléatoire (à la main):
"""
# ╔═╡ e14f9977-d2fd-4d05-84d6-614008dc0c4a
[
ARROW_SYMBOLS[2] ARROW_SYMBOLS[1] ARROW_SYMBOLS[1] SUCCESS_SYMBOL
ARROW_SYMBOLS[1] WALL_SYMBOL ARROW_SYMBOLS[1] DEATH_SYMBOL
ARROW_SYMBOLS[1] ARROW_SYMBOLS[3] ARROW_SYMBOLS[2] ARROW_SYMBOLS[1]
]
# ╔═╡ 486c93ab-9cb9-4df4-b702-bbe12a961647
md"""
Via nos probabilités de mouvements on peut alors constituer ``P``:
"""
# ╔═╡ ab2d705d-fc00-43b2-bb6d-2a3d4ba9dab1
begin
P = [
[
0.8, 0.1, 0.0, 0.0,
0.1, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.7, 0.2, 0.1, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.7, 0.1, 0.1,
0.0, 0.1, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.0, 0.0, 0.0,
0.8, 0.0, 0.0,
0.1, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.1, 0.0,
0.0, 0.7, 0.1,
0.0, 0.0, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.1, 0.0, 0.0,
0.8, 0.1, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.1, 0.8, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.7, 0.0,
0.0, 0.1, 0.1, 0.1,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.1,
0.0, 0.0, 0.7, 0.2,
],
]
P = sparse(reduce(hcat, P)')
end
# ╔═╡ b7ae89c9-3c1b-4f5c-af5b-164d95ccca41
md"""
On peut alors calculer ``V``:
"""
# ╔═╡ 03c17428-5ab9-42e7-bf79-92eb846f11cb
begin
V = Matrix(I(length(REWARDS)) - DISCOUNT_FACTOR * P) \ REWARDS
local V_display = copy(V)
insert!(V_display, 6, 0)
V_display = permutedims(reshape(V_display, 4, 3))
V_display = sparse(V_display)
end
# ╔═╡ c65d0dbc-ecd7-4320-9b3a-a1b9c0545f9a
md"""
### Bonus
On contrôle que ``V`` vérifie l'équation de Bellman en calculant une itération de l'équation de Bellman:
$V_{\text{next}} = R + \gamma P V$
et en observant que ``V`` est un point fixe:
$V_{\text{next}} = V$
On calcule alors ``V_\text{next}``:
"""
# ╔═╡ ad547684-bcbe-44f4-9fc1-f327d2db4584
begin
V_next = REWARDS + Matrix(DISCOUNT_FACTOR * P) * V
local V_display = copy(V_next)
insert!(V_display, 6, 0)
V_display = permutedims(reshape(V_display, 4, 3))
V_display = sparse(V_display)
end
# ╔═╡ d3703ab8-912c-417d-acd9-29590ec1134b
if isapprox(V_next, V)
Markdown.MD(Markdown.Admonition("correct", "V est un point fixe", [md"L'équation de Bellman est vérifiée"]));
else
Markdown.MD(Markdown.Admonition("danger", "V n'est pas un point fixe", [md"L'équation de Bellman n'est vérifiée"]));
end
# ╔═╡ 1319b304-5126-4825-8076-e113e4dd3635
md"""
## Question 2
> Evaluate now the policy using Iterative Policy Evaluation (lecture 2, 2nd part, slides 11/35), and verify that the algorithm converges to the result obtained in 1. To stop iterating, you can take as a criterion that the difference between two iterations must be smaller than some small ``\delta``. Due to the contraction principle, the initial vector can be arbitrary.
"""
# ╔═╡ 3ea3f177-c576-4b9e-a54b-c427e29a8491
md"""
On initialise ``V_\text{random} \in [0, 1]^{11}`` aléatoirement.
On souhaite vérifier que ``V_\text{random}`` converge vers ``V`` par l'évaluation itérative de la politique ``P``.
"""
# ╔═╡ e94fe8a6-274b-4121-b1fc-063d3710c2f7
begin
Random.seed!(RANDOM_SEED)
V_random = rand(length(REWARDS))
local diffs = Vector{Float64}()
for _ in 1:MAX_ITERATIONS
local V_old = V_random
V_random = REWARDS + Matrix(DISCOUNT_FACTOR * P) * V_random
append!(diffs, norm(V_random - V_old))
if isapprox(V_random, V_old)
break
end
end
plot(
diffs,
labels = "",
xlabel = L"n",
ylabel = L"|| V_{n+1} - V_n ||^2",
yticks=[10.0^-x for x in 0:10],
linewidth=2,
yaxis=:log,
title="Iterative Policy Evaluation convergence",
)
end
# ╔═╡ 80090d5f-d56c-4844-a04f-444ed49e5f34
if isapprox(V_random, V, rtol=1e-5)
Markdown.MD(Markdown.Admonition("correct", "L'évaluation itérative des politiques est vérifiée", [md"``V_\text{random}`` converge vers ``V``"]));
else
Markdown.MD(Markdown.Admonition("danger", "L'évaluation itérative des politiques n'est pas vérifiée", [md"``V_\text{random}`` ne converge pas vers ``V``"]));
end
# ╔═╡ 98362798-aae4-4540-9e98-cc7371802552
md"""
## Question 3
> To verify that the Bellman operator is a contraction, take two initial vectors, and calculate the max of their differences. Then, apply the iterative policy evaluation to these 2 vectors as done in the previous item, and plot the maximum of their differences as you keep iterating. Observe what happens with the difference as you iterate, and explain it.
"""
# ╔═╡ 30874daf-7b0e-4335-9a50-d19389cf1620
md"""
On initialise ``V_{r1}, V_{r2} \in [0, 1]^{11}`` aléatoirement.
On souhaite vérifier que ``V_{r1}`` converge vers ``V_{r2}`` (et aussi vers ``V``) par l'évaluation itérative de la politique ``P``.
"""
# ╔═╡ c005a3f8-765c-4a50-90ef-73a5a72eee01
begin
Random.seed!(RANDOM_SEED)
V_random1 = rand(length(REWARDS))
V_random2 = rand(length(REWARDS))
local diffs = Vector{Float64}()
for _ in 1:MAX_ITERATIONS
V_random1 = REWARDS + Matrix(DISCOUNT_FACTOR * P) * V_random1
V_random2 = REWARDS + Matrix(DISCOUNT_FACTOR * P) * V_random2
append!(diffs, norm(V_random1 - V_random2))
if isapprox(V_random1, V_random2)
break
end
end
plot(
diffs,
labels = "",
xlabel = L"n",
ylabel = L"|| V_{r1} - V_{r2} ||^2",
yticks=[10.0^-x for x in 0:10],
linewidth=2,
yaxis=:log,
title="Bellman's operator contraction",
)
end
# ╔═╡ 1b43e9e5-d7d2-4b5e-a2b2-3a8b8eda6d62
if isapprox(V_random1, V_random2, rtol=0.01)
Markdown.MD(Markdown.Admonition("correct", "On vérifie que l'opérateur de Bellman est une contraction", [md"``V_{r1}`` converge vers ``V_{r2}``"]));
else
Markdown.MD(Markdown.Admonition("danger", "On ne vérifie pas que l'opérateur de Bellman est une contraction", [md"``V_{r1}`` ne converge pas vers ``V_{r2}``"]));
end
# ╔═╡ add0221b-e352-4559-a722-c45a64f573f9
md"""
# Optimization
"""
# ╔═╡ 84e07dce-bf6d-4ac1-bfa4-65414fe1d787
md"""
## Question 1
> Write down the Bellman equation that characterizes the optimal policy.
"""
# ╔═╡ df13fa05-14de-409b-a0b1-5bba5eff432e
md"""
Bellman Optimality Equation for ``V_\star`` and ``\pi_\star``:
$V_\star(s) = \max_{a \in A} \left( r(s,a) + \gamma \sum_{s' \in S} p(s' | s, a) V_\star(s') \right)$
$\pi_\star(s) \in \mathop{\mathrm{argmax}}_{a \in A} \left( r(s,a) + \gamma \sum_{s' \in S} p(s' | s, a) V_\star(s') \right)$
"""
# ╔═╡ ac490e4a-ce20-4288-a04f-c224df5ade1a
md"""
## Question 2
> Solve numerically the optimal value function by Value Iteration Algorithm (lecture 2, 2nd part, slides 15/35). Verify that the solution you obtain satisfies the Bellman equation.
"""
# ╔═╡ 33890f22-d3f6-4bcf-870d-756f7ff250a9
md"""
``P_g`` la politique du déplacement toujours à gauche:
"""
# ╔═╡ cf9fb8a8-6c93-4c43-9f01-5f198f0cf4aa
begin
P_g = [
[
0.8, 0.1, 0.0, 0.0,
0.1, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.7, 0.2, 0.1, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.7, 0.1, 0.1,
0.0, 0.1, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.0, 0.0, 0.0,
0.8, 0.0, 0.0,
0.1, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.1, 0.0,
0.0, 0.7, 0.1,
0.0, 0.0, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.1, 0.0, 0.0,
0.8, 0.1, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.7, 0.2, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.1, 0.0,
0.0, 0.7, 0.1, 0.1,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.1,
0.0, 0.0, 0.7, 0.2,
],
]
P_g = sparse(reduce(hcat, P_g)')
end
# ╔═╡ dc87b85f-c87c-4302-9124-194bd799f1fd
md"""
``P_h`` la politique du déplacement toujours en haut:
"""
# ╔═╡ b2595dec-aa5b-462b-b0f8-3555c1231b2f
begin
P_h = [
[
0.8, 0.1, 0.0, 0.0,
0.1, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.8, 0.1, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.1, 0.7, 0.1,
0.0, 0.1, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.7, 0.0, 0.0, 0.0,
0.2, 0.0, 0.0,
0.1, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.7, 0.0,
0.0, 0.1, 0.1,
0.0, 0.0, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.7, 0.0, 0.0,
0.2, 0.1, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.1, 0.8, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.7, 0.0,
0.0, 0.1, 0.1, 0.1,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.7,
0.0, 0.0, 0.1, 0.2,
],
]
P_h = sparse(reduce(hcat, P_h)')
end
# ╔═╡ 70edf811-adb0-4ae8-941a-b298d85a6e0e
md"""
``P_b`` la politique du déplacement toujours en bas:
"""
# ╔═╡ 875673f1-08c9-4713-bbc2-85b0a7a0cb0a
begin
P_b = [
[
0.2, 0.1, 0.0, 0.0,
0.7, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.8, 0.1, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.1, 0.1, 0.1,
0.0, 0.7, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.0, 0.0, 0.0,
0.2, 0.0, 0.0,
0.7, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.1, 0.0,
0.0, 0.1, 0.1,
0.0, 0.0, 0.7, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.1, 0.0, 0.0,
0.8, 0.1, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.1, 0.8, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.1, 0.0,
0.0, 0.1, 0.7, 0.1,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.1,
0.0, 0.0, 0.1, 0.8,
],
]
P_b = sparse(reduce(hcat, P_b)')
end
# ╔═╡ 2deaac7c-ad14-43b0-9cd5-9f0ec12d324c
md"""
``P_d`` la politique du déplacement toujours à droite:
"""
# ╔═╡ b5c93b6f-933c-41b4-8399-44cc0fa07fab
begin
P_d = [
[
0.2, 0.7, 0.0, 0.0,
0.1, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.2, 0.7, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.1, 0.1, 0.7,
0.0, 0.1, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.1, 0.0, 0.0, 0.0,
0.8, 0.0, 0.0,
0.1, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.1, 0.0,
0.0, 0.1, 0.7,
0.0, 0.0, 0.1, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.1, 0.0, 0.0,
0.2, 0.7, 0.0, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.1, 0.2, 0.7, 0.0,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.1, 0.0,
0.0, 0.1, 0.1, 0.7,
],
[
0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.1,
0.0, 0.0, 0.1, 0.8,
],
]
P_d = sparse(reduce(hcat, P_d)')
end
# ╔═╡ 8015bdbb-82dd-48da-905d-a25e5c864298
md"""
Pour trouver la politique optimal, on peut procéder de la façon suivante:
Initialiser ``V_\star`` (random). \
Tant qu'on a pas convergé (ou atteint `MAX_ITERATIONS`):
- Calculer pour chaque direction (gauche, haut, bas, droite) le vecteur correspondant à la fonction valeur de la politique associée à la direction.
- Sélectionner notre nouvel `V_optimal` comme le maximum par ligne de nos vecteurs issus des fonctions valeur (des quatre directions).
- Vérifier la convergeance par comparaison avec l'itération précédente.
Par application de cet algorithme on obtient alors ``V_\star``:
"""
# ╔═╡ 3d7d0b11-5b99-4b1f-ab06-3366678eece8
begin
Random.seed!(RANDOM_SEED)
V_optimal = rand(length(REWARDS))
pi = zeros(length(REWARDS))
for _ in 1:MAX_ITERATIONS
local V_g = REWARDS + Matrix(DISCOUNT_FACTOR * P_g) * V_optimal
local V_h = REWARDS + Matrix(DISCOUNT_FACTOR * P_h) * V_optimal
local V_b = REWARDS + Matrix(DISCOUNT_FACTOR * P_b) * V_optimal
local V_d = REWARDS + Matrix(DISCOUNT_FACTOR * P_d) * V_optimal
local V_new = maximum.(eachrow([V_g V_h V_b V_d]))
if isapprox(V_new, V_optimal)
pi = argmax.(eachrow([V_g V_h V_b V_d]))
break
else
V_optimal = V_new
end
end
local V_display = copy(V_optimal)
insert!(V_display, 6, 0)
V_display = permutedims(reshape(V_display, 4, 3))
V_display = sparse(V_display)
end
# ╔═╡ 664bb753-ccce-4c7a-8b11-76261a3b80d2
md"""
## Question 3
> Explain how you can infer the optimal action in every state from the optimal value function ``V_\star(s)``. Represent in a 2D matrix the optimal policy.
"""
# ╔═╡ df01ea55-b289-4c13-8a6b-780ce068e44c
md"""
La politique optimale se trouve en sélectionnant la direction la plus favorable dans chaque état:
"""
# ╔═╡ d7ff1cb5-d2b4-4597-bcef-0f74f2e7e0db
begin
pi_symbols = [ARROW_SYMBOLS[i] for i in pi]
insert!(pi_symbols, 6, WALL_SYMBOL)
pi_symbols[4] = SUCCESS_SYMBOL
pi_symbols[8] = DEATH_SYMBOL
permutedims(reshape(pi_symbols, 4, 3))
end
# ╔═╡ 40b7e793-d869-4b68-83a1-6bd7d20a3941
md"""
## Question 4
> Compare the performances obtained with the random policy and the optimal one, how can you conclude that the optimal policy performs better ?
"""
# ╔═╡ dce3978b-1334-426e-80cc-9cfe63989909
md"""
À partir ``\pi^\star`` on peut aussi trouver ``P^\star`` la matrice de notre politique optimale:
"""
# ╔═╡ 7aae25dc-38cf-40d5-a7da-44d13d397194
begin
P_star = sparse(zeros(11, 11))
for i in 1:11
if pi[i] == 1
P_star[i, :] = P_g[i, :]
elseif pi[i] == 2
P_star[i, :] = P_h[i, :]
elseif pi[i] == 3
P_star[i, :] = P_b[i, :]
else
P_star[i, :] = P_d[i, :]
end
end
P_star
end
# ╔═╡ b075f5fc-85ac-45a0-8e27-605d3dac0e97
begin
Random.seed!(RANDOM_SEED)
V_Prandom = rand(length(REWARDS))
V_Poptimal = rand(length(REWARDS))
ratio = Vector{Float64}()
convergence_random = Vector{Float64}()
convergence_optimal = Vector{Float64}()
for _ in 1:MAX_ITERATIONS
V_Prandom = REWARDS + Matrix(DISCOUNT_FACTOR * P) * V_Prandom
V_Poptimal = REWARDS + Matrix(DISCOUNT_FACTOR * P_star) * V_Poptimal
append!(convergence_optimal, norm(V_Poptimal-V_optimal))
append!(convergence_random, norm(V_Prandom-V))
append!(ratio, norm(V_Poptimal./V_Prandom))
end
end
# ╔═╡ 1fe62967-a9ea-4f6a-817e-666a900c8f92
plot(
[convergence_optimal, convergence_random],
labels = ["Optimal" "Random"],
xlabel = L"n",
ylabel = L"|| V^\star - \ \ V^r ||^2",
yticks=[10.0^-x for x in 0:20],
linewidth=2,
yaxis=:log,
title="Optimal vs Random: Convergence",
)
# ╔═╡ f31ce9b6-8399-4263-bad7-20c859116fa9
begin
plot(
ratio,
labels = "",
xlabel = L"n",
ylabel = L"|| V^\star / \ \ V^r ||^2",
linewidth=2,
title="Optimal vs Random: Ratio",
ylims=[0, Inf]
)
end
# ╔═╡ 05373383-0c51-49f2-8a62-b06a6225d659
md"""
## Question 5
> **Policy Iteration I**: We are now going to calculate the optimal policy using Policy Iteration (lecture 2, 2nd part, slides 23/35 and 24/35). You can start with the random policy for which you calculated its performance in the **Performance Prediction** section. Carry out a one-step improvement (or greedy step) on the random policy. Represent in a 2D matrix the policy you obtain. How can we verify that it is a better policy than the random one?
"""
# ╔═╡ 81572e40-4cde-4a13-84aa-5c5d6a9dbde3
md"""
0. Initialization: choose a policy ``\pi_0``
On reprend ici notre politique aléatoire ``P^{\pi_0}`` de la [question 1 partie 1](#beb410a8-03e2-4f18-8ccd-941cc926ee12):
"""
# ╔═╡ 4b264154-944d-498b-a998-a4b07f77918e
begin
P_pi_0 = P
P_pi_0
end
# ╔═╡ a68a3d33-f4df-456e-af13-9b39e14dbc13
md"""
2. Policy Evaluation: Compute iteratively ``V_{\pi_k} = (I − \gamma P^{\pi_k} )^{-1} R^{\pi_k}``
(on calcule uniquement ``V_{\pi_0}`` dans cette question)
"""
# ╔═╡ c3a6ab2c-7a3e-458f-a108-e6e81aa3def1
begin
V_pi_0 = Matrix(I(length(REWARDS)) - DISCOUNT_FACTOR * P_pi_0) \ REWARDS
local V_display = copy(V_pi_0)
insert!(V_display, 6, 0)
V_display = permutedims(reshape(V_display, 4, 3))
V_display = sparse(V_display)
end
# ╔═╡ ea457cd9-0db5-433f-9d57-1e875a160990
md"""
3. Policy improvement: Compute ``\pi_{k+1} = \text{greedy}(V_{\pi_k})``
(On calcule donc ici uniquement ``\pi_1`` )
"""
# ╔═╡ 3d62d11d-383c-4060-b697-be0c0155ce95
begin
local V_g = REWARDS + Matrix(DISCOUNT_FACTOR * P_g) * V_pi_0
local V_h = REWARDS + Matrix(DISCOUNT_FACTOR * P_h) * V_pi_0
local V_b = REWARDS + Matrix(DISCOUNT_FACTOR * P_b) * V_pi_0
local V_d = REWARDS + Matrix(DISCOUNT_FACTOR * P_d) * V_pi_0
local pi_1 = argmax.(eachrow([V_g V_h V_b V_d]))
P_pi_1 = sparse(zeros(11, 11))
for i in 1:11
if pi_1[i] == 1
P_pi_1[i, :] = P_g[i, :]
elseif pi_1[i] == 2
P_pi_1[i, :] = P_h[i, :]
elseif pi_1[i] == 3
P_pi_1[i, :] = P_b[i, :]
else
P_pi_1[i, :] = P_d[i, :]
end
end
P_pi_1
end
# ╔═╡ 245f3394-d5e3-4d2c-96a6-ce5ea0bc7d84
md"""
Stop if ``V_{\pi_{k+1}} = V_{\pi_k}``, else repeat
(Ici on s'arrête comme le dit l'énoncé pour k=1)
"""
# ╔═╡ 4f597447-f321-4a8f-adf0-3fd655ab203c
begin
diff_star_pi_0 = sum(abs.(P_star - P_pi_0))
diff_star_pi_1 = sum(abs.(P_star - P_pi_1))
md"""
On peut vérifier que ``\pi_1`` est meilleur que ``\pi_0`` en calculant:
``||\pi_\star - \pi_1||_\text{F} = `` $(diff_star_pi_1)
``||\pi_\star - \pi_0||_\text{F} = `` $(diff_star_pi_0)
"""
end
# ╔═╡ d599c370-6cb5-4bc3-a333-d41e207c39dc
if diff_star_pi_1 <= diff_star_pi_0
Markdown.MD(Markdown.Admonition("correct", "On a une meilleur politique après une itération", [md"``||\pi_\star - \pi_1||_\text{F} \leq ||\pi_\star - \pi_0||_\text{F}``"]));
else
Markdown.MD(Markdown.Admonition("danger", "On n'a pas une meilleur politique après une itération", [md"``||\pi_\star - \pi_1||_\text{F} \nleq ||\pi_\star - \pi_0||_\text{F}``"]));
end
# ╔═╡ 4e8e49b2-60ea-4dc6-906b-d459c7983b34
md"""
## Question 6
> **Policy Iteration II**: Continue iterating the Prediction and the greedy steps until convergence to the optimal policy.
"""
# ╔═╡ 362a3786-f85d-44b9-b369-ecbf4e5194e9
begin
P_pi_k = P_pi_0
local diffs = Vector{Float64}()
for k in 1:MAX_ITERATIONS
V_pi_k = Matrix(I(length(REWARDS)) - DISCOUNT_FACTOR * P_pi_k) \ REWARDS
local V_g = REWARDS + Matrix(DISCOUNT_FACTOR * P_g) * V_pi_k
local V_h = REWARDS + Matrix(DISCOUNT_FACTOR * P_h) * V_pi_k
local V_b = REWARDS + Matrix(DISCOUNT_FACTOR * P_b) * V_pi_k
local V_d = REWARDS + Matrix(DISCOUNT_FACTOR * P_d) * V_pi_k
local pi_k = argmax.(eachrow([V_g V_h V_b V_d]))
P_pi_k = sparse(zeros(11, 11))
for i in 1:11
if pi_k[i] == 1
P_pi_k[i, :] = P_g[i, :]
elseif pi_k[i] == 2
P_pi_k[i, :] = P_h[i, :]
elseif pi_k[i] == 3
P_pi_k[i, :] = P_b[i, :]
else
P_pi_k[i, :] = P_d[i, :]
end
end
append!(diffs, sum(abs.(P_star - P_pi_k)))
if isapprox(P_star, P_pi_k)
break
end
end
local p = plot(
diffs,
labels = "",
xlabel = L"k",
ylabel = L"||\pi_\star - \pi_k||_F",
linewidth=2,
title="Policy Iteration convergence",
)
xticks!(round(Int,xlims(p)[1]):round(Int,xlims(p)[2]))
end
# ╔═╡ a1eaf48e-f92f-4554-942e-f6303ebaa084
md"""
## Question 7
> Investigate the structure of the optimal policy for different values of ``\gamma``, and explain the results. You might use Value Iteration or Policy Iteration.
"""
# ╔═╡ 8d5b2cc2-2e21-47df-b821-189de5d357a3
begin
local gammas = 0.9:0.001:0.99
local iterations = zeros(length(gammas))
for (i, gamma) in enumerate(gammas)
P_pi_k = P_pi_0
k = 0
while true
V_pi_k = Matrix(I(length(REWARDS)) - gamma * P_pi_k) \ REWARDS
local V_g = REWARDS + Matrix(gamma * P_g) * V_pi_k
local V_h = REWARDS + Matrix(gamma * P_h) * V_pi_k
local V_b = REWARDS + Matrix(gamma * P_b) * V_pi_k
local V_d = REWARDS + Matrix(gamma * P_d) * V_pi_k
local pi_k = argmax.(eachrow([V_g V_h V_b V_d]))
P_pi_k = sparse(zeros(11, 11))
for i in 1:11
if pi_k[i] == 1
P_pi_k[i, :] = P_g[i, :]
elseif pi_k[i] == 2
P_pi_k[i, :] = P_h[i, :]
elseif pi_k[i] == 3
P_pi_k[i, :] = P_b[i, :]
else
P_pi_k[i, :] = P_d[i, :]
end
end
k += 1
if isapprox(P_star, P_pi_k) || k >= MAX_ITERATIONS
break
end
end
iterations[i] = k
end
local p = plot(
gammas,
iterations,
labels = "",
xlabel = L"\gamma",
ylabel = L"k",
linetype=:steppre,
linewidth=2,
title=md"Policy Iteration convergence according to ``\gamma``",
)
yticks!(round.(Int, yticks(p)[1][1]))
end
# ╔═╡ 0c6fd7ed-5180-41bd-9958-29cc9f3ce73b
md"""
On observe qu'il y a convergence de la politique généralement en dessous de 5 itérations. Cependant pour certaines combinaisons d'hyperparamètres on remarque qu'il n'y a pas convergence.
"""
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uuid = "6462fe0b-24de-5631-8697-dd941f90decc"
[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
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[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "d75bda01f8c31ebb72df80a46c88b25d1c79c56d"
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deps = ["LinearAlgebra", "SparseArrays"]
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deps = ["LinearAlgebra"]
git-tree-sha1 = "f9af7f195fb13589dd2e2d57fdb401717d2eb1f6"
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deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
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deps = ["Dates"]
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deps = ["ArgTools", "SHA"]
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[[deps.TensorCore]]
deps = ["LinearAlgebra"]
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deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
[[deps.Tricks]]
git-tree-sha1 = "6bac775f2d42a611cdfcd1fb217ee719630c4175"
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deps = ["Random", "SHA"]
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uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
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deps = ["REPL"]
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deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
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[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
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[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"
[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"
[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"
[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"
[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"
[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"
[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"
[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
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git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
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version = "1.5.2+4"
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
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[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"
[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"
[[deps.Xorg_libxkbfile_jll]]
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git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"
[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"
[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
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[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
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[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
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version = "0.3.9+1"
[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
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git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
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git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
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git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
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[[deps.Zstd_jll]]
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git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
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git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
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git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
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git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
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git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
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git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
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git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
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git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"
[[deps.x265_jll]]
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git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
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git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""
# ╔═╡ Cell order:
# ╠═26fdd17e-f03a-4835-93be-85303fe526d8
# ╟─56ac3473-24f4-42d7-84e1-cfce6a70d8d5
# ╟─ccf4d63e-7ace-11ed-2123-d9dbb62bd308
# ╟─9f2879c1-c22b-4067-ad20-4e4c56cc8d00
# ╟─0a30a68a-068e-41fb-92c4-000869ba7dff
# ╟─07b57746-fba0-49aa-ba17-6dcb0bbe44e5
# ╟─92d6874b-651c-4551-840e-ad5d1e934aeb
# ╟─fe44d7f2-155e-42f2-83c3-dd18aadb3810
# ╟─28b769a6-dd3c-43ab-bae0-646d8ebc35d6
# ╟─3881603c-619b-4976-ac4c-2c7e7f3a6ec7
# ╟─fb797a9b-6a0a-4a77-a9b6-6804f98639bb
# ╟─1e3abda8-6645-48ba-874d-28e1011fc3e3
# ╟─beb410a8-03e2-4f18-8ccd-941cc926ee12
# ╟─133f291f-6f21-4441-86f7-ba190a7d6b1f
# ╟─e14f9977-d2fd-4d05-84d6-614008dc0c4a
# ╟─486c93ab-9cb9-4df4-b702-bbe12a961647
# ╟─ab2d705d-fc00-43b2-bb6d-2a3d4ba9dab1
# ╟─b7ae89c9-3c1b-4f5c-af5b-164d95ccca41
# ╟─03c17428-5ab9-42e7-bf79-92eb846f11cb
# ╟─c65d0dbc-ecd7-4320-9b3a-a1b9c0545f9a
# ╟─ad547684-bcbe-44f4-9fc1-f327d2db4584
# ╠═d3703ab8-912c-417d-acd9-29590ec1134b
# ╟─1319b304-5126-4825-8076-e113e4dd3635
# ╟─3ea3f177-c576-4b9e-a54b-c427e29a8491
# ╠═e94fe8a6-274b-4121-b1fc-063d3710c2f7
# ╟─80090d5f-d56c-4844-a04f-444ed49e5f34
# ╟─98362798-aae4-4540-9e98-cc7371802552
# ╟─30874daf-7b0e-4335-9a50-d19389cf1620
# ╟─c005a3f8-765c-4a50-90ef-73a5a72eee01
# ╟─1b43e9e5-d7d2-4b5e-a2b2-3a8b8eda6d62
# ╟─add0221b-e352-4559-a722-c45a64f573f9
# ╟─84e07dce-bf6d-4ac1-bfa4-65414fe1d787
# ╟─df13fa05-14de-409b-a0b1-5bba5eff432e
# ╟─ac490e4a-ce20-4288-a04f-c224df5ade1a
# ╟─33890f22-d3f6-4bcf-870d-756f7ff250a9
# ╟─cf9fb8a8-6c93-4c43-9f01-5f198f0cf4aa
# ╟─dc87b85f-c87c-4302-9124-194bd799f1fd
# ╟─b2595dec-aa5b-462b-b0f8-3555c1231b2f
# ╟─70edf811-adb0-4ae8-941a-b298d85a6e0e
# ╟─875673f1-08c9-4713-bbc2-85b0a7a0cb0a
# ╟─2deaac7c-ad14-43b0-9cd5-9f0ec12d324c
# ╟─b5c93b6f-933c-41b4-8399-44cc0fa07fab
# ╟─8015bdbb-82dd-48da-905d-a25e5c864298
# ╟─3d7d0b11-5b99-4b1f-ab06-3366678eece8
# ╟─664bb753-ccce-4c7a-8b11-76261a3b80d2
# ╟─df01ea55-b289-4c13-8a6b-780ce068e44c
# ╟─d7ff1cb5-d2b4-4597-bcef-0f74f2e7e0db
# ╟─40b7e793-d869-4b68-83a1-6bd7d20a3941
# ╟─dce3978b-1334-426e-80cc-9cfe63989909
# ╟─7aae25dc-38cf-40d5-a7da-44d13d397194
# ╟─b075f5fc-85ac-45a0-8e27-605d3dac0e97
# ╟─1fe62967-a9ea-4f6a-817e-666a900c8f92
# ╟─f31ce9b6-8399-4263-bad7-20c859116fa9
# ╟─05373383-0c51-49f2-8a62-b06a6225d659
# ╟─81572e40-4cde-4a13-84aa-5c5d6a9dbde3
# ╟─4b264154-944d-498b-a998-a4b07f77918e
# ╟─a68a3d33-f4df-456e-af13-9b39e14dbc13
# ╟─c3a6ab2c-7a3e-458f-a108-e6e81aa3def1
# ╟─ea457cd9-0db5-433f-9d57-1e875a160990
# ╟─3d62d11d-383c-4060-b697-be0c0155ce95
# ╟─245f3394-d5e3-4d2c-96a6-ce5ea0bc7d84
# ╟─4f597447-f321-4a8f-adf0-3fd655ab203c
# ╟─d599c370-6cb5-4bc3-a333-d41e207c39dc
# ╟─4e8e49b2-60ea-4dc6-906b-d459c7983b34
# ╟─362a3786-f85d-44b9-b369-ecbf4e5194e9
# ╟─a1eaf48e-f92f-4554-942e-f6303ebaa084
# ╟─8d5b2cc2-2e21-47df-b821-189de5d357a3
# ╟─0c6fd7ed-5180-41bd-9958-29cc9f3ce73b
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
Reference in a new issue Copy permalink