gradiente-estocastico.jl

### A Pluto.jl notebook ###
# v0.17.1

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

# ╔═╡ e8fb86c8-635c-11ec-283e-f923ac0f4ce3
using ForwardDiff, Plots, PlutoUI, Random

# ╔═╡ bb190999-10f8-4d8c-8409-226ea4eb6e7d
begin
	sigmoide(t) = t ≥ 0 ? 1 / (1 + exp(-t)) : exp(t) / (1 + exp(t))
	struct RedeNeural{T}
		dados_linearizados :: Vector{T}
		W
		b
		camadas
	end

	function RedeNeural(camadas :: Vector{Int})
		num_camadas = length(camadas)
		tam = 0
		for i = 1:num_camadas - 1
			m, n = camadas[[i,i+1]]
			tam += m * n + n
		end
		dados_linearizados = randn(tam)
		RedeNeural(dados_linearizados, camadas)
	end

	function RedeNeural(dados_linearizados :: Vector{T}, camadas :: Vector{Int}) where {T}
		num_camadas = length(camadas)
		W = []
		b = []
		idx_b = 0
		for i = 1:num_camadas - 1
			m, n = camadas[[i,i+1]]
			idx = idx_b .+ (1:m*n)
			push!(W, reshape(view(dados_linearizados, idx), m, n))
			idx = idx_b + m * n .+ (1:n)
			push!(b, view(dados_linearizados, idx))
			idx_b += m * n + n
		end
		return RedeNeural{T}(dados_linearizados, W, b, camadas)
	end

	function (modelo::RedeNeural)(X)
		n = length(modelo.W)
		Y = copy(X)
		for i = 1:n
			Y = sigmoide.(Y * modelo.W[i] .+ modelo.b[i]')
		end
		return Y
	end

	RedeNeural
end

# ╔═╡ 2a8db060-8a8e-460b-b876-a5f3e44bf973
begin
	function objetivo(modelo, β, X, y)
		modelo.dados_linearizados .= β
		ŷ = modelo(X)[:]
		-sum(
			y[i] == 1 ? log(ŷ[i] + 1e-8) : log.(1 + 1e-8 - ŷ[i])
			for i = 1:length(y)
		)
	end

	function gradiente(modelo, β, X, y)
		ForwardDiff.gradient(
			γ -> objetivo(
				RedeNeural(γ, modelo.camadas),
				γ, X, y),
			β
		)
	end
end

# ╔═╡ b0f3bf7f-b919-4755-9804-c688e0ed6a28
begin
	# Plot related
	plt_args = [
		[:m => (3, :lightblue, stroke(1, :blue), :circle)],
		[:m => (3, :pink, stroke(1, :red), :square)],
	]
	function plot_dataset!(plt, X, y)
		u = y |> unique |> sort
		for i = 1:length(u)
			yi = u[i]
			idx = findall(y .== yi)
			scatter!(plt, X[idx,1], X[idx,2]; plt_args[i]...)
		end
		plt
	end
end

# ╔═╡ 55e6fdb2-e13b-48a7-8ae2-5d36aae66249
begin
	Random.seed!(0)
	n = 100
	x = range(0, 1, length=n)
	y = 2x .+ 3 + randn(n) * 0.3

	β₀rg = range(2, 4, length=30)
	β₁rg = range(0, 4, length=30)

	h(β, x) = β[1] + β[2] * x
	L(β, i) = (y[i] - h(β, x[i]))^2 / 2
	L(β) = sum(L(β, i) for i = 1:n)

	"Dataset"
end

# ╔═╡ 2cc7102c-104a-4fba-a9f7-6f2e623ee06b
md"""
# Gradiente Estocástico

- Considere o modelo de regressão linear: $h_{\beta}(x) = \beta_0 + \beta_1 x$.
- O problema de otimização é dado por

$$\min_{\beta} L(\beta) = \frac{1}{2n} \sum_{i=1}^n \big(y_i - h_{\beta}(x_i)\big)^2.$$

- Suponha que vamos otimizar com o método do gradiente.
- O gradiente é

$$\nabla L(\beta) = \frac{1}{n}\sum_{i=1}^n (h_{\beta}(x_i) - y_i) \begin{bmatrix} 1 \\ x_i \end{bmatrix}.$$

- O cálculo de um gradiente passa por todos os dados, e o resultado de um passo é longe de ser bom.
- Então por que fazer tanto trabalho?
- No gradiente estocástico, usamos como direção apenas o gradiente da perda de um dado:

$$\nabla L_i(\beta) = (h_{\beta}(x_i) - y_i) \begin{bmatrix} 1 \\ x_i \end{bmatrix}.$$

- O método do gradiente estocástico pode ser descrito como:
  1. Embaralhe o conjunto $\Omega = \{1, 2, \dots, n\}$
  2. Enquanto não estiver satisfeito:
     1. Para cada $i \in \Omega$
        1. Calcule $d = \nabla L_i(\beta)$
        2. Atualize $\beta \leftarrow \beta - \gamma d$

"""

# ╔═╡ 21b3e924-171a-4df6-a2e3-7d964ff6b16b
md"""
**Visualizações**
"""

# ╔═╡ 53cfa697-028e-460a-9a53-8fd0a6470a56
md"""
i = $(@bind ex1_i Slider(1:n, show_value=true))

β = [$(@bind ex1_β₀ Slider(β₀rg, default=3)), $(@bind ex1_β₁ Slider(β₁rg, default=2))]

γ = $(@bind ex1_γ Slider(0.5:0.01:1.0, show_value=true))
"""

# ╔═╡ 79fe7251-4221-42e2-afb5-dcdb9c6ab29b
let
	i = ex1_i
	β = [ex1_β₀; ex1_β₁]
	γ = ex1_γ
	
	β⁺ = β - γ * (h(β, x[i]) - y[i]) * [1; x[i]]
	βᶠ = β - γ * sum((h(β, x[i]) - y[i]) * [1; x[i]] for i = 1:n) / n
	
	plt = plot(size=(800, 400), layout=grid(1, 2), leg=false)
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)), leg=false)
	scatter!(plt[1], [x[i]], [y[i]], m=(4,:pink,stroke(1,:red)))
	plot!(plt[1], [0, 1], [h(β, 0), h(β, 1)], c=:black, lw=1)
	plot!(plt[1], [0.5, 0.5], [h(β, 0.5), h(β⁺, 0.5)], l=:arrow, c=:pink)
	plot!(plt[1], [0, 1], [h(β⁺, 0), h(β⁺, 1)], c=:red, lw=1)
	xlims!(plt[1], -0.02, 1.02)
	ylims!(plt[1], β₀rg[1], 6)
	
	contourf!(plt[2],
		β₀rg,
		β₁rg,
		(x,y) -> L([x; y]),
		c=:blues,
		lw=0,
		levels=50
	)
	scatter!(plt[2], [β[1]], [β[2]], m=(3,:black))
	plot!(plt[2], [β[1], βᶠ[1]], [β[2], βᶠ[2]], l=:arrow, c=:green)
	scatter!(plt[2], [βᶠ[1]], [βᶠ[2]], m=(4,:lime,stroke(1,:green),:diamond))
	plot!(plt[2], [β[1], β⁺[1]], [β[2], β⁺[2]], l=:arrow, c=:red)
	scatter!(plt[2], [β⁺[1]], [β⁺[2]], m=(4,:pink,stroke(1,:red)))
	xlims!(plt[2], β₀rg[1], β₀rg[end])
	ylims!(plt[2], β₁rg[1], β₁rg[end])
end

# ╔═╡ e6c49612-0212-497a-8c63-7a15b4c231d7
md"""
Próximo da solução os dados "competem", mas longe eles concordam.
"""

# ╔═╡ 62c86483-a142-4909-b4c1-64d35543cf9d
md"""
**Exemplo de execução**
"""

# ╔═╡ 1640848a-e315-4c85-ae2d-65c221afb033
md"""
semente = $(@bind ex2_seed Slider(0:10, show_value=true))

iteracoes = $(@bind ex2_iteracoes Slider(1:1000, show_value=true))

γ = $(@bind ex2_γ Slider(0.1:0.01:1.0, show_value=true))
"""

# ╔═╡ c142a955-f0b1-433f-8e8e-73e25d974070
let
	Random.seed!(ex2_seed)
	B = zeros(2, ex2_iteracoes)
	β = [β₀rg[1]; β₁rg[1]]
	B[:,1] = β
	idx = shuffle(1:n)
	γ = ex2_γ
	for i = 1:ex2_iteracoes-1
		j = idx[mod1(i, n)]
		β .= β - γ * (h(β, x[j]) - y[j]) * [1; x[j]]
		B[:,i+1] .= β
		if mod1(i, n) == n
			idx .= shuffle(1:n)
		end
	end
	full_iteracoes = div(ex2_iteracoes, n) + 1
	Bᶠ = zeros(2, full_iteracoes)
	Bᶠ[:,1] = [β₀rg[1]; β₁rg[1]]
	for k = 1:full_iteracoes-1
		Bᶠ[:,k+1] = Bᶠ[:,k] - γ * sum((h(Bᶠ[:,k], x[i]) - y[i]) * [1; x[i]] for i = 1:n) / n
	end
	
	plt = plot(size=(800, 400), layout=grid(1, 2), leg=false)
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)), leg=false)
	plot!(plt[1], [0, 1], [h(β, 0), h(β, 1)], c=:black, lw=1)
	xlims!(plt[1], -0.02, 1.02)
	ylims!(plt[1], β₀rg[1], 6)

	contourf!(plt[2],
		β₀rg,
		β₁rg,
		(x,y) -> L([x; y]),
		c=:blues,
		lw=0,
		levels=50
	)
	scatter!(plt[2], [β[1]], [β[2]], m=(3,:black))
	plot!(plt[2], Bᶠ[1,:], Bᶠ[2,:], l=:arrow, c=:lime, m=(2,stroke(0),:diamond))
	plot!(plt[2], B[1,:], B[2,:], l=:arrow, c=:red, opacity=0.5)
	min_B = max(1, ex2_iteracoes - n)
	scatter!(plt[2], B[1,min_B:end], B[2,min_B:end], m=(1,:yellow,stroke(0,:orange)))
	xlims!(plt[2], β₀rg[1], β₀rg[end])
	ylims!(plt[2], β₁rg[1], β₁rg[end])
end

# ╔═╡ 7e472f12-684b-4918-899c-c162f9829957
md"""
## Mini batch

- O gradiente completo é caro e lento, mas tem mira boa.
- O gradiente estocástico é barato e rápido, mas com direção ruim.
- O gradiente em mini-batches é um meio termo, que tenta ser suficientemente rápido, com direção suficientemente boa.
- A ideia é dividir o conjunto $\Omega = \{1,2,\dots,n\}$ em $N$ partes, de tamanho próximo e embaralhados: $\Omega = \cup_{k=1}^N \Omega_k$
- O gradiente do batch $\Omega_k$ é

$$\nabla L_{\Omega_k}(x) = \frac{1}{|\Omega_k|} \sum_{i \in \Omega_k} (h_{\beta}(x_i) - y_i) \begin{bmatrix} 1 \\ x_i \end{bmatrix}.$$
- Como usamos $N$ dados, a variância do gradiente é menor.
"""

# ╔═╡ 319564b5-46c2-40d4-a640-f283da1a377a
md"""
**Visualização**
"""

# ╔═╡ 4784f03f-21d6-496b-900a-e8ebb8798b78
begin
	divisores = [x for x = 1:n if n % x == 0]
	md"""
	tamanho do batch = $(@bind ex3_batch_size Slider(divisores, show_value=true))

	semente = $(@bind ex3_seed Slider(0:10, show_value=true))
	
	β = [$(@bind ex3_β₀ Slider(β₀rg, default=3)), $(@bind ex3_β₁ Slider(β₁rg, default=2))]
	
	γ = $(@bind ex3_γ Slider(0.5:0.01:1.0, show_value=true))
	"""
end

# ╔═╡ 2eafa1ee-a1a8-4bde-9797-d4572c68bf5c
md"""
batch = $(@bind ex3_batch Slider(1:div(n, ex3_batch_size), show_value=true))
"""

# ╔═╡ fdd835b5-7856-432a-b4a6-36d43e32d2b5
let
	Random.seed!(ex3_seed)
	batch_size = ex3_batch_size
	num_batches = div(n, batch_size)
	batches = reshape(shuffle(1:n), batch_size, num_batches)
	batch = batches[:,ex3_batch]
	β = [ex3_β₀; ex3_β₁]
	γ = ex3_γ
	
	β⁺ = β - γ * sum((h(β, x[i]) - y[i]) * [1; x[i]] for i = batch) / batch_size
	βᶠ = β - γ * sum((h(β, x[i]) - y[i]) * [1; x[i]] for i = 1:n) / n
	
	plt = plot(size=(800, 400), layout=grid(1, 2), leg=false)
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)), leg=false)
	scatter!(plt[1], x[batch], y[batch], m=(4,:pink,stroke(1,:red)))
	plot!(plt[1], [0, 1], [h(β, 0), h(β, 1)], c=:black, lw=1)
	plot!(plt[1], [0.5, 0.5], [h(β, 0.5), h(β⁺, 0.5)], l=:arrow, c=:pink)
	plot!(plt[1], [0, 1], [h(β⁺, 0), h(β⁺, 1)], c=:red, lw=1)
	xlims!(plt[1], -0.02, 1.02)
	ylims!(plt[1], β₀rg[1], 6)
	
	contourf!(plt[2],
		β₀rg,
		β₁rg,
		(x,y) -> L([x; y]),
		c=:blues,
		lw=0,
		levels=50
	)
	scatter!(plt[2], [β[1]], [β[2]], m=(3,:black))
	plot!(plt[2], [β[1], βᶠ[1]], [β[2], βᶠ[2]], l=:arrow, c=:green)
	scatter!(plt[2], [βᶠ[1]], [βᶠ[2]], m=(4,:lime,stroke(1,:green),:diamond))
	plot!(plt[2], [β[1], β⁺[1]], [β[2], β⁺[2]], l=:arrow, c=:red)
	scatter!(plt[2], [β⁺[1]], [β⁺[2]], m=(4,:pink,stroke(1,:red)))
	xlims!(plt[2], β₀rg[1], β₀rg[end])
	ylims!(plt[2], β₁rg[1], β₁rg[end])
end

# ╔═╡ bd8815ad-0566-4032-9220-2d346706eb46
md"""
**Exemplo de execução**
"""

# ╔═╡ ac91363a-647f-42d6-a8e7-09655e223e9a
md"""
tamanho do batch = $(@bind ex4_batch_size Slider(divisores, show_value=true))

semente = $(@bind ex4_seed Slider(0:10, show_value=true))

iteracoes = $(@bind ex4_iteracoes Slider(1:1000, show_value=true))

γ = $(@bind ex4_γ Slider(0.1:0.01:1.0, show_value=true))
"""

# ╔═╡ 52cb5d59-0749-4ce2-89a2-2f2e2a6a9c47
let
	Random.seed!(ex4_seed)
	B = zeros(2, ex4_iteracoes)
	
	batch_size = ex4_batch_size
	num_batches = div(n, batch_size)
	batches = reshape(shuffle(1:n), batch_size, num_batches)
	
	β = [β₀rg[1]; β₁rg[1]]
	B[:,1] = β
	idx = shuffle(1:n)
	γ = ex4_γ
	for i = 1:ex4_iteracoes-1
		batch = batches[:,mod1(i, num_batches)]
		β .= β - γ * sum((h(β, x[j]) - y[j]) * [1; x[j]] for j in batch) / batch_size
		B[:,i+1] .= β
		if mod1(i, n) == batch_size
			batches .= reshape(shuffle(1:n), batch_size, num_batches)
		end
	end
	full_iteracoes = div(ex4_iteracoes * batch_size, n) + 1
	Bᶠ = zeros(2, full_iteracoes)
	Bᶠ[:,1] = [β₀rg[1]; β₁rg[1]]
	for k = 1:full_iteracoes-1
		Bᶠ[:,k+1] = Bᶠ[:,k] - γ * sum((h(Bᶠ[:,k], x[i]) - y[i]) * [1; x[i]] for i = 1:n) / n
	end
	
	plt = plot(size=(800, 400), layout=grid(1, 2), leg=false)
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)), leg=false)
	plot!(plt[1], [0, 1], [h(β, 0), h(β, 1)], c=:black, lw=1)
	xlims!(plt[1], -0.02, 1.02)
	ylims!(plt[1], β₀rg[1], 6)

	contourf!(plt[2],
		β₀rg,
		β₁rg,
		(x,y) -> L([x; y]),
		c=:blues,
		lw=0,
		levels=50
	)
	scatter!(plt[2], [β[1]], [β[2]], m=(3,:black))
	plot!(plt[2], Bᶠ[1,:], Bᶠ[2,:], l=:arrow, c=:lime, m=(2,stroke(0),:diamond))
	plot!(plt[2], B[1,:], B[2,:], l=:arrow, c=:red, opacity=0.5)
	min_B = max(1, ex4_iteracoes - num_batches)
	scatter!(plt[2], B[1,min_B:end], B[2,min_B:end], m=(1,:yellow,stroke(0,:orange)))
	xlims!(plt[2], β₀rg[1], β₀rg[end])
	ylims!(plt[2], β₁rg[1], β₁rg[end])
end

# ╔═╡ c5b130ee-3c23-4f83-90ab-1d76b902785b
md"""
## Em geral

- Tudo que fizemos funciona para regressão logística e redes reurais definindo

$$L_i(\beta) = \ell(y_i, h_{\beta}(x_i)).$$
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[compat]
ForwardDiff = "~0.10.24"
Plots = "~1.25.3"
PlutoUI = "~0.7.27"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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# ╟─bd8815ad-0566-4032-9220-2d346706eb46
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# ╟─c5b130ee-3c23-4f83-90ab-1d76b902785b
# ╟─00000000-0000-0000-0000-000000000001
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