### A Pluto.jl notebook ###
# v0.16.0

using Markdown
using InteractiveUtils

# ╔═╡ 7e7dc954-866d-11eb-14df-c37b4fe266bf
using Calculus

# ╔═╡ 80e8c8fc-85c7-11eb-191f-23c76544a1ae
using Plots; import PlotlyJS; plotlyjs()

# ╔═╡ 8c0015c6-868d-11eb-1633-eb919d27d3ac
using PlutoUI

# ╔═╡ 0af7ed02-d4ab-4aab-8ed5-f03eb8cbadce
md"[Reference Chapter](https://www.deeplearningbook.org/contents/numerical.html)"

# ╔═╡ ddeed08a-8885-11eb-3681-fb6118238332
md"""
!!! example "Example 1"
	
	$y = x^4 - 3x^3 - 1$
	
	its derivative with respect to $x$ :

	$\frac{dy}{dx} = 4x^3 - 9x^2$
	
	equating them to zero to find their minimum:

	$x^2(4x - 9) = 0$

	We see that $y$ is minimum at $x=2.25$
"""

# ╔═╡ f123d468-8630-11eb-3daf-d35082dc5458
md"# Gradient Descent"

# ╔═╡ 946e6300-85c7-11eb-0e3d-abc305941caf
f(x) = x^4 - 3x^3 - 1

# ╔═╡ a4135446-85c7-11eb-1ca5-835e8460acd8
let
	x = -3:0.001:4.5
	plot(x, f)
end

# ╔═╡ c51e07a0-85c8-11eb-3247-c345ab59be62
function gradient_descent(α, x_0, f)
	x = copy(x_0)
	t = copy(x_0)
	df(x) = Calculus.gradient(f, x)
	for i = 1:500
		x = x - α * df(x)
		t = [t; x]
		if(round.(df(x), digits=5) == zeros(length(x_0)))
			break;
		end
	end
	return reshape(t, length(x_0), :)'
end

# ╔═╡ ed02fa0c-acf3-4a1a-b594-19747a69e757
md"""
#### What happened here?

I just took the gradient function from the book, and made it into a function in julia. It returns all the values of x, so we can can visualise the descent later.
"""

# ╔═╡ 2ba433f4-85cc-11eb-2043-0b75815bfec6
points = gradient_descent(0.01, 3.0, f)[1:end]

# ╔═╡ 4c51b95e-85cd-11eb-3432-47fc75f137eb
let
	x = -2:0.01:3
	plot(x, f)
	plot!(points, f, line = (:dash, :green), marker=(:circle,2))
end

# ╔═╡ 11908454-8630-11eb-1430-d37daf6d38d8
md"# Newton's Method"

# ╔═╡ 4e5269ea-8633-11eb-1a9e-db09365517f1
g(x) = x[1]^3 + x[2]^3 - 6x[1]*x[2] 

# ╔═╡ 7408d648-8894-11eb-35f2-09ab2d5fbe52
md"""
!!! example "Example 2"
	
	$z = x^3 + y^3 - 6xy$
	
	its derivatives with respect to $x$ and $y$:

	$\frac{dz}{dx} = 3x^2 - 6y, \frac{dz}{dy} = 3y^2 - 6x$
	
	equating them to zero to find their minimum:

	$x^2 = 2y \;and\;y^2  = 2x$

	$\frac{y^4}{4} = 2y \;and\;\frac{x^4}{4} = 2x$

	$y^4 = 8y \;and\;x^4 = 8x$

	$y = 2 \;and\;x = 2$

	We see that $z$ is minimum at $x=y=2$
"""

# ╔═╡ 28a59930-8631-11eb-301d-6147e94972da
function newton(x_0, g)
	x = vec(Float64.(x_0))
	t = x_0
	for i in 1:500
		x = x - (inv(Calculus.hessian(g, x)) * Calculus.derivative(g, x))
		t = [t; x']
		if(round.(Calculus.derivative(g, x), digits=9) == zeros(length(x_0)))
			break
		end
	end
	return t
end

# ╔═╡ f8338402-866b-11eb-0d5c-9d66645ec5a2
newpoints = newton([4 4], g)

# ╔═╡ 5ed42dac-9016-11eb-3ba1-935c00ec9c99
f1(x,y) = x^3 + y^3 - 6x*y

# ╔═╡ c51c1cc8-8689-11eb-3bea-ad122713fa65
let
	x = y = -3:0.5:3
	surface(x, y, f1, fillalpha=0.5)	
	plot!(newpoints[:,1], newpoints[:,2], f1.(newpoints[:,1],newpoints[:,2]), line=(:dot, 4), marker=(:circle, 1))
end

# ╔═╡ 6845fc55-81aa-423a-a08a-732ba0583a86
md"""
**Note: _You can find many such pre-defined methods in_** [Optim.jl](https://julianlsolvers.github.io/Optim.jl/stable/)
"""

# ╔═╡ ff78ac97-d8ba-45dd-abe5-e2518272de6c
md"In both the above methods, try setting the inital values to $-1.0$ for the Gradient Descent method, and $[-2, -2]$ for Newton's method. Lets see that below."

# ╔═╡ 2109fed6-b891-4be7-a078-cca774d9eed1
md"##### For Gradient Descent"

# ╔═╡ afbec578-7aa8-465d-8ab5-1cdcec6b4750
saddlepoints1 = gradient_descent(0.01, -1.0, f)

# ╔═╡ e1653b17-92e7-4b43-bc41-065d00d8c8f6
let
	x = -2:0.01:3
	plot(x, f, linecolor = :gray)
	plot!(saddlepoints1, f, linestyle = :dash, marker=(:circle,1), linecolor=:red)
end

# ╔═╡ ad1b745d-551d-4e9b-a943-577a1ab34dfb
md"##### For Newton's Method"

# ╔═╡ 842c8484-816d-42ac-be11-573cd39373a0
saddlepoints2 = newton([-2 -2], g)

# ╔═╡ cbc8ecb8-03c5-4e0f-aea1-0ad3807917d1
let
	x = y = -3:0.5:3
	surface(x, y, f1, fillalpha=0.5, c = :winter)
	plot!(saddlepoints2[:,1], saddlepoints2[:,2], f1.(saddlepoints2[:,1],saddlepoints2[:,2]), line=(:dash, :red, 2), marker=(:circle,1))
end

# ╔═╡ a60233dd-ef16-4101-a97e-f4db59ca67d8
md"""
Thats not the global minimum. Those points are called **Saddle Points**.
"""

# ╔═╡ 10a3c5bc-8cfa-11eb-0ff7-195329f7b738
md"## Try Yourself"

# ╔═╡ 87052830-9016-11eb-1082-111250655ad8
md"Try the linear least squares example from the book. Make the function and try plugging it into Newton's step method. As given, you should get your output within $1$ step. Also try to use Optim.jl to optimize this funciton again."

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version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[deps.WebIO]]
deps = ["AssetRegistry", "Base64", "Distributed", "FunctionalCollections", "JSON", "Logging", "Observables", "Pkg", "Random", "Requires", "Sockets", "UUIDs", "WebSockets", "Widgets"]
git-tree-sha1 = "adc25e225bc334c7df6eec3b39496edfc451cc38"
uuid = "0f1e0344-ec1d-5b48-a673-e5cf874b6c29"
version = "0.8.15"

[[deps.WebSockets]]
deps = ["Base64", "Dates", "HTTP", "Logging", "Sockets"]
git-tree-sha1 = "f91a602e25fe6b89afc93cf02a4ae18ee9384ce3"
uuid = "104b5d7c-a370-577a-8038-80a2059c5097"
version = "1.5.9"

[[deps.Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "eae2fbbc34a79ffd57fb4c972b08ce50b8f6a00d"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.3"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+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"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[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]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
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"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
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"
"""

# ╔═╡ Cell order:
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# ╟─7408d648-8894-11eb-35f2-09ab2d5fbe52
# ╠═28a59930-8631-11eb-301d-6147e94972da
# ╠═f8338402-866b-11eb-0d5c-9d66645ec5a2
# ╠═8c0015c6-868d-11eb-1633-eb919d27d3ac
# ╠═5ed42dac-9016-11eb-3ba1-935c00ec9c99
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# ╠═e1653b17-92e7-4b43-bc41-065d00d8c8f6
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# ╠═842c8484-816d-42ac-be11-573cd39373a0
# ╠═cbc8ecb8-03c5-4e0f-aea1-0ad3807917d1
# ╟─a60233dd-ef16-4101-a97e-f4db59ca67d8
# ╟─10a3c5bc-8cfa-11eb-0ff7-195329f7b738
# ╟─87052830-9016-11eb-1082-111250655ad8
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
