{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"from bokeh.plotting import figure, show, output_notebook"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## algorithm"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def gradient_descent(F, dF, x, steps=100, lr=0.001):\n",
" loss = []\n",
" \n",
" for _ in range(steps):\n",
" dx = dF(x)\n",
" x -= lr * dx\n",
" loss.append(F(x))\n",
"\n",
" return x, loss"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def rmsprop(F, dF, x, steps=100, lr=0.001, decay=.9, eps=1e-8):\n",
" loss = []\n",
" dx_mean_sqr = np.zeros(x.shape, dtype=float)\n",
"\n",
" for _ in range(steps):\n",
" dx = dF(x)\n",
" dx_mean_sqr = decay * dx_mean_sqr + (1 - decay) * dx ** 2\n",
" x -= lr * dx / (np.sqrt(dx_mean_sqr) + eps)\n",
" loss.append(F(x))\n",
" \n",
" return x, loss"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def rmsprop_momentum(F, dF, x, steps=100, lr=0.001, decay=.9, eps=1e-8, mu=.9):\n",
" loss = []\n",
" dx_mean_sqr = np.zeros(x.shape, dtype=float)\n",
" momentum = np.zeros(x.shape, dtype=float)\n",
"\n",
" for _ in range(steps):\n",
" dx = dF(x)\n",
" dx_mean_sqr = decay * dx_mean_sqr + (1 - decay) * dx ** 2\n",
" momentum = mu * momentum + lr * dx / (np.sqrt(dx_mean_sqr) + eps)\n",
" x -= momentum\n",
" loss.append(F(x))\n",
"\n",
" return x, loss"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## function"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def F(x):\n",
" residual = A @ x - np.eye(len(A), dtype=float)\n",
" return np.sum(residual ** 2)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def dF(x):\n",
" return 2 * A.T @ (A @ x - np.eye(len(A), dtype=float))"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"A = np.array([\n",
" [2, 5, 1, 4, 6],\n",
" [3, 5, 0, 0, 0],\n",
" [1, 1, 0, 3, 8],\n",
" [6, 6, 2, 2, 1],\n",
" [8, 3, 5, 1, 4],\n",
"], dtype=float)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## optimization"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[ 0.79, -0.01, 0.18, 0.19, -0.08],\n",
" [-0.01, 0.8 , 0. , 0.2 , -0.07],\n",
" [ 0.18, 0. , 0.85, -0.15, 0.07],\n",
" [ 0.19, 0.2 , -0.15, 0.66, 0.13],\n",
" [-0.08, -0.07, 0.07, 0.13, 0.95]]), 0.54691984767143453)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X, loss1 = gradient_descent(F, dF, A * 0, steps=300)\n",
"(A @ X).round(2), loss1[-1]"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[ 0.84, -0.05, 0.1 , 0.1 , -0.06],\n",
" [-0.04, 0.82, 0.03, 0.19, -0.03],\n",
" [ 0.12, 0.03, 0.9 , -0.08, 0.03],\n",
" [ 0.15, 0.2 , -0.12, 0.75, 0.06],\n",
" [-0.08, -0.09, 0.04, 0.1 , 0.97]]), 0.32396954419819657)"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X, loss2 = rmsprop(F, dF, A * 0, steps=300)\n",
"(A @ X).round(2), loss2[-1]"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(array([[ 0.99, 0.01, 0. , -0.01, 0. ],\n",
" [-0. , 1. , 0. , -0. , 0. ],\n",
" [-0. , 0.01, 1. , -0.01, 0. ],\n",
" [-0.01, 0.01, 0. , 0.99, 0. ],\n",
" [-0.01, 0.01, 0. , -0.01, 1. ]]), 0.00062303887772378397)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X, loss3 = rmsprop_momentum(F, dF, A * 0, steps=300)\n",
"(A @ X).round(2), loss3[-1]"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"output_type": "display_data"
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" \"<div style='background-color: #fdd'>\\n\"+\n",
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" \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
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" \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
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"\n",
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" return false;\n",
" }\n",
"\n",
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"\n",
" var inline_js = [\n",
" function(Bokeh) {\n",
" Bokeh.set_log_level(\"info\");\n",
" },\n",
" \n",
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" },\n",
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" function(Bokeh) {\n",
" \n",
" document.getElementById(\"d8b01c18-bc74-42bd-8485-9e68ed5f812a\").textContent = \"BokehJS is loading...\";\n",
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" console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-0.12.5.min.css\");\n",
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" console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.5.min.css\");\n",
" Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.5.min.css\");\n",
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" ];\n",
"\n",
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" inline_js[i](window.Bokeh);\n",
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" display_loaded();\n",
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" setTimeout(run_inline_js, 100);\n",
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" console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
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" cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
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" if (window._bokeh_is_loading === 0) {\n",
" console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
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" console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
" run_inline_js();\n",
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" }\n",
"}(this));"
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{
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" <div class=\"bk-root\">\n",
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" \"<ul>\\n\"+\n",
" \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
" \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
" \"</ul>\\n\"+\n",
" \"<code>\\n\"+\n",
" \"from bokeh.resources import INLINE\\n\"+\n",
" \"output_notebook(resources=INLINE)\\n\"+\n",
" \"</code>\\n\"+\n",
" \"</div>\"}};\n",
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" setTimeout(display_loaded, 100)\n",
" }\n",
" }\n",
" \n",
" function run_callbacks() {\n",
" window._bokeh_onload_callbacks.forEach(function(callback) { callback() });\n",
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" console.info(\"Bokeh: all callbacks have finished\");\n",
" }\n",
" \n",
" function load_libs(js_urls, callback) {\n",
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" return null;\n",
" }\n",
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" var render_items = [{\"docid\":\"ea5831f7-2cea-46a4-a8a9-b4a1eb57f3e9\",\"elementid\":\"845ef23a-3d24-4d6c-a171-8af1f2db5ae0\",\"modelid\":\"ad8f06e8-e58e-462d-bf8b-54a431c8d480\"}];\n",
" \n",
" Bokeh.embed.embed_items(docs_json, render_items);\n",
" };\n",
" if (document.readyState != \"loading\") fn();\n",
" else document.addEventListener(\"DOMContentLoaded\", fn);\n",
" })();\n",
" },\n",
" function(Bokeh) {\n",
" }\n",
" ];\n",
" \n",
" function run_inline_js() {\n",
" \n",
" if ((window.Bokeh !== undefined) || (force === true)) {\n",
" for (var i = 0; i < inline_js.length; i++) {\n",
" inline_js[i](window.Bokeh);\n",
" }if (force === true) {\n",
" display_loaded();\n",
" }} else if (Date.now() < window._bokeh_timeout) {\n",
" setTimeout(run_inline_js, 100);\n",
" } else if (!window._bokeh_failed_load) {\n",
" console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
" window._bokeh_failed_load = true;\n",
" } else if (force !== true) {\n",
" var cell = $(document.getElementById(\"845ef23a-3d24-4d6c-a171-8af1f2db5ae0\")).parents('.cell').data().cell;\n",
" cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
" }\n",
" \n",
" }\n",
" \n",
" if (window._bokeh_is_loading === 0) {\n",
" console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
" run_inline_js();\n",
" } else {\n",
" load_libs(js_urls, function() {\n",
" console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
" run_inline_js();\n",
" });\n",
" }\n",
" }(this));\n",
"</script>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"output_notebook()\n",
"\n",
"plot = figure()\n",
"plot.line(x=range(len(loss1)), y=loss1, color='steelblue', legend='gd')\n",
"plot.line(x=range(len(loss2)), y=loss2, color='green', legend='rmsprop')\n",
"plot.line(x=range(len(loss3)), y=loss3, color='red', legend='rmsprop+momentum')\n",
"\n",
"show(plot)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}