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antmarakis
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Knowledge Notebook: Version Space Learning (aimacode#598)
* Update knowledge.ipynb * Update test_knowledge.py
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‎knowledge.ipynb

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@@ -29,7 +29,8 @@
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"## CONTENTS\n",
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"\n",
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"* Overview\n",
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"* Current-Best Learning"
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"* Current-Best Learning\n",
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"* Version-Space Learning"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[{'Species': 'Cat', 'Rain': '!No'}, {'Coat': 'Yes', 'Species': 'Dog', 'Rain': 'Yes'}, {'Coat': 'Yes', 'Species': 'Cat'}]\n"
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"[{'Species': 'Cat', 'Rain': '!No'}, {'Coat': 'Yes', 'Rain': 'Yes'}, {'Coat': 'Yes'}]\n"
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]
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}
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],
@@ -304,6 +305,27 @@
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"![restaurant](images/restaurant.png)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"With the function `r_example` we will build the dictionary examples:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"def r_example(Alt, Bar, Fri, Hun, Pat, Price, Rain, Res, Type, Est, GOAL):\n",
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" return {'Alt': Alt, 'Bar': Bar, 'Fri': Fri, 'Hun': Hun, 'Pat': Pat,\n",
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" 'Price': Price, 'Rain': Rain, 'Res': Res, 'Type': Type, 'Est': Est,\n",
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" 'GOAL': GOAL}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
@@ -315,60 +337,25 @@
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": 7,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"restaurant = [\n",
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" {'Alt': 'Yes', 'Bar': 'No', 'Fri': 'No', 'Hun': 'Yes', 'Pat': 'Some',\n",
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" 'Price': '$$$', 'Rain': 'No', 'Res': 'Yes', 'Type': 'French', 'Est': '0-10',\n",
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" 'GOAL': True},\n",
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"\n",
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" {'Alt': 'Yes', 'Bar': 'No', 'Fri': 'No', 'Hun': 'Yes', 'Pat': 'Full',\n",
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" 'Price': '$', 'Rain': 'No', 'Res': 'No', 'Type': 'Thai', 'Est': '30-60',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'Yes', 'Fri': 'No', 'Hun': 'No', 'Pat': 'Some',\n",
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" 'Price': '$', 'Rain': 'No', 'Res': 'No', 'Type': 'Burger', 'Est': '0-10',\n",
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" 'GOAL': True},\n",
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"\n",
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" {'Alt': 'Yes', 'Bar': 'No', 'Fri': 'Yes', 'Hun': 'Yes', 'Pat': 'Full',\n",
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" 'Price': '$', 'Rain': 'Yes', 'Res': 'No', 'Type': 'Thai', 'Est': '10-30',\n",
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" 'GOAL': True},\n",
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"\n",
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" {'Alt': 'Yes', 'Bar': 'No', 'Fri': 'Yes', 'Hun': 'No', 'Pat': 'Full',\n",
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" 'Price': '$$$', 'Rain': 'No', 'Res': 'Yes', 'Type': 'French', 'Est': '>60',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'Yes', 'Fri': 'No', 'Hun': 'Yes', 'Pat': 'Some',\n",
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" 'Price': '$$', 'Rain': 'Yes', 'Res': 'Yes', 'Type': 'Italian', 'Est': '0-10',\n",
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" 'GOAL': True},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'Yes', 'Fri': 'No', 'Hun': 'No', 'Pat': 'None',\n",
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" 'Price': '$', 'Rain': 'Yes', 'Res': 'No', 'Type': 'Burger', 'Est': '0-10',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'No', 'Fri': 'No', 'Hun': 'Yes', 'Pat': 'Some',\n",
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" 'Price': '$$', 'Rain': 'Yes', 'Res': 'Yes', 'Type': 'Thai', 'Est': '0-10',\n",
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" 'GOAL': True},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'Yes', 'Fri': 'Yes', 'Hun': 'No', 'Pat': 'Full',\n",
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" 'Price': '$', 'Rain': 'Yes', 'Res': 'No', 'Type': 'Burger', 'Est': '>60',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'Yes', 'Bar': 'Yes', 'Fri': 'Yes', 'Hun': 'Yes', 'Pat': 'Full',\n",
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" 'Price': '$$$', 'Rain': 'No', 'Res': 'Yes', 'Type': 'Italian', 'Est': '10-30',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'No', 'Bar': 'No', 'Fri': 'No', 'Hun': 'No', 'Pat': 'None',\n",
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" 'Price': '$', 'Rain': 'No', 'Res': 'No', 'Type': 'Thai', 'Est': '0-10',\n",
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" 'GOAL': False},\n",
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"\n",
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" {'Alt': 'Yes', 'Bar': 'Yes', 'Fri': 'Yes', 'Hun': 'Yes', 'Pat': 'Full',\n",
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" 'Price': '$', 'Rain': 'No', 'Res': 'No', 'Type': 'Burger', 'Est': '30-60',\n",
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" 'GOAL': True}\n",
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" r_example('Yes', 'No', 'No', 'Yes', 'Some', '$$$', 'No', 'Yes', 'French', '0-10', True),\n",
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" r_example('Yes', 'No', 'No', 'Yes', 'Full', '$', 'No', 'No', 'Thai', '30-60', False),\n",
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" r_example('No', 'Yes', 'No', 'No', 'Some', '$', 'No', 'No', 'Burger', '0-10', True),\n",
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" r_example('Yes', 'No', 'Yes', 'Yes', 'Full', '$', 'Yes', 'No', 'Thai', '10-30', True),\n",
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" r_example('Yes', 'No', 'Yes', 'No', 'Full', '$$$', 'No', 'Yes', 'French', '>60', False),\n",
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" r_example('No', 'Yes', 'No', 'Yes', 'Some', '$$', 'Yes', 'Yes', 'Italian', '0-10', True),\n",
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" r_example('No', 'Yes', 'No', 'No', 'None', '$', 'Yes', 'No', 'Burger', '0-10', False),\n",
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" r_example('No', 'No', 'No', 'Yes', 'Some', '$$', 'Yes', 'Yes', 'Thai', '0-10', True),\n",
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" r_example('No', 'Yes', 'Yes', 'No', 'Full', '$', 'Yes', 'No', 'Burger', '>60', False),\n",
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" r_example('Yes', 'Yes', 'Yes', 'Yes', 'Full', '$$$', 'No', 'Yes', 'Italian', '10-30', False),\n",
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" r_example('No', 'No', 'No', 'No', 'None', '$', 'No', 'No', 'Thai', '0-10', False),\n",
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" r_example('Yes', 'Yes', 'Yes', 'Yes', 'Full', '$', 'No', 'No', 'Burger', '30-60', True)\n",
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"]"
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]
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},
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[{'Type': '!Thai', 'Fri': '!Yes', 'Alt': 'Yes'}, {'Fri': 'No', 'Type': 'Burger', 'Pat': '!None', 'Alt': 'No'}, {'Fri': 'Yes', 'Est': '10-30', 'Pat': 'Full', 'Rain': 'Yes', 'Res': 'No', 'Bar': 'No', 'Price': '$'}, {'Fri': 'No', 'Est': '0-10', 'Pat': 'Some', 'Res': 'Yes', 'Type': 'Italian', 'Alt': 'No'}, {'Fri': 'No', 'Pat': 'Some', 'Res': 'Yes', 'Type': 'Thai', 'Hun': 'Yes', 'Alt': 'No', 'Price': '$$'}, {'Fri': 'Yes', 'Pat': 'Full', 'Rain': 'No', 'Alt': 'Yes', 'Type': 'Burger', 'Hun': 'Yes', 'Bar': 'Yes', 'Price': '$'}]\n"
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"[{'Res': '!No', 'Fri': '!Yes', 'Alt': 'Yes'}, {'Bar': 'Yes', 'Fri': 'No', 'Rain': 'No', 'Hun': 'No'}, {'Bar': 'No', 'Price': '$', 'Fri': 'Yes'}, {'Res': 'Yes', 'Price': '$$', 'Rain': 'Yes', 'Alt': 'No', 'Est': '0-10', 'Fri': 'No', 'Hun': 'Yes', 'Bar': 'Yes'}, {'Fri': 'No', 'Pat': 'Some', 'Price': '$$', 'Rain': 'Yes', 'Hun': 'Yes'}, {'Est': '30-60', 'Res': 'No', 'Price': '$', 'Fri': 'Yes', 'Hun': 'Yes'}]\n"
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]
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}
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],
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"source": [
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"It might be quite complicated, with many disjunctions if we are unlucky, but it will always be correct, as long as a correct hypothesis exists."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## [VERSION-SPACE LEARNING](https://github.com/aimacode/aima-pseudocode/blob/master/md/Version-Space-Learning.md)\n",
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"\n",
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"### Overview\n",
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"\n",
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"**Version-Space Learning** is a general method of learning in logic based domains. We generate the set of all the possible hypotheses in the domain and then we iteratively remove hypotheses inconsistent with the examples. The set of remaining hypotheses is called **version space**. Because hypotheses are being removed until we end up with a set of hypotheses consistent with all the examples, the algorithm is sometimes called **candidate elimination** algorithm.\n",
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"\n",
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"After we update the set on an example, all the hypotheses in the set are consistent with that example. So, when all the examples have been parsed, all the remaining hypotheses in the set are consistent with all the examples. That means we can pick hypotheses at random and we will always get a valid hypothesis."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"### Implementation\n",
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"\n",
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"The set of hypotheses is represented by a list and each hypothesis is represented by a list of dictionaries, each dictionary a disjunction. For each example in the given examples we update the version space with the function `version_space_update`. In the end, we return the version-space.\n",
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"\n",
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"Before we can start updating the version space, we need to generate it. We do that with the `all_hypotheses` function, which builds a list of all the possible hypotheses (including hypotheses with disjunctions). The function works like this: first it finds the possible values for each attribute (using `values_table`), then it builds all the attribute combinations (and adds them to the hypotheses set) and finally it builds the combinations of all the disjunctions (which in this case are the hypotheses build by the attribute combinations).\n",
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"\n",
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"You can read the code for all the functions by running the cells below:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource version_space_learning"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource version_space_update"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource all_hypotheses"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource values_table"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource build_attr_combinations"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource build_h_combinations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Example\n",
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"\n",
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"Since the set of all possible hypotheses is enormous and would take a long time to generate, we will come up with another, even smaller domain. We will try and predict whether we will have a party or not given the availability of pizza and soda. Let's do it:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"party = [\n",
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" {'Pizza': 'Yes', 'Soda': 'No', 'GOAL': True},\n",
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" {'Pizza': 'Yes', 'Soda': 'Yes', 'GOAL': True},\n",
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" {'Pizza': 'No', 'Soda': 'No', 'GOAL': False}\n",
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"]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Even though it is obvious that no-pizza no-party, we will run the algorithm and see what other hypotheses are valid."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"True\n",
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"True\n",
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"False\n"
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]
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}
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],
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"source": [
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"V = version_space_learning(party)\n",
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"for e in party:\n",
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" guess = False\n",
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" for h in V:\n",
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" if guess_value(e, h):\n",
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" guess = True\n",
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" break\n",
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"\n",
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" print(guess)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The results are correct for the given examples. Let's take a look at the version space:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"959\n",
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"[{'Pizza': 'Yes'}, {'Soda': 'Yes'}]\n",
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"[{'Pizza': 'Yes'}, {'Pizza': '!No', 'Soda': 'No'}]\n",
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"True\n"
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]
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}
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],
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"source": [
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"print(len(V))\n",
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"\n",
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"print(V[5])\n",
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"print(V[10])\n",
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"\n",
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"print([{'Pizza': 'Yes'}] in V)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"There are almost 1000 hypotheses in the set. You can see that even with just two attributes the version space in very large.\n",
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"\n",
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"Our initial prediction is indeed in the set of hypotheses. Also, the two other random hypotheses we got are consistent with the examples (since they both include the \"Pizza is available\" disjunction)."
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]
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}
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],
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"metadata": {

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