diff --git a/learning.ipynb b/learning.ipynb index 16bb4bd6b..0e4d97934 100644 --- a/learning.ipynb +++ b/learning.ipynb @@ -11,7 +11,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -1778,6 +1778,275 @@ "source": [ "The Perceptron didn't fare very well mainly because the dataset is not linearly separated. On simpler datasets the algorithm performs much better, but unfortunately such datasets are rare in real life scenarios." ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## AdaBoost\n", + "\n", + "### Overview\n", + "\n", + "**AdaBoost** is an algorithm which uses **ensemble learning**. In ensemble learning the hypotheses in the collection, or ensemble, vote for what the output should be and the output with the majority votes is selected as the final answer.\n", + "\n", + "AdaBoost algorithm, as mentioned in the book, works with a **weighted training set** and **weak learners** (classifiers that have about 50%+epsilon accuracy i.e slightly better than random guessing). It manipulates the weights attached to the the examples that are showed to it. Importance is given to the examples with higher weights.\n", + "\n", + "All the examples start with equal weights and a hypothesis is generated using these examples. Examples which are incorrectly classified, their weights are increased so that they can be classified correctly by the next hypothesis. The examples that are correctly classified, their weights are reduced. This process is repeated *K* times (here *K* is an input to the algorithm) and hence, *K* hypotheses are generated.\n", + "\n", + "These *K* hypotheses are also assigned weights according to their performance on the weighted training set. The final ensemble hypothesis is the weighted-majority combination of these *K* hypotheses.\n", + "\n", + "The speciality of AdaBoost is that by using weak learners and a sufficiently large *K*, a highly accurate classifier can be learned irrespective of the complexity of the function being learned or the dullness of the hypothesis space." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Implementation\n", + "\n", + "As seen in the previous section, the `PerceptronLearner` does not perform that well on the iris dataset. We'll use perceptron as the learner for the AdaBoost algorithm and try to increase the accuracy. \n", + "\n", + "Let's first see what AdaBoost is exactly:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def AdaBoost(L, K):\n",
+       "    """[Figure 18.34]"""\n",
+       "    def train(dataset):\n",
+       "        examples, target = dataset.examples, dataset.target\n",
+       "        N = len(examples)\n",
+       "        epsilon = 1. / (2 * N)\n",
+       "        w = [1. / N] * N\n",
+       "        h, z = [], []\n",
+       "        for k in range(K):\n",
+       "            h_k = L(dataset, w)\n",
+       "            h.append(h_k)\n",
+       "            error = sum(weight for example, weight in zip(examples, w)\n",
+       "                        if example[target] != h_k(example))\n",
+       "            # Avoid divide-by-0 from either 0% or 100% error rates:\n",
+       "            error = clip(error, epsilon, 1 - epsilon)\n",
+       "            for j, example in enumerate(examples):\n",
+       "                if example[target] == h_k(example):\n",
+       "                    w[j] *= error / (1. - error)\n",
+       "            w = normalize(w)\n",
+       "            z.append(math.log((1. - error) / error))\n",
+       "        return WeightedMajority(h, z)\n",
+       "    return train\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(AdaBoost)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "AdaBoost takes as inputs: **L** and *K* where **L** is the learner and *K* is the number of hypotheses to be generated. The learner **L** takes in as inputs: a dataset and the weights associated with the examples in the dataset. But the `PerceptronLearner` doesnot handle weights and only takes a dataset as its input. \n", + "To remedy that we will give as input to the PerceptronLearner a modified dataset in which the examples will be repeated according to the weights associated to them. Intuitively, what this will do is force the learner to repeatedly learn the same example again and again until it can classify it correctly. \n", + "\n", + "To convert `PerceptronLearner` so that it can take weights as input too, we will have to pass it through the **`WeightedLearner`** function." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "psource(WeightedLearner)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The `WeightedLearner` function will then call the `PerceptronLearner`, during each iteration, with the modified dataset which contains the examples according to the weights associated with them." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example\n", + "\n", + "We will pass the `PerceptronLearner` through `WeightedLearner` function. Then we will create an `AdaboostLearner` classifier with number of hypotheses or *K* equal to 5." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "WeightedPerceptron = WeightedLearner(PerceptronLearner)\n", + "AdaboostLearner = AdaBoost(WeightedPerceptron, 5)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "iris2 = DataSet(name=\"iris\")\n", + "iris2.classes_to_numbers()\n", + "\n", + "adaboost = AdaboostLearner(iris2)\n", + "\n", + "adaboost([5, 3, 1, 0.1])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "That is the correct answer. Let's check the error rate of adaboost with perceptron." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Error ratio for adaboost: 0.046666666666666634\n" + ] + } + ], + "source": [ + "print(\"Error ratio for adaboost: \", err_ratio(adaboost, iris2))" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "It reduced the error rate considerably. Unlike the `PerceptronLearner`, `AdaBoost` was able to learn the complexity in the iris dataset." + ] } ], "metadata": { diff --git a/tests/test_learning.py b/tests/test_learning.py index 8a21d6462..3c6d02d28 100644 --- a/tests/test_learning.py +++ b/tests/test_learning.py @@ -218,3 +218,19 @@ def test_random_weights(): assert len(test_weights) == num_weights for weight in test_weights: assert weight >= min_value and weight <= max_value + + +def test_adaboost(): + iris = DataSet(name="iris") + iris.classes_to_numbers() + WeightedPerceptron = WeightedLearner(PerceptronLearner) + AdaboostLearner = AdaBoost(WeightedPerceptron, 5) + adaboost = AdaboostLearner(iris) + tests = [([5, 3, 1, 0.1], 0), + ([5, 3.5, 1, 0], 0), + ([6, 3, 4, 1.1], 1), + ([6, 2, 3.5, 1], 1), + ([7.5, 4, 6, 2], 2), + ([7, 3, 6, 2.5], 2)] + assert grade_learner(adaboost, tests) > 5/6 + assert err_ratio(adaboost, iris) < 0.1