From 4b6467f055d4289c231daeb306677bfefe9959de Mon Sep 17 00:00:00 2001 From: rasbt Date: Sat, 2 Apr 2016 18:45:17 -0400 Subject: [PATCH] softmax regression bonus notebook --- README.md | 1 + code/README.md | 2 + code/bonus/README.md | 2 + .../logistic_regression_schematic_2.png | Bin 0 -> 77694 bytes code/bonus/images/softmax_schematic_1.png | Bin 0 -> 108835 bytes code/bonus/scikit-model-to-json.ipynb | 2 +- code/bonus/softmax-regression.ipynb | 923 ++++++++++++++++++ 7 files changed, 929 insertions(+), 1 deletion(-) create mode 100644 code/bonus/images/logistic_regression_schematic_2.png create mode 100644 code/bonus/images/softmax_schematic_1.png create mode 100644 code/bonus/softmax-regression.ipynb diff --git a/README.md b/README.md index d53b408f..7b149b8b 100644 --- a/README.md +++ b/README.md @@ -90,6 +90,7 @@ Simply click on the `ipynb`/`nbviewer` links next to the chapter headlines to vi - A Simple Barebones Flask Webapp Template [[view directory](./code/bonus/flask_webapp_ex01)][[download as zip-file](https://github.com/rasbt/python-machine-learning-book/raw/master/code/bonus/flask_webapp_ex01/flask_webapp_ex01.zip)] - Reading handwritten digits from MNIST into NumPy arrays [[GitHub ipynb](./code/bonus/reading_mnist.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/reading_mnist.ipynb)] - Scikit-learn Model Persistence using JSON [[GitHub ipynb](./code/bonus/scikit-model-to-json.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/scikit-model-to-json.ipynb)] +- Multinomial logistic regression / softmax regression [[GitHub ipynb](./code/bonus/softmax-regression.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/softmax-regression.ipynb)]
diff --git a/code/README.md b/code/README.md index 86dd8363..636fc72b 100644 --- a/code/README.md +++ b/code/README.md @@ -32,6 +32,8 @@ Simply click on the `ipynb`/`nbviewer` links next to the chapter headlines to vi - An Extended Nested Cross-Validation Example [[dir](./bonus)] [[ipynb](./bonus/nested_cross_validation.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/nested_cross_validation.ipynb)] - A Simple(r) Barebones Flask Webapp Template [[view directory](./bonus/flask_webapp_ex01)][[download as zip-file](https://github.com/rasbt/python-machine-learning-book/raw/master/code/bonus/flask_webapp_ex01/flask_webapp_ex01.zip)] - Reading handwritten digits from MNIST into NumPy arrays [[GitHub ipynb](./bonus/reading_mnist.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/reading_mnist.ipynb)] +- Scikit-learn Model Persistence using JSON [[GitHub ipynb](./bonus/scikit-model-to-json.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/scikit-model-to-json.ipynb)] +- Multinomial logistic regression / softmax regression [[GitHub ipynb](./bonus/softmax-regression.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/softmax-regression.ipynb)] diff --git a/code/bonus/README.md b/code/bonus/README.md index 6cbcb09e..77f73baf 100644 --- a/code/bonus/README.md +++ b/code/bonus/README.md @@ -16,3 +16,5 @@ A collection of additional notebooks and code examples to clarify and explain co - Reading handwritten digits from MNIST into NumPy arrays [[GitHub ipynb](./reading_mnist.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/reading_mnist.ipynb)] - Scikit-learn Model Persistence using JSON [[GitHub ipynb](./scikit-model-to-json.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/scikit-model-to-json.ipynb)] + +- Multinomial logistic regression / softmax regression [[GitHub ipynb](./softmax-regression.ipynb)] [[nbviewer](http://nbviewer.ipython.org/github/rasbt/python-machine-learning-book/blob/master/code/bonus/softmax-regression.ipynb)] diff --git a/code/bonus/images/logistic_regression_schematic_2.png b/code/bonus/images/logistic_regression_schematic_2.png new file mode 100644 index 0000000000000000000000000000000000000000..5d99287df2f33bb486b9683427d335a01cc55265 GIT binary patch literal 77694 zcmeFYRY080wk-;cy9aj*Zo%CG1b26LcL?qpB)F46aCdiicXxN^Hu=|Dd!4gi@8jX2 z>8>vMs-}%OW_6gHj2Hqe4lD==2!e#Thyn-*7&-_Ds1P(Hu*GaR4cLH1X)Y`*Cm}3M zBbkMMMA;j7Tsc-yaa}yFz^Myn6q(-@bpx$#QMw?0h?|J-e#N4N}1r-_;!;KLZjN z_*F~;*UXWdi_>wt4+Pu}1TtM9VFv@F%-Gx;DW9tQ13}RkO)MriteO7V6nbpAMxkX zTxIQ?UszkAyBSZ)wuw?Q9Zc?d@)+w97$K>_pT&$q5%e>Cx}4f#CfwhXI(4&~HD2=QIt! 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z6GDu40u_KJhQQ_((dOr-US2Fm&7R_xe1@RfU2+ti;x@~|?%{RLH_-Y*(fufH@;SOR z+`SCR1j9timzA4--{DeYpJx2b2(=j6GRX0+n$&eNt9jznVRzY-O)c;i3eusRf7zzv zJpVU@{Rb5Rz5f6aPHcu8D8l(SI>LeYo=;*Eq?Z5v4vh>zO1)^;%sSHrot`={^j5q1 zhMrZuIa?lF2pz50?5ihubJ;Cv_J~#C$Lj!ICQ+b~gi&KnWllzrsEKGa~n=WrWfe)L4I1{oHjU>r^CkH$pKucQ;` zhw`;=joi`er)kn6!Ie4R(rQ)2h5P%Z0gOO3Z6GYCpp5|!OdKDYH70sE1Tak)``n*w z5&P}nRqH~}{_$>4ng{@g@}+>SDvWt02iAvLy$q~qvRkbd?aZ`OroF9l&h?Qqw@KAN zYwvsz7mYq>=XZkzA%Iwlp6t6``>b#5w2t&!4x(*uJ$S 2 \n", + "[ 0.21290077 0.32728332 **0.45981591**] -> 2 \n", + "[ **0.42860913** 0.33380113 0.23758974] -> 0 \n", + "[ **0.44941979** 0.32962558 0.22095463]] -> 0 " + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "predicted class labels: [2 2 0 0]\n" + ] + } + ], + "source": [ + "def to_classlabel(z):\n", + " return z.argmax(axis=1)\n", + "\n", + "print('predicted class labels: ', to_classlabel(smax))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we can see, our predictions are terribly wrong, since the correct class labels are `[0, 1, 2, 2]`. Now, in order to train our logistic model (e.g., via an optimization algorithm such as gradient descent), we need to define a cost function $J(\\cdot)$ that we want to minimize:\n", + "\n", + "$$J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum_{i=1}^{n} H(T_i, O_i),$$\n", + "\n", + "which is the average of all cross-entropies over our $n$ training samples. The cross-entropy function is defined as\n", + "\n", + "$$H(T_i, O_i) = -\\sum_m T_i \\cdot log(O_i).$$\n", + "\n", + "Here the $T$ stands for \"target\" (i.e., the *true* class labels) and the $O$ stands for output -- the computed *probability* via softmax; **not** the predicted class label." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Cross Entropy: [ 1.22245465 1.11692907 1.43720989 1.50979788]\n" + ] + } + ], + "source": [ + "def cross_entropy(output, y_target):\n", + " return - np.sum(np.log(output) * (y_target), axis=1)\n", + "\n", + "xent = cross_entropy(smax, y_enc)\n", + "print('Cross Entropy:', xent)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Cost: 1.32159787159\n" + ] + } + ], + "source": [ + "def cost(output, y_target):\n", + " return np.mean(cross_entropy(output, y_target))\n", + "\n", + "J_cost = cost(smax, y_enc)\n", + "print('Cost: ', J_cost)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In order to learn our softmax model -- determining the weight coefficients -- via gradient descent, we then need to compute the derivative \n", + "\n", + "$$\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b}).$$\n", + "\n", + "I don't want to walk through the tedious details here, but this cost derivative turns out to be simply:\n", + "\n", + "$$\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum^{n}_{i=0} \\big[\\mathbf{x}^{(i)}\\ \\big(O_i - T_i \\big) \\big]$$\n", + "\n", + "We can then use the cost derivate to update the weights in opposite direction of the cost gradient with learning rate $\\eta$:\n", + "\n", + "$$\\mathbf{w}_j := \\mathbf{w}_j - \\eta \\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b})$$ \n", + "\n", + "for each class $$j \\in \\{0, 1, ..., k\\}$$\n", + "\n", + "(note that $\\mathbf{w}_j$ is the weight vector for the class $y=j$), and we update the bias units\n", + "\n", + "\n", + "$$\\mathbf{b}_j := \\mathbf{b}_j - \\eta \\bigg[ \\frac{1}{n} \\sum^{n}_{i=0} \\big(O_i - T_i \\big) \\bigg].$$ \n", + " \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As a penalty against complexity, an approach to reduce the variance of our model and decrease the degree of overfitting by adding additional bias, we can further add a regularization term such as the L2 term with the regularization parameter $\\lambda$:\n", + " \n", + "L2: $\\frac{\\lambda}{2} ||\\mathbf{w}||_{2}^{2}$, \n", + "\n", + "where \n", + "\n", + "$$||\\mathbf{w}||_{2}^{2} = \\sum^{m}_{l=0} \\sum^{k}_{j=0} w_{i, j}$$\n", + "\n", + "so that our cost function becomes\n", + "\n", + "$$J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum_{i=1}^{n} H(T_i, O_i) + \\frac{\\lambda}{2} ||\\mathbf{w}||_{2}^{2}$$\n", + "\n", + "and we define the \"regularized\" weight update as\n", + "\n", + "$$\\mathbf{w}_j := \\mathbf{w}_j - \\eta \\big[\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}) + \\lambda \\mathbf{w}_j \\big].$$\n", + "\n", + "(Please note that we don't regularize the bias term.)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# SoftmaxRegression Code" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Bringing the concepts together, we could come up with an implementation as follows:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "# Sebastian Raschka 2016\n", + "# Implementation of the mulitnomial logistic regression algorithm for\n", + "# classification.\n", + "\n", + "# Author: Sebastian Raschka \n", + "#\n", + "# License: BSD 3 clause\n", + "\n", + "import numpy as np\n", + "\n", + "\n", + "class SoftmaxRegression(object):\n", + "\n", + " \"\"\"Softmax regression classifier.\n", + "\n", + " Parameters\n", + " ------------\n", + " eta : float (default: 0.01)\n", + " Learning rate (between 0.0 and 1.0)\n", + " epochs : int (default: 50)\n", + " Passes over the training dataset.\n", + " l2_lambda : float\n", + " Regularization parameter for L2 regularization.\n", + " No regularization if l2_lambda=0.0.\n", + " minibatches : int (default: 1)\n", + " Divide the training data into *k* minibatches\n", + " for accelerated stochastic gradient descent learning.\n", + " Gradient Descent Learning if `minibatches` = 1\n", + " Stochastic Gradient Descent learning if `minibatches` = len(y)\n", + " Minibatch learning if `minibatches` > 1\n", + " random_seed : int (default: None)\n", + " Set random state for shuffling and initializing the weights.\n", + " zero_init_weight : bool (default: False)\n", + " If True, weights are initialized to zero instead of small random\n", + " numbers following a standard normal distribution with mean=0 and\n", + " stddev=1.\n", + "\n", + " Attributes\n", + " -----------\n", + " w_ : 2d-array, shape=[n_features, n_classes]\n", + " Weights after fitting.\n", + " cost_ : list\n", + " List of floats, the average cross_entropy for each epoch.\n", + "\n", + " \"\"\"\n", + " def __init__(self, eta=0.01, epochs=50,\n", + " l2_lambda=0.0, minibatches=1,\n", + " random_seed=None,\n", + " zero_init_weight=False,\n", + " print_progress=0):\n", + "\n", + " self.random_seed = random_seed\n", + " self.eta = eta\n", + " self.epochs = epochs\n", + " self.l2_lambda = l2_lambda\n", + " self.minibatches = minibatches\n", + " self.zero_init_weight = zero_init_weight\n", + "\n", + " def _one_hot(self, y, n_labels):\n", + " mat = np.zeros((len(y), n_labels))\n", + " for i, val in enumerate(y):\n", + " mat[i, val] = 1\n", + " return mat.astype(float)\n", + "\n", + " def _net_input(self, X, W, b):\n", + " return (X.dot(W) + b)\n", + "\n", + " def _softmax(self, z):\n", + " return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T\n", + "\n", + " def _cross_entropy(self, output, y_target):\n", + " return - np.sum(np.log(output) * (y_target), axis=1)\n", + "\n", + " def _cost(self, cross_entropy):\n", + " return np.mean(cross_entropy)\n", + "\n", + " def _to_classlabels(self, z):\n", + " return z.argmax(axis=1)\n", + "\n", + " def fit(self, X, y, init_weights=True, n_classes=None):\n", + " \"\"\"Learn weight coefficients from training data.\n", + "\n", + " Parameters\n", + " ----------\n", + " X : {array-like, sparse matrix}, shape = [n_samples, n_features]\n", + " Training vectors, where n_samples is the number of samples and\n", + " n_features is the number of features.\n", + " y : array-like, shape = [n_samples]\n", + " Target values.\n", + " init_weights : bool (default: True)\n", + " (Re)initializes weights to small random floats if True.\n", + " n_classes : int (default: None)\n", + " A positive integer to declare the number of class labels\n", + " if not all class labels are present in a partial training set.\n", + " Gets the number of class labels automatically if None.\n", + " Ignored if init_weights=False.\n", + "\n", + " Returns\n", + " -------\n", + " self : object\n", + "\n", + " \"\"\"\n", + " if init_weights:\n", + " if n_classes:\n", + " self._n_classes = n_classes\n", + " else:\n", + " self._n_classes = np.max(y) + 1\n", + " self._n_features = X.shape[1]\n", + " self.w_ = self._init_weights(\n", + " shape=(self._n_features, self._n_classes),\n", + " zero_init_weight=self.zero_init_weight,\n", + " seed=self.random_seed)\n", + " self.b_ = self._init_weights(\n", + " shape=self._n_classes,\n", + " zero_init_weight=self.zero_init_weight,\n", + " seed=self.random_seed)\n", + " self.cost_ = []\n", + "\n", + " n_idx = list(range(y.shape[0]))\n", + " y_enc = self._one_hot(y, self._n_classes)\n", + "\n", + " # random seed for shuffling\n", + " if self.random_seed:\n", + " np.random.seed(self.random_seed)\n", + "\n", + " for i in range(self.epochs):\n", + " if self.minibatches > 1:\n", + " n_idx = np.random.permutation(n_idx)\n", + "\n", + " minis = np.array_split(n_idx, self.minibatches)\n", + " for idx in minis:\n", + "\n", + " # givens:\n", + " # w_ -> n_feat x n_classes\n", + " # b_ -> n_classes\n", + "\n", + " # net_input, softmax and diff -> n_samples x n_classes:\n", + " net = self._net_input(X[idx], self.w_, self.b_)\n", + " softm = self._softmax(net)\n", + " diff = softm - y_enc[idx]\n", + "\n", + " # gradient -> n_features x n_classes\n", + " grad = np.dot(X[idx].T, diff)\n", + "\n", + " # update in opp. direction of the cost gradient\n", + " self.w_ -= (self.eta * grad +\n", + " self.eta * self.l2_lambda * self.w_)\n", + " self.b_ -= np.mean(diff, axis=0)\n", + "\n", + " # compute cost of the whole epoch\n", + " net = self._net_input(X, self.w_, self.b_)\n", + " softm = self._softmax(net)\n", + " cross_ent = self._cross_entropy(output=softm, y_target=y_enc)\n", + " cost = self._cost(cross_ent)\n", + " self.cost_.append(cost)\n", + "\n", + " return self\n", + "\n", + " def predict_proba(self, X):\n", + " \"\"\"Predict class probabilities of X from the net input.\n", + "\n", + " Parameters\n", + " ----------\n", + " X : {array-like, sparse matrix}, shape = [n_samples, n_features]\n", + " Training vectors, where n_samples is the number of samples and\n", + " n_features is the number of features.\n", + "\n", + " Returns\n", + " ----------\n", + " Class probabilties : array-like, shape= [n_samples, n_classes]\n", + "\n", + " \"\"\"\n", + " net = self._net_input(X, self.w_, self.b_)\n", + " softm = self._softmax(net)\n", + " return softm\n", + "\n", + " def predict(self, X):\n", + " \"\"\"Predict class labels of X.\n", + "\n", + " Parameters\n", + " ----------\n", + " X : {array-like, sparse matrix}, shape = [n_samples, n_features]\n", + " Training vectors, where n_samples is the number of samples and\n", + " n_features is the number of features.\n", + "\n", + " Returns\n", + " ----------\n", + " class_labels : array-like, shape = [n_samples]\n", + " Predicted class labels.\n", + "\n", + " \"\"\"\n", + " probas = self.predict_proba(X)\n", + " return self._to_classlabels(probas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1 - Gradient Descent" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from mlxtend.data import iris_data\n", + "from mlxtend.evaluate import plot_decision_regions\n", + "from mlxtend.classifier import SoftmaxRegression\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Loading Data\n", + "\n", + "X, y = iris_data()\n", + "X = X[:, [0, 3]] # sepal length and petal width\n", + "\n", + "# standardize\n", + "X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()\n", + "X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()\n", + "\n", + "lr = SoftmaxRegression(eta=0.005, epochs=200, minibatches=1, random_seed=1)\n", + "lr.fit(X, y)\n", + "\n", + "plot_decision_regions(X, y, clf=lr)\n", + "plt.title('Softmax Regression - Stochastic Gradient Descent')\n", + "plt.show()\n", + "\n", + "plt.plot(range(len(lr.cost_)), lr.cost_)\n", + "plt.xlabel('Iterations')\n", + "plt.ylabel('Cost')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "### Predicting Class Labels" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Last 3 Class Labels: [2 2 2]\n" + ] + } + ], + "source": [ + "y_pred = lr.predict(X)\n", + "print('Last 3 Class Labels: %s' % y_pred[-3:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Predicting Class Probabilities" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Last 3 Class Labels:\n", + " [[ 4.99921674e-06 7.23245885e-02 9.27670412e-01]\n", + " [ 2.50487208e-07 1.20047952e-02 9.87994954e-01]\n", + " [ 2.14388120e-04 2.95955727e-01 7.03829884e-01]]\n" + ] + } + ], + "source": [ + "y_pred = lr.predict_proba(X)\n", + "print('Last 3 Class Labels:\\n %s' % y_pred[-3:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2 - Stochastic Gradient Descent" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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fwIH5FW4KIxJVVcCIEU3/vEzCTQsXts6leklmUCRylJ/8BLjySuCGG2zq5kSR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\nFMaOtUayQwdzH089ZT8dLtz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fOnZpJHk0GGoAAAAASUVORK5CYII=\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from mlxtend.data import iris_data\n", + "from mlxtend.evaluate import plot_decision_regions\n", + "from mlxtend.classifier import SoftmaxRegression\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Loading Data\n", + "\n", + "X, y = iris_data()\n", + "X = X[:, [0, 3]] # sepal length and petal width\n", + "\n", + "# standardize\n", + "X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()\n", + "X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()\n", + "\n", + "lr = SoftmaxRegression(eta=0.005, epochs=200, minibatches=len(y), random_seed=1)\n", + "lr.fit(X, y)\n", + "\n", + "plot_decision_regions(X, y, clf=lr)\n", + "plt.title('Softmax Regression - Stochastic Gradient Descent')\n", + "plt.show()\n", + "\n", + "plt.plot(range(len(lr.cost_)), lr.cost_)\n", + "plt.xlabel('Iterations')\n", + "plt.ylabel('Cost')\n", + "plt.show()" + ] + } + ], + "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": "3.5.1" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}