{ "metadata": { "name": "4.3_Logistic_Regression" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": "# Logistic and Softmax Regression (classification) #\n\nLogistic Regression is an algorithm for classification (blame the stat guys for that misnomer). Neural networks are built from these (shallow ones, at least). I'm going to do everything in the multiclass case. 4.3.3 (IRLS) is not covered in this notebook.\n\nLogistic regression is discriminative, and the loss function (*cross entropy*) is given by:\n\n$-\\sum_{n=1}^{N}{\\sum_{k=1}^{K}{t_{nk} ln(y_{nk})}}$\n\n\nWhere $N$ is the number of training examples, $K$ is the number of classes, $t_{nk}$ is 1 if training example $n$ is class $k$ and zero otherwise (a one-hot representation), and $y_{nk}$ is the prediction of our model of the probability that training example $n$ is class $k$.\n\nOur model predicts a conditional probability $p(t=k|x;W)$. As we saw with linear basis function models, we can easily work with linear combinations of nonlinear functions of our input. In other words, we first apply a nonlinear transformation to our data and then learn a linear decision boundary. \n\nBecause it's important, I'm going to add a regularization penalty of the form\n\n$\\lambda tr(WW^{T})$, where $W$ is a *matrix* of weights and $tr$ is the trace function. In other words, we penalize the size of each weight vector for the $K$ classes. The dimensions of our weight matrix, $W$, will be $(K,D)$, where $D$ is the dimensionality of our input after being mapped to the new space. Thus $p(t=k|x;W) = W\\phi(x)$, where $\\phi$ is the mapping (basis functions). \n\nEnough, let's see some code." }, { "cell_type": "code", "collapsed": false, "input": "#imports\n%pylab inline\nfrom sklearn.datasets import make_classification, load_iris\nfrom sklearn.preprocessing import LabelBinarizer\nfrom sklearn.linear_model import LogisticRegression #for comparison\nfrom sklearn.cross_validation import train_test_split", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "Populating the interactive namespace from numpy and matplotlib\n" } ], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": "class LogisticClassifier(object):\n \"\"\"\n Multiclass logistic regression with regularization. Trained with gradient descent + momentum (if desired). \n \"\"\"\n \n def __init__(self, basis=None):\n \"\"\"\n Instantiate a logistic regression model. Specify a custom basis function here,\n it should accept an array and output a (preferably higher dimensional) array.\n The default is the identity function plus a bias term.\n \"\"\"\n self.W = array([])\n self.A = None #the mixing matrix for basis mapping.\n self.basis=basis\n if basis == 'poly':\n self.basisfunc = self.poly_basis\n elif basis == 'rbf':\n self.basisfunc = self.rbf_basis\n elif basis == 'sigmoid':\n self.basisfunc = self.sigmoid_basis\n elif basis == 'rectifier':\n self.basisfunc = self.rectifier_basis\n else:\n self.basisfunc = self.identity\n \n \n def identity(self, x):\n #identity basis function + a bias\n return hstack((x,1))\n \n def poly_basis(self, x):\n #polynomial basis\n degree = 2\n #first mix the components of x in a higher dimension\n xn = dot(self.A,x)\n return self.identity(hstack(tuple(sum(xn**i for i in range(degree)))))\n \n def rbf_basis(self, x):\n #in this case, use the mixing matrix as centroids.\n return self.identity(hstack(tuple(exp(-norm(x-mu)) for mu in self.A)))\n \n def sigmoid_basis(self, x):\n #just like a neural network layer.\n xn = dot(self.A, x)\n return self.identity((1+exp(-xn))**-1)\n \n def rectifier_basis(self, x):\n #used in the latest neural nets\n xn = dot(self.A, x)\n return self.identity(maximum(xn, 0))\n \n def basismap(self, X):\n #if X is an observation matrix (examples by dimensions),\n #return each row mapped to a higher dimsional space\n new_dimensions = self.basisfunc(X[0,:]).shape[0]\n Xn = zeros((X.shape[0], new_dimensions))\n for i,xi in enumerate(X):\n Xn[i,:] = self.basisfunc(xi)\n return Xn\n \n def fit(self, X, Y, itrs=100, learn_rate=0.1, reg=0.1,\n momentum=0.5, report_cost=False, proj_layer_size=10):\n \"\"\"\n Fit the model. \n X - observation matrix (observations by dimensions)\n Y - one-hot target matrix (examples by classes)\n itrs - number of iterations to run\n learn_rate - size of step to use for gradient descent\n reg - regularization penalty (lambda above)\n momentum - weight of the previous gradient in the update step\n report_cost - if true, return the loss function at each step (expensive).\n proj_layer_size - number of dimensions in the projection (mixing) layer. Higher -> more variance\n \"\"\"\n \n #first map to a new basis\n if self.basis != 'rbf':\n self.A = uniform(-1, 1, (proj_layer_size, X.shape[1]))\n else:\n #use the training examples as bases\n self.A = X[permutation(X.shape[0])[:proj_layer_size],:]\n Xn = self.basismap(X)\n \n #set up weights\n self.W = uniform(-0.1, 0.1, (Y.shape[1], Xn.shape[1]))\n \n #optimize\n costs = []\n previous_grad = zeros(self.W.shape) #used in momentum\n for i in range(itrs):\n grad = self.grad(Xn, Y, reg) #compute gradient\n self.W = self.W - learn_rate*(grad + momentum*previous_grad) #take a step, use previous gradient as well.\n previous_grad = grad\n \n if report_cost:\n costs.append(self.loss(X,Y,reg))\n \n return costs\n \n def softmax(self, Z):\n #returns sigmoid elementwise\n Z = maximum(Z, -1e3)\n Z = minimum(Z, 1e3)\n numerator = exp(Z)\n return numerator / sum(numerator, axis=1).reshape((-1,1))\n \n def predict(self, X):\n \"\"\"\n If the model has been trained, makes predictions on an observation matrix (observations by features)\n \"\"\"\n Xn = self.basismap(X)\n return self.softmax(dot(Xn, self.W.T))\n \n def grad(self, Xn, Y, reg):\n \"\"\"\n Returns the gradient of the loss function wrt the weights. \n \"\"\"\n #Xn should be the design matrix\n Yh = self.softmax(dot(Xn, self.W.T))\n return -dot(Y.T-Yh.T,Xn)/Xn.shape[0] + reg*self.W\n \n def loss(self, X, Y, reg):\n #assuming X is the data matrix\n Yh = self.predict(X)\n return -mean(mean(Y*log(Yh))) - reg*trace(dot(self.W,self.W.T))/self.W.shape[0]\n ", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": "I use this plotting code a lot. It required a little modification" }, { "cell_type": "code", "collapsed": false, "input": "#nice plotting code from before\ndef plot_contour_scatter(X, Y, model, title_text, binarizer):\n #sample from a lattice (for the nice visualization)\n x1, x2 = meshgrid(arange(-5,5,0.1), arange(-5,5,0.1))\n Xnew = vstack((x1.ravel(), x2.ravel())).T\n \n Z = model.predict(Xnew).reshape((Xnew.shape[0],-1))\n Zc = binarizer.inverse_transform(Z)\n y = binarizer.inverse_transform(Y)\n\n #plot - contour plot and scatter superimposed\n contourf(arange(-5,5,0.1), arange(-5,5,0.1), Zc.reshape(x1.shape),\n cmap ='Paired',levels=arange(0,4,0.1))\n c_dict = {0:'b', 1:'g', 2:'r', 3:'y', 4:'m', 5:'k', 6:'c'}\n colorsToUse= [c_dict[yi] for yi in y]\n scatter(X[:,0], X[:,1], c=colorsToUse)\n title(title_text)\n show()", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": "Let's try some toy examples. First we need to get some data." }, { "cell_type": "code", "collapsed": false, "input": "# #make data\nX,Y = make_classification(n_features=2, n_informative=2, n_redundant=0,\n n_repeated=0, n_classes=3, n_clusters_per_class=1)\n\n#make Y into a one-hot matrix\nlb = LabelBinarizer()\nY = lb.fit_transform(Y)", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": "Now we can fit our model. Let's try with no basis functions first." }, { "cell_type": "code", "collapsed": false, "input": "#fit\nmodel = LogisticClassifier()\ncosts = model.fit(X,Y,report_cost=True, momentum=0.9, learn_rate=0.1,itrs=500,reg=0.01)\nplot(range(len(costs)), costs)\ntitle('loss over time')\nshow()\nplot_contour_scatter(X,Y, model, 'Logistic Regression, linear basis', lb)", "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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mMTyzYBFbVizlw+fexFJ2rP1lQtM80R12ROMmwS0aDK9ADxwWHVoDLYCVztn+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\nYsaKPaxPz8duMNBPg1uPbjt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"text": "" } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": "How often is it wrong?\n" }, { "cell_type": "code", "collapsed": false, "input": "def error_rate(Yh, Y):\n return sum( not_equal(argmax(Yh,axis=1), argmax(Y,axis=1))) / float(Yh.shape[0]) \n\nprint error_rate(model.predict(X), Y)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "0.14\n" } ], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": "Ok. How about a polynomial basis?" }, { "cell_type": "code", "collapsed": false, "input": "#fit\nmodel = LogisticClassifier(basis='poly')\ncosts = model.fit(X,Y,report_cost=True, momentum=0.9, learn_rate=0.1,itrs=100,reg=0.01)\nplot(range(len(costs)), costs)\ntitle('loss over time')\nshow()\nplot_contour_scatter(X,Y, model, 'Logistic Regression, polynomial basis', lb)", "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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t/PXObrpNvfWnCE4/m8TqRZ9SmFdIi25daHt/31I1YJOhhF3rf0a1ZgBuWC0P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\nlvk/GBwxoEt1760AAAAASUVORK5CYII=\n", "text": "" } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": "Not bad - if we zoomed out it would be clear that the decision boundary isn't linear anymore. Let's use a sigmoid basis." }, { "cell_type": "code", "collapsed": false, "input": "#fit\nmodel = LogisticClassifier(basis='sigmoid')\ncosts = model.fit(X,Y,report_cost=True, momentum=0.9, learn_rate=0.2,itrs=400,reg=1e-3)\nplot(range(len(costs)), costs)\ntitle('loss over time')\nshow()\nplot_contour_scatter(X,Y, model, 'Logistic Regression, sigmoid basis', lb)", "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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m4IHNeh/FBTEc3rERvcFAfnYmUdEtCalrL9O+TesRWib2CTBhQC4HtzYgLTGe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\nZuw6zh8nMvB3ceKd9o1o7CtblGc5Rds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"text": "" } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": "Looks nice. The sigmoid basis is important because it's the same thing that each layer of a neural network does (ignoring recent results in deep learning). Each layer in a neural net (Bishop ch5) performs a matrix multiply and then an elementwise sigmoid. In neural nets, however, we use the chain rule to figure out the gradient of the error w.r.t. the layer 1 weight matrix. In this example, it's the same thing but the first layer is static. \n\nWe can also simulate a rbf network by using a rbf basis." }, { "cell_type": "code", "collapsed": false, "input": "#fit\nmodel = LogisticClassifier(basis='rbf')\ncosts = model.fit(X,Y,report_cost=True, momentum=0.9, learn_rate=0.15,itrs=300,reg=0.01)\nplot(range(len(costs)), costs)\ntitle('loss over time')\nshow()\nplot_contour_scatter(X,Y, model, 'Logistic Regression, RBF basis', lb)", "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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WOvDq0G5kcw6uOIHZJFu66f+los0bwdV2/IZk9/9WkXolCJPhB0BCkpbx7eKF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\nHiGobefbOlbqlVi+ff9dTMb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wgqywFigGvjHL7EnKId9gqvDY/xnaiY6dA3mlaRMW2qgp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\nK1aswOVyUVdXF448IiLShZCK2+/3U1xczJgxY8K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"text": "" } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": "Highly nonlinear decision boundaries here. This classifier is basically the same as a RBF network. \n\nModern neural nets (deep ones, that is) use an activation function called a rectifier. " }, { "cell_type": "code", "collapsed": false, "input": "#fit\nmodel = LogisticClassifier(basis='rectifier')\ncosts = model.fit(X,Y,report_cost=True, momentum=0.9, learn_rate=0.2,itrs=100,reg=0.01)\nplot(range(len(costs)), costs)\ntitle('loss over time')\nshow()\nplot_contour_scatter(X,Y, model, 'Logistic Regression, linear rectifier basis', lb)", "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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eiqcs34yDqxGD4z/d/4QuOLo6hZyjeXiFehA1rDmnDueT+ncmju6ORA4Jw9Gt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\nMRS4HdgNmIFEA6wug++x9SQpX5JhPDD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"text": "" } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": "## Real Data ##\n\nSklearn includes the iris dataset for use in validating algorithms. Let's give it a shot." }, { "cell_type": "code", "collapsed": false, "input": "iris_data = load_iris()\nX = iris_data.data\nY = iris_data.target\nY = lb.fit_transform(Y)\nXtrain, Xtest, Ytrain, Ytest = train_test_split(X,Y,test_size=0.25)\n#ugh, sklearn doesn't take one-hot representations\nYtrain_sk = lb.inverse_transform(Ytrain)\nYtest_sk = lb.inverse_transform(Ytest)", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": "Let's compare our implementation with the implementation already in sklearn" }, { "cell_type": "code", "collapsed": false, "input": "sklearn_lr = LogisticRegression()\nlinear_lr = LogisticClassifier()\nlinear_lr.fit(Xtrain,Ytrain, momentum=0.5, learn_rate=0.1,itrs=1000,reg=0.01)\nsklearn_lr.fit(Xtrain, Ytrain_sk)\n\n#performance\nprint 'ours:', error_rate(linear_lr.predict(Xtest), Ytest)\nprint 'sklearn:', error_rate(sklearn_lr.predict_proba(Xtest), Ytest)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "ours: 0.0526315789474\nsklearn: 0.105263157895\n" } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": "The difference isn't significant, the data set is too small. Let's use the make_classification utility to run some more tests." }, { "cell_type": "code", "collapsed": false, "input": "X,Y = make_classification(n_samples=1000, n_features=20, n_informative=10,\n n_redundant=10, n_classes=5, n_clusters_per_class=1)\nY = lb.fit_transform(Y)\nXtrain, Xtest, Ytrain, Ytest = train_test_split(X,Y,test_size=0.25)\n#ugh, sklearn doesn't take one-hot representations\nYtrain_sk = lb.inverse_transform(Ytrain)\n", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": "And now we can compare them." }, { "cell_type": "code", "collapsed": false, "input": "sklearn_lr = LogisticRegression()\nlinear_lr = LogisticClassifier()\nlinear_lr.fit(Xtrain, Ytrain, momentum=0.5, learn_rate=0.1,\n itrs=1000, reg=0.01)\nsklearn_lr.fit(Xtrain, Ytrain_sk)\n\nprint 'ours:', error_rate(linear_lr.predict(Xtest), Ytest)\nprint 'sklearn:', error_rate(sklearn_lr.predict_proba(Xtest), Ytest)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "ours: 0.284\nsklearn: 0.296\n" } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": "We can also see how our different basis functions perform" }, { "cell_type": "code", "collapsed": false, "input": "#fit a sigmoid basis logistic regressor\nsigmoid_lr = LogisticClassifier(basis='sigmoid')\nsigmoid_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a polynomial basis logistic regressor\npoly_lr = LogisticClassifier(basis='poly')\npoly_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rectifier basis logistic regressor\nrect_lr = LogisticClassifier(basis='rectifier')\nrect_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rbf basis logistic regressor\nrbf_lr = LogisticClassifier(basis='rbf')\nrbf_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\nprint 'ours (linear):', error_rate(linear_lr.predict(Xtest), Ytest)\nprint 'ours (sigmoid):', error_rate(sigmoid_lr.predict(Xtest), Ytest)\nprint 'ours (polynomial):', error_rate(poly_lr.predict(Xtest), Ytest)\nprint 'ours (rectifier):', error_rate(rect_lr.predict(Xtest), Ytest)\nprint 'ours (rbf):', error_rate(rbf_lr.predict(Xtest), Ytest)\nprint 'sklearn:', error_rate(sklearn_lr.predict_proba(Xtest), Ytest)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "ours (linear): 0.284\nours (sigmoid): 0.256\nours (polynomial): " }, { "output_type": "stream", "stream": "stdout", "text": "0.396\nours (rectifier): 0.212\nours (rbf): " }, { "output_type": "stream", "stream": "stdout", "text": "0.816\nsklearn: 0.296\n" } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": "Very strong performance by the rectifier and sigmoid layers, very poor performance for RBF and polynomial layers. Let's run it again, on different data." }, { "cell_type": "code", "collapsed": false, "input": "X,Y = make_classification(n_samples=1000, n_features=20, n_informative=10,\n n_redundant=10, n_classes=9, n_clusters_per_class=2)\nY = lb.fit_transform(Y)\nXtrain, Xtest, Ytrain, Ytest = train_test_split(X,Y,test_size=0.25)\n#ugh, sklearn doesn't take one-hot representations\nYtrain_sk = lb.inverse_transform(Ytrain)\n", "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": "sklearn_lr = LogisticRegression()\nlinear_lr = LogisticClassifier()\n\nlinear_lr.fit(Xtrain, Ytrain, momentum=0.5, learn_rate=0.1,\n itrs=1000, reg=0.01)\nsklearn_lr.fit(Xtrain, Ytrain_sk)\n\n#fit a sigmoid basis logistic regressor\nsigmoid_lr = LogisticClassifier(basis='sigmoid')\nsigmoid_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a polynomial basis logistic regressor\npoly_lr = LogisticClassifier(basis='poly')\npoly_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rectifier basis logistic regressor\nrect_lr = LogisticClassifier(basis='rectifier')\nrect_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rbf basis logistic regressor\nrbf_lr = LogisticClassifier(basis='rbf')\nrbf_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n\nprint 'ours (linear):', error_rate(linear_lr.predict(Xtest), Ytest)\nprint 'ours (sigmoid):', error_rate(sigmoid_lr.predict(Xtest), Ytest)\nprint 'ours (polynomial):', error_rate(poly_lr.predict(Xtest), Ytest)\nprint 'ours (rectifier):', error_rate(rect_lr.predict(Xtest), Ytest)\nprint 'ours (rbf):', error_rate(rbf_lr.predict(Xtest), Ytest)\nprint 'sklearn:', error_rate(sklearn_lr.predict_proba(Xtest), Ytest)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "ours (linear): 0.636\nours (sigmoid): 0.648\nours (polynomial): " }, { "output_type": "stream", "stream": "stdout", "text": "0.724\nours (rectifier): 0.532\nours (rbf): " }, { "output_type": "stream", "stream": "stdout", "text": "0.908\nsklearn: 0.644\n" } ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": "Again, we see a similar distribution of scores. I'm starting to see why the rectifier is becoming more common than the sigmoid. One more time!" }, { "cell_type": "code", "collapsed": false, "input": "X,Y = make_classification(n_samples=1000, n_features=20, n_informative=10,\n n_redundant=10, n_classes=9, n_clusters_per_class=2)\nY = lb.fit_transform(Y)\nXtrain, Xtest, Ytrain, Ytest = train_test_split(X,Y,test_size=0.25)\n#ugh, sklearn doesn't take one-hot representations\nYtrain_sk = lb.inverse_transform(Ytrain)\n\nsklearn_lr = LogisticRegression()\nlinear_lr = LogisticClassifier()\n\nlinear_lr.fit(Xtrain, Ytrain, momentum=0.5, learn_rate=0.1,\n itrs=1000, reg=0.01)\nsklearn_lr.fit(Xtrain, Ytrain_sk)\n\n#fit a sigmoid basis logistic regressor\nsigmoid_lr = LogisticClassifier(basis='sigmoid')\nsigmoid_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a polynomial basis logistic regressor\npoly_lr = LogisticClassifier(basis='poly')\npoly_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rectifier basis logistic regressor\nrect_lr = LogisticClassifier(basis='rectifier')\nrect_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n#fit a rbf basis logistic regressor\nrbf_lr = LogisticClassifier(basis='rbf')\nrbf_lr.fit(Xtrain, Ytrain, momentum=0.9, learn_rate=0.05,\n itrs=1000, reg=0.001, proj_layer_size=50)\n\n\nprint 'ours (linear):', error_rate(linear_lr.predict(Xtest), Ytest)\nprint 'ours (sigmoid):', error_rate(sigmoid_lr.predict(Xtest), Ytest)\nprint 'ours (polynomial):', error_rate(poly_lr.predict(Xtest), Ytest)\nprint 'ours (rectifier):', error_rate(rect_lr.predict(Xtest), Ytest)\nprint 'ours (rbf):', error_rate(rbf_lr.predict(Xtest), Ytest)\nprint 'sklearn:', error_rate(sklearn_lr.predict_proba(Xtest), Ytest)", "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": "ours (linear): 0.592\nours (sigmoid): 0.612\nours (polynomial): " }, { "output_type": "stream", "stream": "stdout", "text": "0.752\nours (rectifier): 0.548\nours (rbf): " }, { "output_type": "stream", "stream": "stdout", "text": "0.904\nsklearn: 0.628\n" } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": "Time to draw some conclusions. First, a linear projection followed by a rectifier activation is a *very* competitive basis function. It won in all of our experiments, usually by a large margin. Both my RBF and polynomial activations are very bad, which is probably a bug on my part but I'm not sure. I looked into it a little and they just seem to be learning *very* slowly. Our linear basis logistic regression is comparable to sklearn's, probably because they're doing the exact same thing. " } ], "metadata": {} } ] }