{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 56 POINTS Programming Assignment 3\n", "In this exercise, we'll use a multiclass classification problem. We'll use the ``20newsgroups`` dataset. Let us load the dataset and convert it to the bag-of-words representation." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sklearn.feature_extraction.text import CountVectorizer\n", "from sklearn.model_selection import train_test_split\n", "from sklearn.datasets import fetch_20newsgroups\n", "from sklearn.datasets import fetch_mldata\n", "from sklearn import preprocessing\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import math" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def Load20Newsgroups():\n", " newsgroups_train = fetch_20newsgroups(subset='train')\n", " newsgroups_test = fetch_20newsgroups(subset='test')\n", " \n", " count_vect = CountVectorizer(max_features=10**3)\n", " X_train = count_vect.fit_transform(newsgroups_train.data).toarray().astype(float)\n", " X_test = count_vect.transform(newsgroups_test.data).toarray().astype(float)\n", " Y_train = newsgroups_train.target\n", " Y_test = newsgroups_test.target\n", " \n", " X_train = preprocessing.normalize(X_train, norm='l2')\n", " X_test = preprocessing.normalize(X_test, norm='l2')\n", " return X_train, Y_train, X_test, Y_test\n", "\n", "X_train, Y_train, X_test, Y_test = Load20Newsgroups()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 20 POINTS Logistic Regression is Smooth\n", "Recall that given a feature vector ${\\bf x}\\in\\mathbb{R}^d$, Logistic Regression (parameterized by $W\\in \\mathbb{R}^{d\\times k}$) outputs a probability distribution such that\n", "$$ \\mathbb{P}(h_W(x)=j) = \\frac{e^{{\\bf w}_j^T{\\bf x}}}{\\sum_{i=1}^k e^{{\\bf w}_i^T{\\bf x}}}. $$\n", "\n", "With respect to a true label $y\\in \\{1,\\dots k\\}$ for the datapoint $x$, the loss for $W$ is defined as \n", "$$l(W; ({\\bf x},y)) = -\\log \\mathbb{P}(h_W({\\bf x})=y).$$\n", "\n", "By extension, the loss for a dataset is the average of the loss values on individual datapoints." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**ACT 1:** For any matrix $W\\in \\mathbb{R}^{d\\times k}$, define \n", "$$ \\|W\\|_{p,q} = \\max_{{\\bf x}\\neq 0} \\frac{\\|{\\bf x}W\\|_q}{\\|{\\bf x}\\|_p}.$$\n", "Prove that $S = \\{W: \\|W\\|_{2,1}\\leq 1\\}$ is a convex set.\n", "\n", "$$ \\|\\lambda W_1 + (1-\\lambda) W_2\\|_{2,1} = \\max_{\\|x\\|_2\\leq 1} \\|\\lambda x^TW_1 + (1-\\lambda) x^TW_2\\|_{1} \\leq \\max_{\\|x\\|_2\\leq 1} \\|\\lambda x^TW_1\\|_1 + \\max_{\\|x\\|_2\\leq 1}\\|(1-\\lambda) x^TW_2\\|_{1}$$\n", "\n", "6 POINTS -- 2 for using the definition, 2 for invoking triangle inequality on $l_1$, 2 for max of sum $\\leq$ sum of max." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**ACT 2:** The behavior of binary logistic regression (parameterized by ${\\bf w}\\in\\mathbb{R}^d$) is simpler:\n", "$$ \\mathbb{P}(h_{\\bf w}({\\bf x})=1) = \\frac{e^{{\\bf w}^T{\\bf x}}}{1+e^{{\\bf w}^T{\\bf x}}}, \\mathbb{P}(h_w(x)=0) = \\frac{1}{1+e^{{\\bf w}^T{\\bf x}}}$$\n", "Given that $\\|{\\bf x}\\|_2 = D$ and $\\|w\\|_2\\leq M$, state (and justify, for a single datapoint) -\n", "1. the smallest value $\\beta\\in\\mathbb{R}$, so that $l(w)$ is $\\beta$-smooth;\n", "3. if $l(w)$ is $\\alpha$-strongly convex for any $\\alpha>0$;\n", "2. the smallest possible value of $x^T\\nabla^2 l(w)x$.\n", "\n", "With $p=\\mathbb{P}(h_{\\bf w}({\\bf x})=1)$,\n", "$$\\nabla^2 l(w) = (p-p^2) xx^T.$$\n", "This is $\\frac{1}{4}D^2$-smooth (1).\n", "\n", "6 POINTS in total.\n", "+ 4 for computing the Hessian.\n", "+ 2 for the answer. Deduct 1 if constant is wrong.\n", "\n", "Not strongly convex (2) since $y^T\\nabla^2 y=0$ if $y^Tx=0$.\n", "\n", "2 POINTS in total.\n", "+ 1 for the answer.\n", "+ 1 for justification.\n", "\n", "The smallest possible value is $(p_* -p_*^2)D^4$ where $p_*= \\frac{1}{1+e^{MD}}$.\n", "\n", "6 POINTS.\n", "+ 4 for the correct answer upto $\\propto D^2$, or wrong constant.\n", "+ 6 for perfect." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 7 POINTS Logistic Regression\n", "The first job is to code up the loss function for multiclass logistic regression, and the gradient for the same." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 6 POINTS in total. 2 for loss (must pass assertion), 4 for reasonable grad.\n", "# ACT (3) Design the function LogLossAndGrad as described in the preamble \n", "# to the previous section. The function must return a tuple (loss, grad) where\n", "# 1. loss is the value of the loss for W on (X, Y),\n", "# 2. grad is a matrix/array of W's shape containing the gradient.\n", "def LogLossAndGrad(W, X, Y):\n", " Loss, Grad = 0, np.zeros_like(W)\n", " Z = np.exp(X.dot(W))\n", " P = Z/Z.sum(axis=1)[:, np.newaxis]\n", " Grad = X.T.dot(P)\n", " for i in range(X.shape[0]):\n", " Loss += -np.log(P[i,Y[i]])\n", " Grad[:, Y[i]] -= X[i]\n", " return Loss/X.shape[0], Grad/X.shape[0]\n", "\n", "assert(0.49 < LogLossAndGrad(np.array([[0,1]]), np.array([[0], [1]]), np.array([1,1]))[0] < 0.51)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 1 POINT for correct implementation.\n", "# ACT (4) Design the 0-1 loss function for multiclass classification.\n", "# The function returns a scalar.\n", "def ZeroOneLoss(W, X, Y):\n", " return np.mean(Y != np.argmax(X.dot(W), axis=1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 11 POINTS Stochastic Gradient Descent\n", "As described [here](http://www.cs.princeton.edu/courses/archive/fall17/cos324/lectures/lect11.pdf), our implementation of SGD will be slightly more mature than in Programming Assignment 2.\n", "\n", "In particular, you're require to implement SGD with \n", "1. **mini-batches:** In PA2, every update to the weight vector required computing the gradient on exactly one datapoint. In the new variant, you're required to estimate the gradient by averaging the gradient on $S$ randomly sampled datapoints.\n", "2. **projection:** Every iterate of SGD must be such that $\\|W\\|_{2,\\infty}\\leq D$.\n", "3. **suffix averaging:** The output weight vector for Stochastic Gradient Descent should be the the average of all iterates from epoch $T_0$ to $T$, as described on slide $13$.\n", "\n", "Your implementation must be parameterized by these $(T, T_0, \\eta, S, D)$, and must return\n", "1. a weight vector $\\overline{W}$,\n", "2. a $T$ length vector of logistic loss of iterate $W_{t+1}$ on epoch $t$,\n", "2. a $T$ length vector of 0-1 loss of iterate $W_{t+1}$ on epoch $t$." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 10 POINTS in total.\n", "# 4 for basic version of GD\n", "# 2 for mini-batching, 2 for projection, 2 for suffix averageing.\n", "# ACT (5) Code up mini-batch stochastic gradient descent.\n", "def MiniSGD(X, Y, LossAndGrad, T, T0, eta, S, D):\n", " n, d, k = X.shape[0], X.shape[1], max(Y)+1\n", " W, Wbar = np.zeros((d,k)), np.zeros((d,k))\n", " LL, ZL = np.zeros(T), np.zeros(T)\n", " batch_per_epoch = math.floor(n/S)\n", " for e in range(T):\n", " if e%5 == 0:\n", " print(\"On epoch %d of %d.\"%(e,T))\n", " perm = np.random.permutation(n)\n", " LLoss, ZLoss = 0, 0\n", " for i in range(batch_per_epoch):\n", " index = perm[i*S:(i+1)*S]\n", " Loss, Grad = LossAndGrad(W, X[index], Y[index])\n", " LLoss, ZLoss = LLoss+Loss, ZLoss+ZeroOneLoss(W, X[index], Y[index])\n", " W -= eta*Grad\n", " for k0 in range(k):\n", " rnorm = np.linalg.norm(W[:,k0])\n", " W[:,k0] = W[:,k0]/rnorm if rnorm>D else W[:,k0]\n", " if e >= T0:\n", " Wbar += W\n", " LL[e], ZL[e] = LLoss/batch_per_epoch, ZLoss/batch_per_epoch\n", " return Wbar/(T-T0), LL, ZL" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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2dLCjuZ33duyjwy9MBirOTMHjdvH5ud5wmJWfTuHEZJssCUzLdXHlvEIeXr2T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\nencXP3+pmv+9vjxk17XAGAd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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 10 Training Error 0.239084\n", "Batch size 10 Test Error 0.419411\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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DraiHXtlDR3do3acUgzHDs6ksGkZlcU7ov+HnFYU5CowYUCiISCDMjJF5WYzM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\nKLZe73sxjdAAAAAASUVORK5CYII=\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 100 Training Error 0.142832\n", "Batch size 100 Test Error 0.363914\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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zFsAkYIS7rzCzLHdfV9N2VQoiwXJ3Fq7bxluzQnsQC9dtwwwGdmrFyCPaMqJP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\npJpKQUREqqkURESkmkpBRESq/X9Hx1niGgMUvwAAAABJRU5ErkJggg==\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 1000 Training Error 0.343468\n", "Batch size 1000 Test Error 0.458046\n" ] } ], "source": [ "# 1 POINT is there some S for which test loss is smaller than asked for.\n", "# ACT (6) Run SGD on 20Newsgroups dataset.\n", "# To do this, adjust eta, D.\n", "# What value of S works best?\n", "eta, T = 5e1, 10\n", "\n", "def PlotText(data, xlab, ylab):\n", " plt.plot(data)\n", " plt.ylabel(ylab)\n", " plt.xlabel(xlab)\n", " plt.show()\n", "\n", "for S in [10, 100, 1000]:\n", " T0, D = T/2, 1e2\n", " W, LL, ZL = MiniSGD(X_train, Y_train, LogLossAndGrad, T, T0, eta, S, D)\n", " PlotText(LL, 'Epoch Number', 'Log Loss')\n", " PlotText(ZL, 'Epoch Number', '0/1 Loss')\n", " print(\"Batch size %d Training Error %f\"%(S, ZeroOneLoss(W, X_train, Y_train)))\n", " print(\"Batch size %d Test Error %f\"%(S, ZeroOneLoss(W, X_test, Y_test)))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Test Error 0.397503983006\n" ] } ], "source": [ "# Not for Production: Scikit-learn implementation\n", "def SKLogCheck():\n", " from sklearn.linear_model import LogisticRegression\n", " lr = LogisticRegression()\n", " lr.fit(X_train, Y_train)\n", " Y_hat = lr.predict(X_test)\n", " print(\"Test Error\", sum(Y_hat != Y_test)/len(Y_test))\n", " \n", "SKLogCheck()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# 17 POINTS Max-margin Multiclass\n", "Recall that given a feature vector $x\\in\\mathbb{R}^d$, the Max-Margin Loss (parameterized by $W\\in \\mathbb{R}^{d\\times k}$) outputs a label\n", "$$ h_W(x) = \\textbf{argmax}_{j\\leq k} W_j^Tx. $$\n", "\n", "With respect to a true label $y\\in \\{1,\\dots k\\}$ for the datapoint $x$, the loss for $W$ is defined as \n", "$$l_\\gamma(W; (x,y)) = [\\gamma + \\max_{j\\neq y} W_j^Tx - W_y^Tx]_+.$$\n", "\n", "By extension, the loss for a dataset is the average of the loss values on individual datapoints." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 6 POINTS in total. 2 for loss (must pass assertion), 4 for reasonable grad.\n", "# ACT (7) Design the function MaxLossAndGrad as described in the slides. \n", "# The function must return a tuple (loss, grad) where\n", "# 1. loss is the value of the loss for W on (X, Y),\n", "# 2. grad is a matrix/array of W's shape containing the sub-gradient\n", "# evaluated in accordance to the max-violation scheme.\n", "def MaxLossAndGrad(W, X, Y, gamma=1.0):\n", " Loss, Grad = 0, np.zeros_like(W).astype(float)\n", " Z = X.dot(W).astype(float)\n", " for i in range(X.shape[0]): \n", " z, y = Z[i], Y[i]\n", " z[y] -= gamma\n", " err = np.max(z - z[y])\n", " if err > 0:\n", " yh = np.argmax(z)\n", " Loss += err\n", " Grad[:,y] -= X[i]\n", " Grad[:,yh] += X[i]\n", " return Loss/X.shape[0], Grad/X.shape[0]\n", "\n", "assert(MaxLossAndGrad(np.array([[0,1]]), np.array([[0], [1]]), np.array([1,1]), 1.0)[0] == 0.5)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 10 Training Error 0.361764\n", "Batch size 10 Test Error 0.495884\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 100 Training Error 0.110306\n", "Batch size 100 Test Error 0.363914\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 1000 Training Error 0.456161\n", "Batch size 1000 Test Error 0.549124\n" ] } ], "source": [ "# 1 POINT is there some S for which test loss is smaller than asked for.\n", "# ACT (8) Run SGD on 20Newsgroups dataset.\n", "# To do this, adjust eta, D.\n", "# What value of S works best?\n", "eta, T = 5e1, 10\n", "\n", "for S in [10, 100, 1000]:\n", " T0, D = T/2, 1e2\n", " W, LL, ZL = MiniSGD(X_train, Y_train, MaxLossAndGrad, T, T0, eta, S, D)\n", " PlotText(LL, 'Epoch Number', 'Log Loss')\n", " PlotText(ZL, 'Epoch Number', '0/1 Loss')\n", " print(\"Batch size %d Training Error %f\"%(S, ZeroOneLoss(W, X_train, Y_train)))\n", " print(\"Batch size %d Test Error %f\"%(S, ZeroOneLoss(W, X_test, Y_test)))" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 4 POINTS in total, for reasonable grad -- has to be different from MaxLossAndGrad.\n", "# ACT (9) Design the function UnifLossAndGrad as described in the slides. \n", "# The function must return a tuple (loss, grad) where\n", "# 1. loss is the value of the loss for W on (X, Y),\n", "# 2. grad is a matrix/array of W's shape containing the sub-gradient\n", "# evaluated in accordance to the uniform violation scheme.\n", "def UnifLossAndGrad(W, X, Y, gamma=1.0):\n", " Loss, Grad = 0, np.zeros_like(W).astype(float)\n", " Z = X.dot(W).astype(float)\n", " for i in range(X.shape[0]): \n", " z, y = Z[i], Y[i]\n", " z[y] -= gamma\n", " err = np.max(z - z[y])\n", " if err > 0:\n", " yh = z - z[y] > 0\n", " Loss += err\n", " Grad[:,y] -= X[i]\n", " Grad[:,yh] += (X[i]/yh.sum())[:, np.newaxis]\n", " return Loss/X.shape[0], Grad/X.shape[0]\n", "\n", "assert(UnifLossAndGrad(np.array([[0,1]]), np.array([[0], [1]]), np.array([1,1]), 1.0)[0] == 0.5)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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EjBuRzuGSMl7+2lolpmGwRNJIdE2O472xQ/jdlWcz7/sCLps4n3+s2NlkWydL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\nZR9P3tSb/76uV8gvD+DarS0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i5/ATQH9gBFAA/KmuG4vILSKSISIZhYWFzRRucDp/SA9+cvFg3lpfwJPvb3c7HGNMiPEl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\nuQnYBVzjy2cwTZO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c69iTXttzgbmnKDvrFMcPAJOaMUxjTANqaiU3PPcF8zN3852z+jRY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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 10 Training Error 0.307584\n", "Batch size 10 Test Error 0.474774\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 100 Training Error 0.112604\n", "Batch size 100 Test Error 0.357408\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 1000 Training Error 0.285310\n", "Batch size 1000 Test Error 0.422862\n" ] } ], "source": [ "# 1 POINT is there some S for which test loss is smaller than asked for.\n", "# ACT (10) Run SGD on 20Newsgroups dataset.\n", "# To do this, adjust eta, D.\n", "# What value of S works best?\n", "eta, T = 5e1, 10\n", "\n", "for S in [10, 100, 1000]:\n", " T0, D = T/2, 1e2\n", " W, LL, ZL = MiniSGD(X_train, Y_train, UnifLossAndGrad, T, T0, eta, S, D)\n", " PlotText(LL, 'Epoch Number', 'Log Loss')\n", " PlotText(ZL, 'Epoch Number', '0/1 Loss')\n", " print(\"Batch size %d Training Error %f\"%(S, ZeroOneLoss(W, X_train, Y_train)))\n", " print(\"Batch size %d Test Error %f\"%(S, ZeroOneLoss(W, X_test, Y_test)))" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# 4 POINTS in total, for reasonable grad -- has to be different from Unif/MaxLossAndGrad.\n", "# ACT (11) Design the function PropLossAndGrad as described in the slides. \n", "# The function must return a tuple (loss, grad) where\n", "# 1. loss is the value of the loss for W on (X, Y),\n", "# 2. grad is a matrix/array of W's shape containing the sub-gradient\n", "# evaluated in accordance to the proportional-violation scheme.\n", "def PropLossAndGrad(W, X, Y, gamma=1.0):\n", " Loss, Grad = 0, np.zeros_like(W).astype(float)\n", " Z = X.dot(W).astype(float)\n", " for i in range(X.shape[0]): \n", " z, y = Z[i], Y[i]\n", " z[y] -= gamma\n", " err = np.max(z - z[y])\n", " if err > 0:\n", " yh = z - z[y] > 0\n", " errh = (z - z[y])[yh]\n", " Loss += err\n", " Grad[:,y] -= X[i]\n", " Grad[:,yh] += np.outer(X[i],errh)/errh.sum()\n", " return Loss/X.shape[0], Grad/X.shape[0]\n", "\n", "assert(PropLossAndGrad(np.array([[0,1]]), np.array([[0], [1]]), np.array([1,1]), 1.0)[0] == 0.5)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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MIq5j6wao0hj3WJAYU4PWzUJ56tpBZBw8yiMfVj+wY0WF8pePN3Hf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\nEmOMMXViQWKMMaZO/h/dewBhkVXqDwAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 10 Training Error 0.298833\n", "Batch size 10 Test Error 0.456585\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 100 Training Error 0.109068\n", "Batch size 100 Test Error 0.367631\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 1000 Training Error 0.330476\n", "Batch size 1000 Test Error 0.468667\n" ] } ], "source": [ "# 1 POINT is there some S for which test loss is smaller than asked for.\n", "# ACT (12) Run SGD on 20Newsgroups dataset.\n", "# To do this, adjust eta, D.\n", "# What value of S works best?\n", "eta, T = 5e1, 10\n", "\n", "for S in [10, 100, 1000]:\n", " T0, D = T/2, 1e2\n", " W, LL, ZL = MiniSGD(X_train, Y_train, PropLossAndGrad, T, T0, eta, S, D)\n", " PlotText(LL, 'Epoch Number', 'Log Loss')\n", " PlotText(ZL, 'Epoch Number', '0/1 Loss')\n", " print(\"Batch size %d Training Error %f\"%(S, ZeroOneLoss(W, X_train, Y_train)))\n", " print(\"Batch size %d Test Error %f\"%(S, ZeroOneLoss(W, X_test, Y_test)))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Test Error 0.345857673925\n" ] } ], "source": [ "# Not for Production: Scikit-learn implementation\n", "def SKMaxCheck():\n", " from sklearn.svm import LinearSVC\n", " lr = LinearSVC(multi_class='crammer_singer')\n", " lr.fit(X_train, Y_train)\n", " Y_hat = lr.predict(X_test)\n", " print(\"Test Error\", sum(Y_hat != Y_test)/len(Y_test))\n", " \n", "SKMaxCheck()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1 POINT Classifying Digits\n", "Let us move to a more exciting dataset ``MNIST``. Here we'll use the tools we've developed in the previous sections to classify hand written digits as belonging to one of 10 possible classes - $\\{0, 1,\\dots 9\\}$." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def LoadMNIST():\n", " mnist = fetch_mldata('MNIST original')\n", " X_train, X_test, Y_train, Y_test = train_test_split(mnist.data, mnist.target, train_size=0.5)\n", " X_train, X_test = X_train.astype(float), X_test.astype(float)\n", " Y_train, Y_test = Y_train.astype(int), Y_test.astype(int)\n", " \n", " X_train = preprocessing.normalize(X_train, norm='l2')\n", " X_test = preprocessing.normalize(X_test, norm='l2')\n", " return X_train, Y_train, X_test, Y_test\n", "\n", "X_train, Y_train, X_test, Y_test = LoadMNIST()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def PlotMNIST(X):\n", " plt.imshow(X.reshape(28, 28), cmap='gray')\n", " plt.show()\n", "\n", "PlotMNIST(X_train[0])" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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Pm3cdAKB7pzTOGZLDecP6cNaJ2XSN09U/LVVX77y/bgczl1Uws6yC8qq95PXM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\n/x+4JZU5EiRNIQAAAABJRU5ErkJggg==\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 10 Training Error 0.072914\n", "Batch size 10 Test Error 0.086314\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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WF6g/JEUlMzB2ArVxyzXBuhHM7FLgPuAadz8Qtz6XWFg86O6Pj/Um7n4vcC/E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\ndfKWcDO5u9MAAAAASUVORK5CYII=\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 100 Training Error 0.066343\n", "Batch size 100 Test Error 0.076886\n", "On epoch 0 of 10.\n", "On epoch 5 of 10.\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Batch size 1000 Training Error 0.085686\n", "Batch size 1000 Test Error 0.086429\n" ] } ], "source": [ "# 1 POINT is there some S for which test loss is smaller than asked for.\n", "# Any method acceptable.\n", "# ACT (13) Run SGD on MNIST dataset.\n", "# You're free to choose any method of your choice --\n", "# LogLoss, MaxLoss, UnifLoss, PropLoss.\n", "# To do this, adjust eta, D.\n", "# What value of S works best?\n", "eta, T = 2e1, 10\n", "\n", "for S in [10, 100, 1000]:\n", " T0, D = T/2, 1e2\n", " W, LL, ZL = MiniSGD(X_train, Y_train, PropLossAndGrad, T, T0, eta, S, D)\n", " PlotText(LL, 'Epoch Number', 'Log Loss')\n", " PlotText(ZL, 'Epoch Number', '0/1 Loss')\n", " print(\"Batch size %d Training Error %f\"%(S, ZeroOneLoss(W, X_train, Y_train)))\n", " print(\"Batch size %d Test Error %f\"%(S, ZeroOneLoss(W, X_test, Y_test)))" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Test Error 0.0925428571429\n", "Test Error 0.0766857142857\n" ] } ], "source": [ "# Not for Production: Scikit-learn implementation\n", "SKLogCheck()\n", "SKMaxCheck()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernel_info": { "name": "python3" }, "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": "3.6.1" }, "nteract": { "version": "0.4.3" } }, "nbformat": 4, "nbformat_minor": 2 }