JNotebooks/tutorial03-overfitting_regularization.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Model Selection, Overfitting and Regularization\n",
    "\n",
    "This tutorial is meant to be a gentle introduction to machine learning concepts. We present a simple polynomial fitting example using a least squares solution, which is a specific case of what is called maximum likelihood, but we will not get into details about this probabilistic view of least squares in this tutorial. We use this example to introduce important machine learning concepts using plain language that should be accessible to undergradiuate and graduate students with a minimum background of calculus.\n",
    "\n",
    "The goals of this tutorial are:\n",
    "  - Explain how to develop an experiment. Split your data into development set (*i.e.*, train and validaion sets) and test set.   \n",
    "  - Introduce how to select your model.\n",
    "  - Introduce the concepts of *over-fitting*, *under-fitting*, and *model generalization*.\n",
    "  - Introduce the concept of *regularization* for reducing *over-fitting*.\n",
    " \n",
    "\n",
    "This tutorial is interactive and it corresponds to an adaptation of the example presented in chapter 1 of the book: **Christopher M. Bishop. 2006. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag New York, Inc., Secaucus, NJ, USA.**\n",
    "\n",
    "## Designing your experiment\n",
    "\n",
    "Machine learning builds models by learning from data. When designing your experiment, you need to split your data into a development set and a test set. The development set is split into 2 sets: a train set and a validation set. The train set is used to learn the parameters of the different models you are fititng (training). The validation set is employed to select hopefully what is the best model among the different models you trained, therefore it has a bias and cannot be used as proof of generalization. The test set is used to see if the selected model generalizes well to unseen data. \n",
    "\n",
    "<img src=\"../Figures/train_val_test.png\" alt=\"Drawing\" style=\"width: 500px;\"/>\n",
    "\n",
    "## Generating synthetic data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Directive to make plots inline as opposed to having pop-up plots \n",
    "%matplotlib inline \n",
    "import numpy as np # Import numpy with nickname np\n",
    "import matplotlib.pylab as plt # plotting library\n",
    "from ipywidgets import * # Interaction library\n",
    "\n",
    "var = 0.2 #Noise variance\n",
    "\n",
    "#Create data set\n",
    "N = 25\n",
    "x = np.linspace(0, 1, N) \n",
    "y_noiseless = np.sin(2*np.pi*x) # signal\n",
    "y = y_noiseless + np.random.normal(0, var, N) #signal + noise -> real measurements always come with noise\n",
    "\n",
    "\n",
    "# Plot entire data set with and without noise\n",
    "plt.figure()\n",
    "plt.plot(x,y_noiseless,linewidth = 2.0,label = r'Data without noise: $sin(2 \\pi x)$')\n",
    "plt.scatter(x,y,color ='red', marker = 'x', label = r'Data with noise')\n",
    "plt.legend(loc = (0.02, 0.18))\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Splitting the data into train, validation, and test sets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Splitting the data in train/validation/test sets - size of each set was choosen arbitrarily\n",
    "train_size = 10\n",
    "val_size = 5\n",
    "test_size = 10\n",
    "\n",
    "indexes = np.arange(N, dtype =int)\n",
    "np.random.seed(seed = 2) # Random seed to keep results always the same\n",
    "np.random.shuffle(indexes) # Shuffling the data before the split\n",
    "\n",
    "# Train set\n",
    "aux = indexes[:train_size]\n",
    "aux = np.sort(aux)\n",
    "x_train = x[aux]\n",
    "y_train = y[aux] \n",
    "\n",
    "#Validation set\n",
    "aux = indexes[train_size: train_size + val_size]\n",
    "aux = np.sort(aux)\n",
    "x_val= x[aux]\n",
    "y_val = y[aux]\n",
    "\n",
    "# Test set\n",
    "aux = indexes[-test_size:]\n",
    "aux = np.sort(aux)\n",
    "x_test = x[aux]\n",
    "y_test = y[aux]\n",
    "\n",
    "# Plot train/val/test sets\n",
    "plt.figure()\n",
    "plt.plot(x,y_noiseless,linewidth = 2.0,label = r'Model no noise: $sin(2 \\pi x)$')\n",
    "plt.scatter(x_train,y_train,color ='red', marker = 'x', label = \"Train set\")\n",
    "plt.scatter(x_val,y_val,color = 'blue',marker = '^' , label = \"Validation set\")\n",
    "plt.scatter(x_test,y_test,color = 'green', marker = 'o', label = \"Test set\")\n",
    "plt.legend(loc = (0.02, 0.18))\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data\n",
    "\n",
    "Observations: $$\\boldsymbol{X} =[x_1,x_2,...,x_N]^T$$\n",
    "Target:  $$\\boldsymbol{T} =[t_1,t_2,...,t_N]^T$$\n",
    "Estimates:  $$\\boldsymbol{Y} =[y_1,y_2,...,y_N]^T$$\n",
    "\n",
    "\n",
    "## Polynomial Model\n",
    "\n",
    "$$y(x,\\boldsymbol{W})= w_0 + w_1x +w_2x^2+...+w_mx^m = \\sum^M_{j=0}w_jx^j$$\n",
    "\n",
    "Weights (*i.e.*, what our model learns):  $$\\boldsymbol{W} =[w_1,w_2,...,w_M]^T$$\n",
    "\n",
    "## Cost Function\n",
    "\n",
    "Quadratic cost function: $$E(\\boldsymbol{W})=\\frac{1}{2}\\sum_{n=1}^N\\{y(x_n,\\boldsymbol{W})-t_n\\}^2$$\n",
    "\n",
    "Computing the derivative of the cost function and making it equal to zero, we can find the vector **W*** that minimizes the error:\n",
    "$$ \\boldsymbol{W}^* = (\\boldsymbol{A}^T\\boldsymbol{A})^{-1}\\boldsymbol{A} ^T\\boldsymbol{T}$$\n",
    "\n",
    "Where **A** is defined by:\n",
    "\n",
    "$$\\boldsymbol{A} = \\begin{bmatrix}\n",
    "    1 & x_{1} & x_{1}^2 & \\dots  & x_{1}^M \\\\\n",
    "    1 & x_{2} & x_{2}^2 & \\dots  & x_{2}^M \\\\\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    1 & x_{N} & x_{N}^2 & \\dots  & x_{N}^M\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Least squares polynomial fitting solution\n",
    "# Implementation of the equation shown above \n",
    "def polynomial_fit(X,T,M):\n",
    "    A = np.power(X.reshape(-1,1),np.arange(0,M+1).reshape(1,-1))\n",
    "    T = T.reshape(-1,1)\n",
    "    W = np.dot(np.linalg.pinv(A),T)\n",
    "    return W.ravel()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the least squares result varying the polynomial degree between  0 a 9. **Which model is a good model?** Look at the plots but also the magnitude of the weights resulting from each polynomial fit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1023380375c9489dbb50a3c4ec914289",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=4, description='M', max=9), Output()), _dom_classes=('widget-interact',)…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotmodel(M)>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotmodel(M):\n",
    "    coefs = polynomial_fit(x_train, y_train, M)[::-1]\n",
    "    print(\"Weights:\\n\", coefs)\n",
    "    p = np.poly1d(coefs)\n",
    "    plt.figure()\n",
    "    plt.plot(x,y_noiseless,linewidth = 1.5,label = r'Data no noise: $sin(2 \\pi x)$')\n",
    "    plt.scatter(x_train,y_train,color='red',label= \"Train set\")\n",
    "    plt.xlabel(\"x\")\n",
    "    plt.ylabel(r'y')\n",
    "    y_fit = p(x_train) \n",
    "    plt.plot(x_train,y_fit,linewidth = 1.0,label =\"Polynomial Fit\")\n",
    "    plt.legend(loc=(0.02,0.02))\n",
    "    plt.show()\n",
    "\n",
    "interact(plotmodel,M=(0,9,1))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Depending on the degree, M, of the polynomial we fit to our data, our model falls under one of these categories:\n",
    "\n",
    "- **Under-fitting**: the model is too inflexible and is not able to capture any patterns in the data.\n",
    "\n",
    "- **Over-fitting**: the model is too flexible. It ends up tuning to the random noise in the data. The model may have a low error in the train set, but it is not expected to generalize well to new (unseen) data.\n",
    "\n",
    "- **Good fit**: The model is able to capture patterns in our data, but it does not get tuned to the random noise in the data. Better chances to generalize to new  (unseen) data.\n",
    "\n",
    "A good exercise is to visually determine whether the model is under-fitting, over-fitting or it is a good model based on the polynomial degree in the interactive plot shown above. \n",
    "\n",
    "## Root mean squared error and Model Selection\n",
    "\n",
    "Root mean squared error is an error measure commonly emplyed in regression problems.\n",
    "\n",
    "$$E_{RMS}=\\sqrt{2E(\\boldsymbol{W^*})/N}$$\n",
    "\n",
    "We will analyze the root mean squared error in the validation set to select our model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model selected was a  polynomial of degree 7\n",
      "Root mean squared test error: 0.363\n"
     ]
    }
   ],
   "source": [
    "# Computes RMS error\n",
    "def rms_error(X,T,W):\n",
    "    p = np.poly1d(W)\n",
    "    T_fit = p(X)\n",
    "    E = np.sqrt(((T - T_fit)**2/T.size).sum())\n",
    "    return E\n",
    "   \n",
    "m = range(10)\n",
    "train = []\n",
    "val = []\n",
    "\n",
    "# Compute RMS error across different polynomial fits\n",
    "for M in m:\n",
    "    W = polynomial_fit(x_train, y_train, M)[::-1]\n",
    "    error_train = rms_error(x_train,y_train,W)\n",
    "    error_val = rms_error(x_val,y_val,W)\n",
    "    train.append(error_train)\n",
    "    val.append(error_val)\n",
    "\n",
    "# Plot the errors\n",
    "plt.figure()\n",
    "plt.plot(m,train,linewidth = 2.0,marker = 'o',markersize = 12,label = r'$E_{RMS}$ Train')\n",
    "plt.plot(m,val,linewidth = 2.0,marker = 'x',markersize = 12,label = r'$E_{RMS}$ Validation')\n",
    "plt.legend(loc = (0.02, 0.05))\n",
    "plt.xlabel(\"Polynomial degree\")\n",
    "plt.ylabel(r'$E_{RMS}$')\n",
    "plt.show()\n",
    "\n",
    "\n",
    "# Model selection - the model with the lowest error in the validation set is selected. Then, the model \n",
    "# generalizability is assessed on the test set. \n",
    "best_M = np.argmin(val)\n",
    "W = polynomial_fit(x_train, y_train, best_M)[::-1]\n",
    "test_error = rms_error(x_test,y_test,W)\n",
    "print(\"Model selected was a  polynomial of degree %d\" %best_M)\n",
    "print(\"Root mean squared test error: %.3f\" %test_error)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cost function with regularization\n",
    "\n",
    "Regularization is a technique to avoid overfitting. Do you remember how the values of the estimated weights increased quickly for polynomial fits with high degrees in the example without regularization? That was the model tuning itself to the noise in the data. Regularization consists in adding a penalty term to the cost function. Let's add a quadratic penalty to the weights we are trying to estimate. The quadratic penalty is called **L2 regularization**. \n",
    "\n",
    "$$E(\\boldsymbol{W})=\\frac{1}{2}\\sum_{n=1}^N\\{y(x_n,\\boldsymbol{W})-t_n\\}^2 +\\frac{\\lambda}{2}||\\boldsymbol{W}||^2$$\n",
    "\n",
    "The above equation also has a well-defined minimum point. Computing its derivative and making it equal to zero, the solution of the equation is given by:\n",
    "\n",
    "$$\\boldsymbol{W}^* = (\\boldsymbol{A}^T\\boldsymbol{A} + \\lambda n\\boldsymbol{I})^{-1}\\boldsymbol{A} ^T\\boldsymbol{T} $$\n",
    "\n",
    "Note that our problem now has two hyper-parameters that we need to set. The polynomial degree (M) and the regularization factor ($\\lambda$). Hyper-parameters are set by the user (*e.g.*, M and $\\lambda$), while parameters are learned by the model (*e.g.*, the weights)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Least square solution with regularization\n",
    "def polynomial_fit_reg(X,T,M,lamb):\n",
    "    N = X.shape[0]\n",
    "    A = np.power(X.reshape(-1,1),np.arange(0,M+1).reshape(1,-1))\n",
    "    lambda_matrix = lamb*N*np.eye(M+1)\n",
    "    T = T.reshape(-1,1)\n",
    "    aux = np.dot(A.T,A) + lambda_matrix\n",
    "    aux = np.linalg.pinv(aux)\n",
    "    aux2 = np.dot(A.T,T)\n",
    "    W = np.dot(aux,aux2)\n",
    "    return W.ravel()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the demo below, we show the influence of $log(\\lambda)$ and $M$ in the polynomial fitting. Note the influence of $\\lambda$  in the estimated weights. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "27766f86d2c44baeb67d87378381b149",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=4, description='M', max=9), FloatSlider(value=-25.0, description='log_la…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.plotmodel2(M, log_lamb)>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plotmodel2(M,log_lamb):\n",
    "    lamb = np.exp(log_lamb)\n",
    "    coefs = polynomial_fit_reg(x_train, y_train, M,lamb)[::-1]\n",
    "    print(\"Weights:\\n\",coefs)\n",
    "    print(\"Lambda\\n\", lamb)\n",
    "    p = np.poly1d(coefs)\n",
    "    plt.figure()\n",
    "    plt.plot(x,y_noiseless,linewidth = 1.5,label = r'Data no noise: $sin(2 \\pi x)$')\n",
    "    plt.scatter(x_train,y_train,color='red',label= \"Train set\")\n",
    "    plt.xlabel(\"x\")\n",
    "    plt.ylabel(r'y')\n",
    "    y_fit = p(x_train) \n",
    "    plt.plot(x_train,y_fit,linewidth = 1.0,label =\"Polynomial Fit\")\n",
    "    plt.legend(loc=(0.02,0.02))\n",
    "    plt.show()\n",
    "interact(plotmodel2,M=(0,9,1),log_lamb = (-40,-9,.1))  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we fit our model to the training data, we do a grid search through different polynomial degrees (M) and different regularization values ($\\lambda$) to search for the model with lowest error in the validation set, which again is the model we select. An alternative to the grid search is to perform a random search for the best set of model hyper-maraters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 7200x6600 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model selected was a  polynome of degree 9 with lambda = 1.026188e-10\n",
      "Root mean squared test error: 0.383\n"
     ]
    }
   ],
   "source": [
    "log_lamb = range(-40,-8) # regularization values\n",
    "M = range(7,10) # different polynomial degrees\n",
    "train = np.zeros((len(log_lamb), len(M)))\n",
    "val = np.zeros((len(log_lamb), len(M)))\n",
    "\n",
    "for (i,m) in enumerate(M):\n",
    "    for (j,l) in enumerate(log_lamb):\n",
    "        lamb = np.exp(l)\n",
    "        coeffs = polynomial_fit_reg(x_train, y_train, m,lamb)[::-1]\n",
    "        train[j,i] = rms_error(x_train,y_train,coeffs)\n",
    "        val[j,i] = rms_error(x_val,y_val,coeffs)\n",
    "    \n",
    "plt.figure(figsize = (24,22), dpi = 300)\n",
    "for (i,m) in enumerate(M):\n",
    "    plt.subplot(2, 2, i + 1)\n",
    "    plt.plot(log_lamb,train[:,i],linewidth = 1.0,marker = 'o',markersize = 12,label = r'$E_{RMS}$ Train')\n",
    "    plt.plot(log_lamb,val[:,i],linewidth = 1.0,marker = 'x',markersize = 12,label = r'$E_{RMS}$ Validation')\n",
    "    plt.legend(loc = (0.02, 0.075))\n",
    "    plt.xlabel(r'$ln\\lambda$')\n",
    "    plt.ylabel(r'$E_{RMS}$')\n",
    "    plt.title(\"Polynomial degree %d\" %m)\n",
    "plt.show()\n",
    "\n",
    "# Model selection\n",
    "best_M_reg = np.unravel_index(val.argmin(), val.shape)\n",
    "W = polynomial_fit_reg(x_train, y_train, M[best_M_reg[1]], np.exp(log_lamb[best_M_reg[0]]))[::-1]\n",
    "test_error = rms_error(x_test,y_test,W)\n",
    "print(\"Model selected was a  polynome of degree %d with lambda = %e\" %(M[best_M_reg[1]], np.exp(log_lamb[best_M_reg[0]])))\n",
    "print(\"Root mean squared test error: %.3f\" %test_error)"
   ]
  },
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   "metadata": {},
   "source": [
    "## Summary \n",
    "\n",
    "That is all folks. In this tutorial, we presented a gentle introduction to model selection, over-fitting and regularization. We saw how to design our experiment by splitting our dataset into a development set (train + validation sets) and a test set. This method is commonly employed when we have very large datasets that may take days to train. For smaller datasets, a procedure called [cross-validation](https://en.wikipedia.org/wiki/Cross-validation_(statistics)#:~:text=Cross%2Dvalidation%2C%20sometimes%20called%20rotation,to%20an%20independent%20data%20set.) is often employed. We also saw that polynomials with high degrees tended to overfit to the data and by adding a regularization term to the cost function, over-fitting can be potentially mitigated. Another way to avoid over-fitting is by collecting more data (see activity suggestions), which is not always feasible. \n",
    "\n",
    "The concepts explained in this tutorial are valid not just for polynomial fits, but also across diffrent machine learning models like neural networks and support vector machines.\n",
    "\n",
    "\n",
    "\n",
    "## Activity suggestions\n",
    "\n",
    "- Use more data for training your model;\n",
    "- Change the input signal;\n",
    "- Change the noise intensity;"
   ]
  }
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