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" }, { "metadata": {}, "cell_type": "markdown", "source": "$\\newcommand{\\reals}{\\mathbb{R}}\n\\newcommand{\\ip}[1]{\\langle #1 \\rangle}\n\\DeclareMathOperator*{\\argmin}{\\arg\\min}\n\\newcommand{\\E}{\\mathbb{E}}$" }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "import numpy as np\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport timeit", "execution_count": 1, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "np.random.seed(101)", "execution_count": 2, "outputs": [] }, { "metadata": {}, "cell_type": "markdown", "source": "# Introduction & Set-up\n## Re-cap from last time\n\nFor a real-valued continuously differentiable function $f \\in \\mathcal{C}^1(\\reals^n)$, the gradient of $f$ — denoted $\\nabla f$ — gives the direction of \"steepest ascent\". This is the direction in which $f$ increases most quickly. This statement can be verified by optimizing the directional derivative of $f$ to see which direction $u \\in \\reals^n$ gives the largest d.d. \n\nThe directional derivative of $f$ at a point $w^0 \\in \\reals^n$ in the direction $u \\in \\reals^n$ is given by $\\nabla f(w^0) u \\equiv \\ip{\\nabla f(w^0), u}$ and denotes the rate of change of $f$ in the direction $u$ when $\\|u\\|_2 = 1$. In this case, it follows from the Cauchy-Schwarz inequality that\n$$\n\\nabla f(w^0) u \n\\leq \\|\\nabla f(w^0)\\|_2 \\|u\\|_2 \n= \\|\\nabla f(w^0)\\|_2 \n$$ \nThe two sides achieve equality precisely when $u = \\nabla f(w^0) / \\|\\nabla f(w^0)\\|_2$.\n\nCorrespondingly, the direction $v \\in \\reals^n$ of steepest **descent** of $f$ at $w^0$ is *opposite* the direction of the gradient, *i.e.,* $v = -\\nabla f(w^0)/\\|\\nabla f(w^0)\\|_2$. Hence, if we're looking to minimize a function $f$, then a seemingly reasonable approach is to *march* in the direction opposite the gradient until [hopefully] we find the smallest value of $f$. \n\n\n\n## Model assumptions \n\nThis is a quick re-cap from last time and for consistency of notation.\n\nThe **data** is comprised of the **features**/**covariates** $X \\in \\reals^{n\\times d}$ and the **labels**/**response** $y \\in \\reals^n$. Suppose for the time being that $f(w; X,y): \\reals^d \\to \\reals$ defines a convex objective (\"cost\") function parametrized by the data. We think of $w$ as a vector of weights or **parameters** for a linear combination of the $d$ features. \n\nFor example, the objective function for $L^2$ regularized least squares regression is given by \n$$\n\\argmin_{w\\in \\reals^d} f(X,y; w,\\lambda) = \\argmin_{w\\in\\reals^d}\\big\\{\\frac{1}{2} \\|Xw - y\\|_2^2 + \\frac{\\lambda}{2}\\|w\\|_2^2\\big\\}\n$$\n\nSince the function $f$ is assumed to be convex, a global minimizer $w^*$ exists. To find $w^*$ we need simply start at some point $w^0$ and *march* for a sufficiently long time in the direction opposite the gradient (*cf.* picture above):\n\\begin{align*}\nw^1 &:= w^0 - \\alpha_0 \\nabla f(w^0)\\\\\nw^2 &:= w^1 - \\alpha_1 \\nabla f(w^1)\\\\\n&\\,\\,\\vdots\\\\\nw^{j+1} &:= w^j - \\alpha_j \\nabla f(w^j)\n\\end{align*}\nIt may not always be sensible while marching to take a step of size equal to the magnitude of the gradient. In general, one instead takes walks in the direction of the gradient scaled by some a parameter $\\alpha_t, t\\geq 0$. But how does one choose $\\alpha_t$? Should they all be the same? Is there an optimal $\\alpha$? " }, { "metadata": {}, "cell_type": "markdown", "source": "## Create fake data" }, { "metadata": {}, "cell_type": "markdown", "source": "Throughout this example, we'll be performing logistic regression on a two-dimensional data set, which will allow us to visualize the results. We'll be examining how long our code takes to fit and predict, by looking at both run-time and number of iterations." }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "n = 500\nd = 2\nX = np.random.randn(n, d)\ny = np.zeros((n,1))\nfor j in range(n):\n if X[j,1] > X[j,0]:\n if np.random.rand(1) >= .9:\n y[j] = 1\n else:\n y[j] = -1\n else:\n if np.random.rand(1) < .1:\n y[j] = -1\n else:\n y[j] = 1", "execution_count": 3, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "plt.scatter(X[:,0], X[:,1], s=10, lw=0, alpha=.8, c=y);", "execution_count": 4, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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XMmnSRpKTCxg7tg3vv98PrVZX6511SkoBp08n06tXE6NWOqwNycn5tGr1A1qtvoJfUtK7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\nGWRnF6K19iSzwA0PkUf3OgTPwYO96NzZhbi4TKZP78Fdd7V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24VunXza1bhqq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\nRK5CYII=\n", "text/plain": "" }, "metadata": {} } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('#{y == 1} = %d\\n#{y == -1} = %d' % ((y == 1).sum(), (y == -1).sum()))", "execution_count": 5, "outputs": [ { "output_type": "stream", "text": "#{y == 1} = 253\n#{y == -1} = 247\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "## Writing a basic Logistic Regressor class" }, { "metadata": {}, "cell_type": "markdown", "source": "Last time, we wrote a very simple gradient descent method. I've re-written it here in a slightly more general form so that it cooperates with what we'll be defining below." }, { "metadata": { "code_folding": [ 0 ], "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def vanillaGradientDescent(funObj, w, X, y, \n objectiveParms={}, \n gdParms={}):\n # optimality tolerance\n optTol = gdParms.get('optTol')\n if optTol is None:\n optTol = 1e-2\n maxEvals = gdParms.get('maxEvals')\n if maxEvals is None:\n maxEvals = 500\n verbose = gdParms.get('verbose')\n if verbose is None:\n verbose = False\n # initial objective and gradient value\n f,g = funObj(w, X, y, **objectiveParms)\n # this took a single function evaluation\n funEvals = 1\n \n # initialize the step size\n alpha = gdParms.get('alpha')\n if alpha is None:\n alpha = .01\n \n # Gradient Descent loop\n while True:\n # Step in search direction\n w = w - alpha*g\n # Update objective and gradient values\n f, g = funObj(w, X, y, **objectiveParms)\n # This cost one more function evaluation\n funEvals += 1\n # Test terminate conditions — largest magnitude of derivative\n optCond = np.linalg.norm(g, np.inf)\n if verbose:\n print('%6d %15.5e %15.5e %15.5e' % (funEvals, alpha, f, optCond))\n if optCond < optTol:\n if verbose:\n print('Problem solved up to optimality tolerance')\n break\n if funEvals >= maxEvals:\n if verbose:\n print('At maximum number of function evaluations')\n break\n return (w, f, funEvals)", "execution_count": 6, "outputs": [] }, { "metadata": {}, "cell_type": "markdown", "source": "### A black box" }, { "metadata": {}, "cell_type": "markdown", "source": "We'll also be using a function [translated and modified from Mark Schmidt's Matlab version] that can be — for now — be treated as a black box. This function is not available, to my knowledge, on his website; however, you can find minFunc [through his website](https://www.cs.ubc.ca/~schmidtm/Software/minFunc.html), which is likely a generalization of findMin. " }, { "metadata": { "code_folding": [ 1 ], "trusted": true, "collapsed": true }, "cell_type": "code", "source": "# checks if a number v is in the appropriate real range\ndef isLegal(v):\n # assumes v is a scalar or vector\n return any(np.imag(x) for x in [v]) and any(np.isnan(x) for x in [v]) and any(np.isinf(x) for x in [v])", "execution_count": 7, "outputs": [] }, { "metadata": { "code_folding": [ 0 ], "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def findMin(funObj, w, X, y, \n objectiveParms={}, \n gdParms={}):\n \"\"\"\n findMin computes the optimal parameter vector w to \n funObj given the data X, y and execution parameters \n objectiveParms & gdParms using gradient descent with \n backtracking via Armijo rule, setting alpha via cubic\n Hermite interpolation (I think)\n \"\"\"\n optTol = gdParms.get('optTol')\n if optTol is None:\n optTol = 1e-2\n maxEvals = gdParms.get('maxEvals')\n if maxEvals is None:\n maxEvals = 500\n verbose = gdParms.get('verbose')\n if verbose is None:\n verbose = False\n gamma = gdParms.get('gamma')\n if gamma is None:\n gamma = 1e-4\n \n f,g = funObj(w, X, y, **objectiveParms)\n funEvals = 1\n \n alpha = 1\n while True:\n # Line-search to find an acceptable value of alpha\n w_new = w - alpha*g\n f_new, g_new = funObj(w_new, X, y, **objectiveParms)\n funEvals += 1\n \n gg = g.T @ g\n while f_new > f - gamma*alpha*gg:\n if verbose:\n print('Backtracking...')\n alpha = alpha**2 * gg/(2*(f_new - f + alpha*gg))\n w_new = w - alpha*g\n f_new, g_new = funObj(w_new, X, y, **objectiveParms)\n funEvals += 1\n # Update step size for next iteration\n dg = g_new - g\n alpha - -alpha*(dg.T @ g)/(dg.T @ dg)\n # Sanity check on step-size\n if (not isLegal(alpha)) or (alpha < 1e-10) or (alpha > 1e10):\n alpha = 1\n # Update parameters/function/gradient\n w = w_new\n f = f_new\n g = g_new\n # Test terminate conditions\n optCond = np.linalg.norm(g, np.inf)\n if verbose:\n print('%6d %15.5e %15.5e %15.5e' % (funEvals, alpha, f, optCond))\n if optCond < optTol:\n if verbose:\n print('Problem solved up to optimality tolerance')\n break\n if funEvals >= maxEvals:\n if verbose:\n print('At maximum number of function evaluations')\n break\n return (w, f, funEvals)", "execution_count": 8, "outputs": [] }, { "metadata": {}, "cell_type": "markdown", "source": "### Logistic Regressor shell class" }, { "metadata": {}, "cell_type": "markdown", "source": "Below, we construct a class called LogisticRegressor which has fit and predict methods. One major difference between this and sklearn's LogisticRegression class object is that I more easily know how to alter these moving parts without breaking the whole thing. This will let us alter the gradient descent method and the objective function so that we can compare run time, iteration cost, *etc.*" }, { "metadata": {}, "cell_type": "markdown", "source": "The logistic regression objective function is given by \n$$\nf(w; X,y) := \\sum_{i=1}^n \\log (1 + \\exp(-y^i w^T x^i)) + \\frac{\\lambda}{2} \\|w\\|_2^2\n$$\nand its gradient is given by \n$$\n\\nabla_w f(w; X,y) = g(w; X,y) = X^T r + \\lambda w, \\quad r_i := -y^i \\sigma(-y^i w^T x^i)\n$$\nwhere $\\sigma(x) := (1 + e^{-x})^{-1}$ is the *logistic function*.\n\n" }, { "metadata": { "code_folding": [ 17, 45, 76, 85 ], "trusted": true, "collapsed": false }, "cell_type": "code", "source": "class LogisticRegressor:\n \"\"\"\n LogisticRegressor takes data [X, y] and optimization parameters and computes a \n logistic regression linear model to perform binary classification.\n X: an n-by-d numpy.ndarray object whose rows are samples/observations of d features\n y: an n-by-1 numpy.ndarray vector whose elements lie in the set {-1, +1}\n objectiveParms: A dictionary of parameters to pass to the objective function \n (e.g., an L2-regularization parameter lam)\n gdParms: A dictionary of parameters to pass to the gradient descent method\n (e.g., the optimality tolerance, the gradient descent step size, etc.)\n kwargs:\n intercept: whether to fit a linear model a.T * X or affine model a.T * X + a0\n maxIter: maximum number of iterations before loop breaks\n verbose: whether to print progress during the optimization\n objective: the objective function to use in the optimization\n gd: the gradient descent method to use in the optimization\n \"\"\"\n def __init__(self,objectiveParms={},gdParms={},**kwargs):\n # # # Optional Parameters # # #\n # Whether to include the intercept term \n # (default: yes)\n self.intercept = kwargs.get('intercept')\n # Whether to print out information along the way \n # (default: yes)\n self.verbose = gdParms.get('verbose')\n if self.verbose is None:\n self.verbose = False\n gdParms['verbose'] = False\n # Objective function for logistic regression \n # (default: L2 regularized LR)\n self.objective = kwargs.get('objective')\n self.objectiveParms = objectiveParms\n if self.objective is None:\n self.objective = LogisticRegressor.simpleObjective\n if (objectiveParms is None) or (objectiveParms.get('lam') is None):\n self.objectiveParms = {'lam': 1}\n # Gradient descent algorithm\n # (default: vanillaGradientDescent)\n self.gd = kwargs.get('gd')\n self.gdParms = gdParms\n if self.gd is None:\n self.gd = vanillaGradientDescent\n if (gdParms is None) or (gdParms.get('alpha') is None):\n self.gdParms = gdParms = {'alpha': .01}\n return\n def fit(self, X, y, w=None):\n \"\"\"\n LogisticRegressor.fit fits the objective function objective to the \n data [X, y] using w as the initial starting point for the parameters\n and lam as the regularization parameter.\n \"\"\"\n # # # Necessary parameters # # #\n self.X = X\n self.n, self.d = self.X.shape\n self.y = y\n self.w = w\n if self.w is None:\n self.w = np.zeros((self.d,1))\n # Whether to fit intercept\n if self.intercept:\n self.X = np.hstack((np.ones((self.n,1)),self.X))\n self.w = np.vstack((np.ones((1,1)),self.w))\n # # # Gradient descent algorithm # # #\n if self.verbose:\n st = timeit.time.clock()\n w, oM, fE = self.gd(self.objective, self.w, \n self.X, self.y, \n self.objectiveParms,\n self.gdParms)\n if self.verbose:\n et = timeit.time.clock()\n print('total elapsed time: %15.5g s' % (et-st))\n self.w = w\n self.objMin = oM\n self.funEvals = fE\n return\n def predict(self, Xhat):\n \"\"\"\n LogisticRegressor.predict computes the predicted values yhat from the\n data Xhat and the parameters w computed from fit.\n \"\"\"\n t,d = Xhat.shape\n if self.intercept:\n Xhat = np.hstack((np.ones((t,1)),Xhat))\n return np.sign(np.dot(Xhat, self.w))\n def simpleObjective(w,X,y,lam):\n \"\"\"\n LogisticRegressor.simpleObjective returns the objective value and its \n gradient given the parameters w, data [X, y] and regularization\n parameter lam. It does this without regard to number of computations\n required for matrix multiplication, etc. (hence, 'simple')\n \"\"\"\n yXw = y*(X @ w)\n nll = np.sum(np.log(1 + np.exp(-yXw))) + .5*lam*np.dot(w.T,w)\n sigmoid = 1/(1+np.exp(-yXw))\n g = - np.dot(X.T, y*(1-sigmoid)) + lam*w\n return (nll, g)", "execution_count": 9, "outputs": [] }, { "metadata": {}, "cell_type": "markdown", "source": "## Test run" }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "LR_vanilla1 = LogisticRegressor(gdParms={'alpha':1e-1})\nLR_vanilla2 = LogisticRegressor(gdParms={'alpha':1e-2})", "execution_count": 10, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false, "scrolled": true }, "cell_type": "code", "source": "%timeit LR_vanilla1.fit(X,y)\n%timeit LR_vanilla2.fit(X,y)", "execution_count": 11, "outputs": [ { "output_type": "stream", "text": "10 loops, best of 3: 40.6 ms per loop\n100 loops, best of 3: 2.28 ms per loop\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('LR_vanilla1.funEvals = {}'.format(LR_vanilla1.funEvals))\nprint('LR_vanilla2.funEvals = {}'.format(LR_vanilla2.funEvals))", "execution_count": 12, "outputs": [ { "output_type": "stream", "text": "LR_vanilla1.funEvals = 500\nLR_vanilla2.funEvals = 28\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "Now compare against the \"black box\" from Mark Schmidt. " }, { "metadata": { "trusted": true, "collapsed": false, "scrolled": true }, "cell_type": "code", "source": "LR_findMin = LogisticRegressor(gd=findMin)\n%timeit LR_findMin.fit(X,y)", "execution_count": 13, "outputs": [ { "output_type": "stream", "text": "100 loops, best of 3: 2.21 ms per loop\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('LR_findMin.funEvals = {}'.format(LR_findMin.funEvals))", "execution_count": 14, "outputs": [ { "output_type": "stream", "text": "LR_findMin.funEvals = 26\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "## Visualizing Progress" }, { "metadata": { "code_folding": [ 0 ], "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def binaryClassifier2DPlot(model):\n \"\"\"\n binaryClassifier2DPlot makes a plot of the two dimensional data X\n used by model, coloured by class membership y. The background of the \n plot is coloured according to the decision function computed by \n fitting the model to the data [X, y].\n \"\"\"\n increment = 500\n X = model.X\n if model.intercept:\n X = X[:, 1:]\n # Get the label values\n yu = np.unique(y).tolist()\n # This sets the axis limits\n plt.scatter(X[y.ravel() == yu[0], 0], X[y.ravel() == yu[0], 1], s=20, lw=1, alpha=.5, c='g', marker='+');\n plt.hold(True)\n plt.scatter(X[y.ravel() == yu[1], 0], X[y.ravel() == yu[1], 1], s=10, lw=0, alpha=.5, c='b', marker='o');\n # Fetch axis limits\n xLim = plt.xlim()\n yLim = plt.ylim()\n # Domain on which to compute the decision function\n domain1 = np.linspace(*xLim, increment+1)\n domain2 = np.linspace(*yLim, increment+1)\n d1, d2 = np.meshgrid(domain1, domain2)\n d12 = np.array([d1.ravel(), d2.ravel()]).T\n # Compute the decision function\n vals = model.predict(d12)\n zData = vals.reshape(d1.shape)\n # Set the colour map\n if all(zData.ravel() == yu[0]):\n cm = [0, .4, 0]\n elif all(zData.ravel() == yu[1]):\n cm = [0, 0, .5]\n else:\n cm = np.array([[0, .4, 0], [0, 0, .5]])\n # Filled contour plot of the decision function\n plt.contourf(d1,d2,zData+np.random.rand(*zData.shape)/1000,colors=cm)\n # Plot the data [X, y] over top (it has been covered by the filled contours)\n plt.scatter(X[y.ravel() == yu[1], 0], X[y.ravel() == yu[1], 1], s=10, lw=0, c=[0,.5,1], marker='o');\n plt.scatter(X[y.ravel() == yu[0], 0], X[y.ravel() == yu[0], 1], s=20, lw=1, c=[0,1,0], marker='+');\n return", "execution_count": 15, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "plt.figure(figsize=(10,10))\nfor j, model, pTitle in zip(range(221,224), \n [LR_vanilla1, LR_vanilla2, LR_findMin], \n ['Vanilla, alpha=1e-1', 'Vanilla, alpha=1e-2', 'findMin']):\n plt.subplot(j)\n binaryClassifier2DPlot(model)\n plt.title(pTitle)\n plt.axis('tight');", "execution_count": 16, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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+TcGE8TYTI3/Al8jtqnKbp/yAa0y6rzSx/P33Jo/XXJOcF7VVTara2mfWAcDN\nbwResB+4+uPJjeYrV2CarazKtnaFXaB4xXF9zTtuTwxg2vVp1chD0LO1Mm1UDM1TxrXUr/xtlNUv\nfc1qx/su/QL8GqaLdp7+0ES/zDIKZj6V5u/XAv/tyZNdz65dx0X1a8dOYOEU4MjDk8/zVhzX1+gN\nOKETT+oXE8ZbS4z8J5eY+Krc2u+b/fo+02jRSEvyPG17cuNs3Jjc+KPF6fC0rhiuReSk9wIHPgGc\nvRe4csfsAcBmQU0difKtz5uPOgQOTMYDzK7Lh6zT62sOfCJ5rcsfuChbdM4VQm+R8BAyRdP6ZV+T\npl+atER1XbTziZ9KzrsbLc4eYGxr5VaV6NdXdwBvXzab56QJ1S8gmfDpJbzNm4Crr3ZrVh79uuTi\nSU6pT8OoX8HUejAwSSGrurcLO7HbNevztXsw5fPd1qNZnVcLpGvWJgtJzZa9SEThuuuAXSuBo8bM\nyiw9MFybRHMWr01eD7ZOL93ZMzfzxrcPz9QztcOHk9PIdcTJFKkvfjH9tW8ZTl+zZk0iQMePp88a\nfaea5yFGG4TURVP65brGp1862dunYaZ+nXxpEm3ajtniw2b06iCS6BTGj7sBrB66y7GE6Je95JWl\nYZo8+nXzLemRL+pXEIw8tYXQGZs5SwqZ9fnaXQ3/mr/5np5F2cmWLlFTRxIDBEwnUpsM906iSXYU\nSieG2+UKXJjJlaZArFqVPLrOdDrycPpr141+9FiSG6BZsyYsgTOGaHRYeEjHaEq/XNf49MuONtka\nZuqXjja5zr0zI2V2tEsf+KtLr/jw6ZddpgBI16wi+rVwSraGUb8yYc7TvOGK+KQlPtrkzVEIvd6+\nzq4O7jqmZeu6SSTKVQwzD67Ik55RZeU4heQM6O29LU+GrAPmPGVcS/3y07R+wdF/aLu+a7SJ08ap\nCFnJ1mmalVe/9FJgTzWMCeN9I0ZiZtG29PXfB3BtYJuuo1VCz7nzkSUSdt2n2OQVrI5C85RxLfVr\nljbo124A58BdYLOMefIdSeUia7MJMH2kVEyKTBo7CBPG+4IZmnaFuNPqq7hqOOm27MJvaejrn5xx\njZmHYC7PafSSHDCd22QmhvuWwUISI3UxuY0b3e2UrTtiCk0PRYeQQujIUl79sj8vq18b4N6Np68x\n9WuDY2yuDTVmcvjinnSNydKwrGKYMfXL9Zrkguap7dghZ1fp/9Dvmq9NAcriKkznPF3luGY9pqvx\n6vwlYNoCWZqoAAAgAElEQVREaQNlGil9bejRK7oAmykm9jWhRw/MWyG3eRsvIa6iu67Psr5bVL/s\niJGvSKarmjgc15r9mkdS3fNxv/mx9WnhFOrXnEPz1FZ822mXMj53feY7n2nJ+o4POynSDnub4qSr\n8ZrY59Od2G23N3v7LgBcdNH09y+6KBESU0zM7bGarKMHqjxOoAqR6NnxB2TOcWlUiH65Ps86Ay9N\nv8yIETAbrTL7cumXbfCWjPdvGCYalnW8ia1hV19N/ZpzaJ7aStZOlLTPXTVSXIdemrjC1DamqKw3\nHu2ZpTZXrqNX7Pf18p4pLubOkZUrJucwAcCBA9OvTTHRO1jsHSiuuiMxznLyUYVIVDleQqogj0Zl\n6ZvrWjsqlaVf5qG9add+y7jGdzSMuVwH+PULmNUwE+rX3ELz1HayQtNpn7vqn9hRKPO1LUC2wHzY\neN+snWLO6rSwXT50lyEAppfsTv1gUnzSFJdly9wHZu7fD9x08/QMzRaqo8fcZ0S53quCqkSiJ4Xn\nSAfJo1Fpn+vntob59AvWa13c0p7w+TTMZ/CWEK5fwKyGUb86oV/cbdcF8mz1dZ0VlbaLxLWt2JXD\n4Pr+DcPktV2GQO+6u+03gZP+DbB36B7rvn3AeW92nwNlmqsi+I4hiLEDRW8Ddp0dVZYW7JDhbruM\na6lf+cijX8CshqXtlLM1LK3cgK8Ne3w3DJOI022/Cfz5+5NDdl0aZuqXrS1V6Zf+rIxGUL9OwN12\n80pWGFqTlXSp2zJnaweN9+0zpAB/uNolMCHF7jRmuYKXf2BSlM7F+ecn15rHq2hcBTjz4JoJxQpX\n79g5Ocwz9vp+B2ZspCfE1C/dnq1hvsmca8ktzSCFatjpaydLdS//QGKcfBpm6tfmTbMRptj6BcTR\nMOpXEFHMk4hcKCLfFJFvichvxmiTIFtUspIuTVx5UPp9k9WOa10Ck3aUgh7H5cPkUZslu86TZlGA\nNVcmz3VYe+1w+tiDK3fMhpDLhpTTqvyWadvMcejI+n7XoYZVQEz9AtwaZu+60/mcLg3L0i/9vj0O\n/Tt0JN1EG6dFSTSrKf3S7cXQMOpXEKXNk4gsA/BxAC8HcB6AXxKRZ5dtt9dk7aTT5In4aEyhMfsx\n8R1foPv2CY+dyLke0wnjwCQH6uY3Jgdq6vcueUoiBDffkpyJtHc4+XyrAq7fPD1rMQ+7LIMZTo+1\nJt/B9f0uQw2LTJX6BcxqmF2GwOzXvlZ/39eXuZnG/h06/UBr2MV3T2vYy1c2o19AXA2jfgVROudJ\nRH4WwKJS6hXj178FQCmlPmBdx5yBvLhyjsoeb2B/zzU7dB1fENK3awnw/3FUEdcitHJFcjCwWQ9K\nP9fr7ue+CXjNGbO5AuZ6f5m1efuYgphr8i1Y36+CruU8hWgY9asATeiXnc+Z1bfZv2tZ7xwk0Xi7\ngrhLw1z6teZK4NXPmNaBhVOSUgWasrlFVWkY9avynKdnAPiu8fofxu+RsviiRPZsLq1ek34tC7Pt\n2zM/cyuv3Yavb1NsPr8wPdtajelaThrzdHEzkdy1pHfLm2bX8zdunD3sUpuqPCFmV5g7pli84/Z4\nbZEqoYZVQZ36ZWqXK2KetjyoNczWr88vjDXMcUi5S8NM/dJ5Q/uvn87X3LwpMU6HD0/eK6pf+vqq\nNOzosdlTIsgJmDBehtCEyKLkqYuicdVvOm07cPpDyaMLLTbmVl57HHbfLkF61ZHkNPK/XztZ7wfc\nR7WY2PWg3v0Vd0Vx871bb50YqLSE7zQxqjo87TKDhLSJKjWsTv3S2uUbh+9oGFPD3jbWr9Ei8Jo1\nwCePJJ8N9wJ7bvQ0Dnc9u4VTZvOGTA1btSqpWQcU1y+AGtYgyyO08Y8AzjRenzF+b4bhcHji+WAw\nwGAwiNB9g+jExDrx1UUxQ856J4o+x+n9W4DRgWQd/sHLAHVk+rv2b/D9Ljsx0y5nsA3A0hbg/QuA\nwkSYXIdmmjOkreuS7byPXptcu3kTMNaYqaJwJu/8MvDXG6fbM83Vrl3JjC/r5PAdO5NrY4qOmRiv\nhu5DkjvKaDTCaDRqehh5CdKwzukXUL+GtUW/9PdMDdsG4NAWYPvCpI+0Q3+1hrn06+rzk+jSqlV+\nDdv08SQypdsqol8ANSwiefQrRs7T45D8U38xgPsA/H8AfkkpdcC6rjs5A2lbXmP2UaRNe11/G4Df\n2g6cfGkyqzq0Jb3PvL9Lz1x9tVP+cu0kN8BcftOisGtXUp13zZpJfpMvn8n83i3PmYTYgSTZ/LP/\nkBSs0+366qDUhRqGn7Y+R3Qw5ylTwzqlX0B7Naxu/QKSopvmgcFmnbo9NwLrXj+7HGZq0bJlfv0C\nps2PWUPppvMmGtZG/QI6qWGtyXlSSv0IwJsB3A7gHgDbbePUOYruEjHxhctdZqQI5s6SQ1uA+051\nC0+M3S9ppQ3sA39dB2Dq8PaBTySPvnymHTuBZS9OjBMw2R587jgv6pLfnWzdfcft8cLZRbfqunIl\nSOughiG/hmXpFxBHw7L0y+yz6G/6luN7uizButfPLqfZGmbrl++MOmCiYTedl7xeFGDzT8/q19Fj\nia7p9soaJ2pYdFhhvAxFo0NA+o4Qk9DZk2/nSeg47F1zrj5d79uVyu3PZSHJV9Bcd90kHO1Cz9Iu\nuTgRJT0Ls8VDz4hWrkjC5SbD8Q6/k94LrFgBHHnY3VcI9k4W0rnIUwid1C8gXoTbfH83ikWw7e9s\n8PTh67OIfunvmxpo1nPyVfLWumCjI+UrV0x0zrebTg0TjXLp19Z1k8+BOBXDqWEAWhR56jVFI05Z\nh01qQmZPrlmXJq3wnD0O+6wnF+Y1rt+hv2fujFFHJkXj9AxKF3czd5xo9Czt5luSawB3xdzh2mRd\n3mecgOSzpSP5EjBNOniYJSFTFIk4ZenXBkzyi0I1zN45Z+Y/hWiYea5daDK6/VuAxDi5zqUz9QuY\n7Kaz0ZFyrXNplbpHZ83ql14i0wnoj14L3HaUGtZCaJ7qJm2Jyy4bEIJ9GOZuq52QcWh8QmWLzFWO\n8erf4doZ86TnJJV47VnPqlWT5/bOOY3vxt+6bvbgYT1jO/WD42tUkmSqv5v36AIWiyNkmlD9WkI+\nDdPoCFKohtnthOjXeswaNj1ml0bs2DmrX2YVbmB255x9jcu4rHv9rH4BwP/xqumSCdSwVkLz1BQ+\nYTHLBqSRNetyHdjr+77GN0u0Z4V2cqVGFpLETiB5lIXpfvQ2XmC69sm+fZNIkylQ5o0PAFde6f4t\nWnR0ftXVVyeHdppcdFGxGZjrNPOs73J2R7pOln4B+TTMjl6bbbk0rIx+LWHWsAHA7793ViNcZQiA\nWf266eYw/XJpg0u/Dh9OSiYcPZp8pqP0dWgY9SsImqemsG/yrJCz6/umULlmXWmF4UIjT+uN9w5i\nNpxujlcdAR77TvL8se8kr5cwPbvSO+/MfIFdu5JH16zoi1+cPF+1CvjlX5q9Rm+hNdv98/cDH/3o\n5Jo1a/zRrSxcxyr4Zn6xDhcmpM2U1a9Q0gprhkbOrxo/6uVAl2G7YZjc59qkHD48KUMQU79c2uDS\nr1WrgF98IrBy5eQ1UDyKFKph1K9gaJ7aQsjOFJcB8uUWuELfLmE5iNlZmT0ubcxc23lNZAFYPi6X\ns/zMSe7T5cPJ7CrvGUxHHp7OjTrnHOBtvzFpy5WfoNt98PD0zO/48ekZmFm4M2S2lZU/wPwC0ldC\nd9bl2d3rihjZGpalX0CiW74NNWaC+MoVE5OyatX0/RtLv4BpbVg4ZfJ+ln5t3DgbRYqpYdSvXMQo\nkknKYBeHS9uZ4ir+ljXTs0Pf9vXaEOmokrljxRQbnQS6YXytrACQY+bzvpcBm62ib6GF3T7yUeDt\nVwGPf3zyetUqYMulwLPHZ7eau0h27Exme8fG7e7aNVt8TjPcOwmVh+xG0eKmr7XHnvU5IV0jj34B\nbg3Lyo3SurOE2e+vdlyjMes3ad1aPb529TIAx6d31qVRVr+efjpw+eWT91asSJbxVq1KzNVHxlHy\nEP0y+4ypYdSvXLBUQVvwbf0F/EXcQnDNuLLyocz2t1nvme391vbZ2iunGQXtdDVg/Z5JnqJvumxB\nGq4txFpIbGExq+cCSW7B3mH4uLLOj+rogZoalirIuJb6NY2rdl2ZEizAdAVyG7ttsxSBOY6L7541\nGq5Ddl2lVfLol6uswcGDwGrjB3zoQ0mUyrxWlzhwGaMqNYz6dQKWKmgLvi23QPrsazWKF7SzcwP0\n9105A3ocdtKmntXZQvb+LcAGK7SrC9oB4113O2aN0/797pvTFSa2d7W4MI8/MEVKh57tUPf7Xjad\nx3D95ul2ssi6psPCQ3pMUf3SEaEYBXj144cxvVyXxkFjHKZ+3fKc5BxNE1MrdP6PbXx8+gWkL+Wb\nrF49nV915OHZa9esSSaOtn6tXDGbixVTw6hfQXDZrk7SzpFyvW+HxA9an+UxUVpg0nbl+WaDG6xr\n0iJPmhO77jYBj9wKnHzR5DNd98Sc9fnCzmYo2WT//uRwYH2NftR1VWxcR8MMx/kCeULwi3tmz3jq\n+EyNEADl9UtPxMzPQzVst/F4FRLzZL6vz6ZzTQi/hdmdeYA78gS4J2Im2tTYhS9dGmbql61N118/\nXcTXpXVm3ShXHzoXK1TDqF/RYOSpDnyF5bKwZ10fNj7Le/SBjjbZsy/9ns84+eq3LK3wGyc1Pp0c\nGJ9H9YvAI5ZImQmJWYmKO3YmYW2dPKnD2UePTd/0urjmmnFJA3sWZvfzvpdNPgsVD/uUce5OIV0n\nln5pDdLk0bAl43G19b5uxxeBsscxXAu850VhuUFA8mgXxNQGSpOmYTpydOutszt+zdMPdJR82Yvd\nO4NdfZhGKETDqF/RYOSpDrRopOUFpGHP1vIkaLq+p79r5g34EspdY1gCZpLFZQF41ZHJ9w9tmT4B\n/dBm4OMfTGqYuEhLVNSzLR3mPn589vvmjOzi5wAnPSM7oTsPrlPG3/ey9GROQrpATP0CimuYmfu5\nDcD3ATzZaidrHL4EcR19MSMzdjRn5YrJIebA9LmbWcnWZt7U/v3puVbvGgJyy2xEyNVHaNSI+hUd\nJozXSd6ltjSKCpmeQdonlxdpS6MTwhcFeMPCxDDZ3DCcDTubr22hOnosyUk4/uXZtvTRLVpArrkm\nOUJhsHVyzXteBPz2C2e/ayaB5j3vyT5lPG2XS5Fw+ByE0JkwnnEt9SuMMhpWVL9c5sm8h295zvT9\n7cI8dxOYTTA3NewdtycmxTwfD5gkdWvz5dIvbXBsPdB9FDmzjvrFhPG5JKbwhB594PqeHckq2haQ\nRJwuPJAYJwD45BHg4pQQsF6C27VrNgwNJDMkM5T8rjsnIqUjT/v2JeZHX6NnZHuHSR4DkBxrcPzL\n/nB00Xom9injruq9QLFwOEPopM3E1C+guO7YuZ+h7biMk9agtcPEOAHTlcRd3HrrRMNsHXnH7clr\nfS8P985WGtdRI33Nxo1u/frhH7v1wM7JyqNh1K9o0DwVJXTdvyp8CZoh37PzDsqIojoCfOnc6bOY\nvrpj+nBgE338gCkawERQhntnxeyW5yR5AB/5aHKju0RLi8CuXcm2XY1LWMqc92QnW+r2TIoIGwvU\nkbppm4aFjsfO/QzRL99ynTnxMjXMzIc00Qbh6qv9+rVyBXDljsmEUg2B896c6JM2Kvb9vmvXrH7p\ngp0xNYz6FQ2ap6LkTdiOQZa45BmTLwnUZ3rSPnvwsuRR3/SP3OpeunOVEtCicd6bJ2dIadHRszAZ\nTpbfdJK4mTxu7rY7emyybReYTbjU+GZcabi2IbsoImw8wJPUTd0a1pR+mcbJdc+ahXNHi8khv75S\nKiH69ei1wAMPTMyYPtbF3ODi0jBbv8xIe50aRv0KgjlPebETr/MUrCyLb20/z5jsvAWzTbPA5Uzx\nyx3jsgOOz8zvPvZdYPkzPW1YxTJd6+x6Td6VwGliF8O0kx3NRE7z+rw5TnZ/vsKbLpgzwJynNtKU\nhjWhX6ZxMnOVfAnbhw8nER/fff2235hEhL75TWD7TdOfmzlFehnPpV9mn8BkB7FG60BamYK85NUw\n6hdznqJStNhbGUK2CoeOyVdtXBaM2kyXTs/gtHGyPzOvObQF+N5TE+PkasNsX/PM8bVmfoFek9c3\noEt4XDNAe63dnOWVDSnb3184Jay9IiLScuEhHaBuDWtKv1zGCZjcs3YU50MfSl8qM8++A5Ljod72\nG279ApJ72WecfAUxze+aj01oGPUrFZqnopRJss6LKXa+w3v1Z4D/ZHGXgOm27NpMetlNFibGCUjq\nNekjV05/KHnUHH/A3YbdvmbVquQmNmuP+MTG5Oix2cM2gXQRKBNStr9/5OHehahJB6lLw2yzZuOq\n02RrWF792maURFk4Zbo45f790xtONFn3tZ34DSQalle/gKR/G7P0gQ01rHVw2W6e0OHtK6zXGm2s\nXOFx/V7Wtl6xSg2YS22P7EzqNclCYpw09506/R27DZM//GByQOaqVcDP/hbw8g9MPhuuDROflSuA\n245OznLSlXvTwtllQ966X3tpsIeiw2W7jGupX270kpt51pxvuc5XjiBEv0zjZC/H6ZMJzNIB9llw\nWee+Acmhvhs/MltaoIh+HTgAnHtutjZRw6IQS79onpomtHZKiNC4roH13kFM71ZJI80kTZkqTx6U\njf7Ovn3JyeG6uq5deyQENUwSMl31VWx0HShNnkM989IDMaJ5yriW+uW/NmuyZ19jHwAcol96uc6+\n7/Xhu4D7APEsbPOycErSXtX65fotVWkY9WuKynKeRGSziOwTkR+JyPPLtNVbQneYpOUp7E65xn4v\n1DgB/qU8YLLDDpjNb3Jh5iScfz5wzLhBdZ5A1jr+4p7kj7kr77w3J8+zbviqQtTmmHtW56QLUMNK\nkveIFZeG2acX+A4ADtEvM8/JtVSlMXfYheQQuXKOdHuh+gUAe24spl+A+8iWslC/ClP2eJa7Afwi\n3McukjTMGVaeIwpceQr29w46rima32AfsxKCa9lOGzEdeTJv/q3rgJvuBw58In0WONw72fYbOtuz\nj3aJiV0dnUcdzCPUsCIU1S9gVotc37M1LES/XLWc8hz6bRJyNIomVL8AYHBvolt5olX2rryiS3a+\ndqlfhSgVeVJKHVRKfRtA9+PUsQvKFd3xEnLdasd7eXbU2FEkl3HyRaVcieSaQ1umi8RpVq4ALvnd\n5LlrFmhGm3T1X7tSrgtztpi2iyYveqeOXR2dCZhzR280rC36pb+bha1hWd9Jq+XkuhfTErB9ERiz\nrlIe/QJmNWx0VsYPMtq2d+XFKEBp77YDqF85iZLzJCJ7ALxdKfX1lGvmO2eg7PlvPvLkDIS05cuL\nSkviNjFrpeiIU9p3zc+yEsk1P7h7MuM5783Tu1VufiNw6VPdfRXJLdBblLPqt4RiztaA2QRO5gxM\nMS85T1kaRv3yEFO/dHsuDUvTINM4ZZ2VaeNKoM7KLTL7yKNfQDkNA8rrF5Be04r6NUWafmUu24nI\nHQDMfw0CQAG4VimVazFoOByeeD4YDDAYDPJ8vRnKhKdDiN2Wa7edXTzOJ0R2rZSTLwUe+w6w/Ex/\nUrjZzhM/NXlu50iZfZgznuteNlmGO+m96TduSLTJZPOmRHR0ATpX4blQ9GxvauzXhQl03n7mWLxG\noxFGo1HTw5giloZRvxzErhG1hMl4fcUvTf2yI07m/Zk2ydHY95pZQsAVgbH7MPXr1A9O51W5KKNh\nt95aXL/02PW4gcQ4mUn0sZhjDcujX4w8hVLVzC0mrlnb561o0CM7wyqFu/BFkoDwqBMALDyY3LiH\nDyfn1W3elJwFdf3muFtwXTPIItt9005R18TYRhyjjRpg5CmjLepXMfLq1we+OduGmR/kIm2XWuiO\nNl1lXOsXEJ7z5MNlOFzj2bixmEZQw07Qit12dj8R22ofdRbFzItZMM7OQ5jKTdrprhRucmhLYnz0\ndx77zvi7nkiSJm1nnoksTPKPdJHM889Pap4UzUdy5Si4chryVulduWK2MrE+y8oUhhiHYvbwYM0W\n0l0Na7N+AZPlvzz65bpHdF6See+H5vKEFKI0q4yvWjXJfzzwieS9IveuL8fKHo9uP7Qf/Tk1rBJK\n7bYTkYsAfAzAkwEsichdSqlXRBlZ24gdng4lJKfAPlncFsqpHXNGCNxncNSR6e8se0pSPTyL0J15\n5q47XenWtYslBNfp5LoNe6dN2o4ZG9cM1jyE2CRPuz5itEFy0xsNa7N+AdMaFqJfaffI0WOz975Z\nVy6NkN15rvu06L2bpl+u8eTVL11AWEMNiwaLZLYJl9CkhdvTEsTTCE0eB9IPCw5F92e29ZG/iVfp\nNm+oOG9BOmD24M4i7YZcPwf5Al1ctsuC+hWIrWFZy4VFNOz3M3IjbWJV5jaXzFwGq+i9W7V+aQNF\nDQPACuPdJK3SbpqohOYz5DFN+vrQPCZf2ycM087pM/LKJipWfcyAb9dOFe23NDfAB81TxrV91S8g\n7OiVtO9lkdc4heYxpemHKwp93XXJY1FdqFO/QiqYx+hjTmhjzhMpynrMHnqZp45KSD5DWv0lH6F5\nTL62p3bvbUoMFJDkUV199fQav+sEcx++HKeYmDVdygik7/0e5QaQHrBhxbSGAfnqQIVo2A/uzl8B\nOySPKa2ytl1nSbfjOljYrv3ko279KtM+NcwLzVMb+Op2YDR2w/bsK0RUsoTJLkHgO0olLYHct2SX\n1rZtvg5tBr731KT0ATC56WwxCRWzqm/arMNB00j7DWVPSCekTZy2HfjLo8B/Hf9bN6NMoYnqIUt1\nIfd9WhK5KzqSpSf2varLk9jfMe/3LugXQA3LgOapabT5GGxNDJQsTIe7Q3KYsqoHh0SQ0iJTeXfZ\nmeOxzdfxBybXu3aQ2JVvF06Z7q/pmzbk/KcQgUwTdELmBXPy9JKdwNKK+Pp1w7B8BCktudxud3HP\n9DV2FDprF9y86xdADQuAOU9NY+cGmIQmgJfNeZIFYMt/SsoFAOm5TT7MHXkh4zFzF+y187QKuJoy\na/hlisyFnmw+p/kAITDnKePaPukXAFy8KTFONrH0yy6E6ctZuu3oRMPS7k0XC6dMcjBDq4CbY3Gd\nPFCVfhX9fh79AjqrYUwY7xqmgISaoaK77Xz9b1XTx7KEopPCL9g8LaJZ40kTxYVTkrwoTV4x9FFW\nEPJ8v+W7TopC85RxbR/1SxaATx5JdCumfrkO+/WhhomGhewqM9H39LlvmpxRByTVwLeuC2/HvN+r\n0i9zvEU0LPbOvjmECeNdY7fneRp5ksp9mMnqiwK85JbJ8p0vN8rEDNt/dQfwhoXp8Sx7Stg47BtU\n138C4oW3Y+Qb5AlVd0x0CPGijkx0q079AqYP3V0U4I03JyYh5P42NeHAJ5IjooAk8qSNk7305sO8\n36vQL6C8huVdaqOGeaF5agtLnueatLyAMtWDTQF7w0JigIDEEIXsznPlPOnxPOVe4Gn3J49F2LET\nWPZi/42eVzjMfAO7WFyetigohMyyZD3a+DTMp1+hUaet6ybLbFvV5LSCkNweVw6Sef7c234jiSC9\n7TfCxmKijcp5b3Z/XmTy5suZon7VDs3TvODLiwLyz9hcEaXdmDZCmrTdeRo7KVxHnPSuuuVnAo87\nO+cgx7zrTvf7oYmPNjt2TorGhezuI4TEwadhLv1KM04+ozBcOzEWmpDojB2NMSNO9lFSeTl6DBju\nnX0/puZQvxqB5qntuGpAFUUW/LvqtIA9eNn0+4/sDMt/sq85/sDkXLzjR4Cn/l2+GlNmKF4Np3fA\nlAldr1wxfc6Tvbuvh/VKCKmUvBrmM06usiYmW9clZQRMfMeR2LiuOfJwkvSteUXOU3t8GlZWv9J2\nJ1O/aoPmqe3EygvQpimr3pN9EOehzQU7BPDAWcD9Pw4sW0jv04UZijfzDwD/ob8h2N/Nk5tAYSIk\nP3k0zGectGnKMgr20nyexHEX118/eZ7XnPg0rCn9AqhhEeFuu3kh9IBNF/YxK0D2WXV5j3JJI+18\nvKy8hsU9ieiknaFUZPdJ3uMROrpttwjcbZdxLfXLTZaGpUWc7LMms+7DmLvEyt77i3uA972sWf0C\nqGFjuNuub4Qe+OvCTupOqxhuficWWVXK02ZDW9f5Q/V6xlYkbG0Lja92jH5kaJyQcmQZJ9995ary\nnfU//5hJ0Vk71LL04J6PN6tf+jk1LCo0T10h6+w608DkMUa+/ITQ5TeNr8+sZMeQm76KLcHmuJqu\nCkxIl7lhmK0DRc+atKuFa4rs1HUxD/oFUMMqgOapC4SeXVckmuTaIVPkkGEXspAvh8HeSaMFAoh7\nTIBL8Hp+FAEhlREaFSnyP/wqd7qFjLst+gVQwyJD8zTPaJMUcnZdXnw7ZEKNWgjmuNNmQ/qmByai\nZwtETHyzNM7WCImLPqYpdlSkip1uNqHjbot+6c9IFJgwPq+4krDLJnm7vu86auG0HcDJm7KTzkNY\nD2Dze7Nvate5TBs3VpsA2cGjCWLAhPGMa6lfYfzg7un7N8b9ZrfhOqcuduL0u78C/PYLs8dF/WoF\nTBjvM77oTxnj5FuKs6v/nrZ9bJx2ljdOQLIsGFqHxZ5N7dgJfOhD1YWhKTyEVINryb7s/eZajjOr\nhVeFr5CvCfWrc9A8zSOxl+nSluKWfNdtKrZkp79jLgvaRTB92Gv2mzclRyeEnmNlwt0mhDTHtqvj\nLtX5luPsg33LLtuZ16cV8nWRpl9223nGQRqB5mleydr+n4dQM1bWtJnRLbNwnl0EMw3zLCdTBPMk\ngGYljIYYOUJIMXQ9p5gJzKH5R2Xyq2zdSCvkmzZOYFa/Lrk4XMOoX62glHkSkQ+KyAERuUtEdorI\nE2INjAQQal5CjnQJNWNFTZsvulX0UGNTBDUhM8mQmadrhw7pJNSwmrELYYaYl1AzEGrGipi2NN0o\nsrRiKgcAACAASURBVDRoV0I3j4xK0zDqV2soG3m6HcB5SqnnAvg2gHeUHxKJTtqhwiahZqzIMqEv\narUE4PJh/vaAiQjmmUmmzTzzhuFJF6CG1UXWaQI+8piB0EhS3mXCNN0IjZrbaP26+ZZwDaN+tYZo\nu+1E5CIAm5RSr/F8zt0qdbMe08ZpN4of8VIGcxefb0dgUWHViaZ5E07Trnft0CEn6OpuuzQNo36V\npMj9vbhn2jgN1xY3KmUpqjOh7drP83zPhvqVSiz9Wh5tRMCvAShZNZFEZWn8x1VuoC7skgoxj30p\ns+U4TaTq2KFD2gg1rAqKToy2rkv+qCFwUkA5k6qo6kw4u908v4/61TiZ5klE7gDwVPMtAArAtUqp\n3eNrrgVwTCn1ubS2hsPhieeDwQCDwSD/iPtMkTpOp20HRgeA086Nk1yeBzvP6cHL4pkne+1/1654\n4trU7LYDjEYjjEajpocxRSwNo34VwDRORaM2N78xSZBu4kDbqnSG+tVK8uhX6WU7EflVAP8bgBcp\npR5NuY5h7zK4imJmIQvJ7jbNfafGjfyEEDruIrNTnhLeCF1btgvRMOpXQfR9XfRedRWXrDsCVVfk\nidRCLP0qZZ5E5EIA/yeAFyql/iXjWopPUcqYoCKmq8j40sYTUsSzbM6T7zWJTpfMU6iGUb8KoO/p\nsgaoapMRohl6V1tsbXH1TQ2rlLbkPH0MwEoAd4gIAPy5UupNJdskNnqnmjZBeaJHh7bEXS6zCTFn\nT/wDoyr55rj9myLDmRzJDzWsCszJkN4hpu/NvMZgx864y1omoZpx0UVJOYHY2mL/JmrY3FCqVIFS\n6t8rpc5SSj1//IeiUxVlimJWZZxCDgnW5+AByeNpO8LaLlL5N9aBn6Q3UMMqwBVFLlsQswrjFKoZ\nl1wcXofJbLuq8ZBWwArj80Td+UpZx69kVRyXhYlx0oQc65JVQddF7JPZKVyE5Cdt+b2Jpai0+zhE\nM1aumBinUIroV+h4QqF+VU60Ok+ZHTFnYL7IkyuVlvOk23nsu8DyZ6a3d8OwfH5ErANGGTr30qWc\np1CoXwEUzVusitD7OEszdDtAWFtlE9zLahj1K5VY+sXIE5klZDnOJC0ippcbHzgzbNmx7OyrqgNG\nCSF+2mac8tzHWZqhlxxDlh1jRI/KRpyoX7UQs0gm6QplEtR97ZmPWVSZIJpF2eRWQvpG24wTEP8+\nzvN96lcv4LId8VOkKGcZ2iTC3C7shct2Gdf2Tb/adN/a9PU+7uvvDoDLdmRC1rJaUepOUC96QHAV\nUHgIySaGcapyaamv93Fff3eN0DzNO6dtTwponlbhkVxVmTNCyPwSwzgV3ZmWF+b+kMjQPM0zeRO7\ni1CHOSOEzBexIk51JDfXZdBIr6B5mmey6iyVpQ5zRgiZL2LlOMWuzeaCu89IRdA8zTtlKo9nUbU5\nc9GmvCdCSLWUrTyeRR0GjfQSliroAlmmpsyuuarPxiOEzA9V7KwLMTRldo81WTqAdBZGnrpOjJwl\nGidCSFMlCWLkLNE4kcjQPHUZ5iwRQuYZ5iyRlkLz1GWayFkihHSPpqJOzFkiLYUVxvtA3ZXCY9Dm\nqsU9hxXGM67tmn614V5kxWwSCVYYJ+H02TgxzE9IcdpgnID+GifqV2uheeoz65seQMWwOB4hxWmL\ncfKxuKfpEVQL9avV0Dz1mQ1ND8BDjFpPTDQlpDhtN04AMNzb9Aiqg/rVemie+sh6ANvGz7ehmxEo\nJpoSUoy2G6fFPYAaJs/VsJsRKOpX6ymVMC4i7wbwCwCOA7gfwK8qpb7nuba/CZdtZRuAK5oehIdY\nAs5E0+h0KWE8VMN6o19tN04magjIsOlRVAv1KzptSRj/oFLqp5RSzwPweQDhoyLNs7vpAaQQ65gW\nCg9Jhxo2rwzXNj2C6qF+tZZS5kkp9ZDxcgHJ7I3MC0tND4CQZqGGGcxT1AkAtq5regSkx5Q+205E\n/jOA1wJ4EAD/NZM4zJuQk7mFGgbeb4TkJDPyJCJ3iMhfG3/uHj9uAACl1DuVUmcC+EMAb6l6wKQn\nxFq2I72HGkYIiU1m5Ekp9dLAtj4H4AsAhr4LhsPJR4PBAIPBILBpQsg8MBqNMBqNmh7GFLE0rLP6\nxagTIQDy6VfZ3XY/oZT6b+PnbwHwH5RSl3iu7cduFeIn7zExFPVW0rHddkEa1kn94v2VH+5+m3ti\n6VfZnKf3i8g5SJIs7wXwhpLtka5y2nbg5EuTA4oPbWl6NIRoqGEkjM2bkoKV+/YBO3Y2PRrSMKXM\nk1Jqc6yBkA4jC4lxApLHBy+bv/P2SCfprYYx6pQPu+L3rl2MQPUcVhgn1aOOJBEnIHkMNU5MGick\nPjRO+WHFb2JRulQBIUEc2sKIEyFNQ+NUnB07GXEiJ2DkidQHjRMhzUHjVB4aJzKG5okQQgghJAc0\nT4QQ0nUYdSIkKjRPhBDSZWicCIkOzRMhhHQVGidCKoHmiRBCugiNEyGVQfNE2g1rPRFCCGkZNE+E\nENI1GHUipFJongghpEvQOBFSOTRPhBDSFWicCKkFmidCCOkCNE6E1AbNE2k/TBonJB0aJ0JqheaJ\nEELmGRonQmqH5okQQuYVGidCGoHmiRBCCCEkBzRPZD5g3hMh0zDqREhj0DwRQsi8QeNESKPQPJH5\ngP+zIIQQ0hKimCcRebuIHBeRJ8Voj5AZuGxHKmSuNIwTCUIap7R5EpEzALwUwL3lh0MIIfUyVxpG\n40RIK4gRefoIgKsjtENIOow+kWqYDw2jcSKkNZQyTyKyEcB3lVJ3RxoPIX74Pw8SmbnRMP7bJ6RV\nLM+6QETuAPBU8y0ACsA7AVyDJNxtfkYIIa2BGkYIiU2meVJKvdT1voicD+BsAH8lIgLgDAB/KSIv\nUEo94PrOcDg88XwwGGAwGLj73KayhkX6yLamB0CyGI1GGI1GTQ9jilga1qh+8d8+IZWTR79EqTg3\nuoj8HYDnK6UOeT5XsfoihMwHIgKl1FxEc9I0jPpFSP9I06+YdZ4UGPImhMwv1DBCSBDRIk+ZHXHm\nRkjvmKfIUxrUL0L6R12RJ0IIIYSQzkPzRAghhBCSA5onQgghhJActNI8NbHVmX2yz3nqr099ziN9\n+G/Th9/IPrvVZ8z+aJ7YJ/ucw/761Oc80of/Nn34jeyzW3123jwRQgghhLQVmidCCCGEkBzUWuep\nlo4IIa2iK3Wemh4DIaR+fPpVm3kihBBCCOkCXLYjhBBCCMkBzRMhhBBCSA5ongghhBBCctB68yQi\nbxeR4yLypBr6ereI/JWIfENEviQiT6uhzw+KyAERuUtEdorIE2roc7OI7BORH4nI8yvs50IR+aaI\nfEtEfrOqfqw+PyUi94vIX9fU3xkicqeI3CMid4vIr9fQ50ki8rXxv9O7RWSx6j7H/S4Tka+LyK46\n+usKdWkY9auSvmrVMOpX5X1H07BWmycROQPASwHcW1OXH1RK/ZRS6nkAPg+gjv+otwM4Tyn1XADf\nBvCOGvq8G8AvAthbVQcisgzAxwG8HMB5AH5JRJ5dVX8GN477rIvHAFyllDoPwM8BuLLq36mUehTA\nuvG/0+cCeIWIvKDKPse8FcD+GvrpDDVrGPUrIg1pGPWrWqJpWKvNE4CPALi6rs6UUg8ZLxcAHK+h\nz/+qlNL9/DmAM2ro86BS6tsAqtxC/gIA31ZK3auUOgZgO4BfqLA/AIBS6k8BHKq6H6O/7yml7ho/\nfwjAAQDPqKHfh8dPTwKwHECl22bHJuCVAH6vyn46SG0aRv2KTu0aRv2qjtga1lrzJCIbAXxXKXV3\nzf3+ZxH5DoBfBvDbdfYN4NcAfLHmPqviGQC+a7z+B9RwUzaJiJyNZCb1tRr6WiYi3wDwPQB3KKX+\nouIutQlgbZNAmtAw6ldUeqVhHdcvILKGLY/RSFFE5A4ATzXfQvLD3gngGiThbvOzKvu8Vim1Wyn1\nTgDvHK9vvwXAsOo+x9dcC+CYUupzZfsL7ZPEQ0ROBbADwFutCEAljGf7zxvnmNwqImuUUpUsqYnI\nqwDcr5S6S0QGqH7GPzfUrWHUL+pXFXRZv4BqNKxR86SUeqnrfRE5H8DZAP5KRARJKPgvReQFSqkH\nqujTwecAfAERxCerTxH5VSThxBeV7Su0zxr4RwBnGq/PGL/XOURkORLh+axS6o/r7Fsp9UMR2QPg\nQlSXj3QBgI0i8koAJwN4vIh8Rin12or6mxvq1jDqV630QsN6oF9ABRrWymU7pdQ+pdTTlFLPUkr9\nOJJw6fPKGqcsROQnjJcXIVn/rRQRuRBJKHHjOJGubqqKIvwFgJ8QkbNEZCWALQDq2qUlqDc68mkA\n+5VSv1NHZyLyZBFZNX5+MpLoxjer6k8pdY1S6kyl1LOQ/He8k8YpnSY0jPoVnaY0jPoVmSo0rJXm\nyYFCPf+Y3i8ify0idwF4CZLM/Kr5GIBTAdwx3kL5iao7FJGLROS7AH4WwJKIRM9TUEr9CMCbkezG\nuQfAdqVUHWL+OQB/BuAcEfmOiLy+4v4uAPArAF403nr79fH/UKrkdAB7xv9OvwbgNqXUFyruk5Sj\nDg2jfkWkCQ2jfs0PPNuOEEIIISQH8xJ5IoQQQghpBTRPhBBCCCE5oHkihBBCCMkBzRMhhBBCSA5o\nngghhBBCckDzRAghhBCSA5onQgghhJAc0DwRQgghhOSA5okQQgghJAc0T4QQQgghOaB5IoQQQgjJ\nAc0TIYQQQkgOaJ4IIYQQQnJA80QIIYQQkgOaJ0IIIYSQHNA8EUIIIYTkgOaJEEIIISQHNE+EEEII\nITmgeSKEEEIIyQHNEyGEEEJIDmieCCGEEEJyQPNECCGEEJIDmidCCCGEkBzQPJEpROQcEfmGiPxQ\nRB4TkWsLtvM6EfmTAt/7goi8pkifhBBCSB0sb3oApHX87wDuVEo9L0JbSj8RkeMAHgDwdKXU8fF7\nywH8E4AfU0o9DgCUUq+M0C8hhBBSGYw8EZuzANxTUduHALzCeP0KAD+oqC9CCCGkEmieyAlE5MsA\n1gH4+HjZ7g9F5N3jz9aKyHdF5CoRuV9E/lFEftX47pNEZJeIHBaRPwfw7xxdfBbA64zXrwXwB9YY\n9ojIr42fv05E/kREPiQiPxCRvxGRC+P+akIIISQfNE/kBEqpFwP4EwBXKqWeAOCodcnTADwewNMB\n/EcA14vIqvFnnwDwMICnArgMwK/ZzQO4FcALReQJIvJEAD8P4I8zhvUCAAcA/BiADwH4VIGfRggh\nhESD5om4EM/7RwG8Ryn1I6XUFwE8BGC1iCwD8L8CeJdS6r8rpe6BFVEa898B7AKwBcCl4+ePZozl\nXqXUp5VSatzm00TkKfl/EiGEEBIHmieSh3/Ryd5jHgZwKoB/C+BxAP7B+Oxe67vakH0WyXLdawB8\nJqDP7+knSqlHxu2cmm/YhBBCSDxonkgM/hnAjwA803jvTNeFSqk/AXA6gKcopb5aw9gIIYSQqNA8\nkdKMo1E7AQxF5GQRWYPpxHCb9QB+wXjtWyYkhBBCWgfNE7FR2Zc4r30LkmTy+wB8evzHea1S6oBS\n6oCnnaz+84yPEEIIiY4kebiEEEIIISQERp4IIYQQQnJA80QIIYQQkgOaJ0IIIYSQHNR2MLCIMLmK\nkB6ilOJuSkJIp6g18qSUCvqzuLgYfG2sP+yTfc5Tf/PSJyGEdBEu2xFCCCGE5IDmiRBCCCEkB600\nT4PBgH2yz7nqsw+/sak+CSGkbdRWJFNEFHMgCOkXIgLFhHFCSMdoZeSJEEIIIaStRDNPIrJMRL4u\nIrtitUkIIYQQ0jZiRp7eCmB/xPYIIYQQQlpHFPMkImcAeCWA34vRHiGEEEJIW4kVefoIgKsBMCOc\nEEIIIZ2m9PEsIvIqAPcrpe4SkQGA0jtr5ApuziGkNDcMK2lWqcVK2iWEkHkhxtl2FwDYKCKvBHAy\ngMeLyGeUUq+1LxwOhyeeDwYD1owhpGOMRiOMRqOmh0EIIZUStc6TiKwF8Hal1EbHZ8F1nhh5IqQk\nFUWdgHyRJ9Z5IoR0EdZ5IqRrVGicCCGExFm2O4FSai+AvTHbJITkgMaJEEIqh5EnQroCjRMhhNQC\nzRMhXYDGiRBCaoPmiZB5h8aJEEJqheaJkHmGxokQQmqH5okQQgghJAc0T4TMK4w6EUJII9A8ETKP\n0DgRQkhj0DwRQgghhOSA5omQeYNRJ0IIaRSaJ0LmCRonQghpHJonQuYFGidCCGkFNE+EzAM0ToQQ\n0hponghpOzROhBDSKmieCGkzNE6EENI6aJ4IIYQQQnJA80RIW2HUiRBCWgnNEyFthMaJEEJaC80T\nIYQQQkgOaJ4IaRuMOhFCSKuheSKkTdA4EUJI61letgEROQnAVwCsHLe3Qym1tWy7hPQOGidCCJkL\nSkeelFKPAlinlHoegOcCeIWIvKD0yEg3Wd/0AFoKjRMhhMwNUZbtlFIPj5+ehCT6pGK0SzrIhqYH\nQAghhJQjinkSkWUi8g0A3wNwh1LqL2K0SzrEegDbxs+3gREoE0adCCFkrogVeTo+XrY7A8DPiMia\nGO2SDrEE4Irx8yvGrwmNEyGEzCGlE8ZNlFI/FJE9AC4EsN/+fDgcnng+GAwwGAxidk/mgd1ND6BF\ndNA4jUYjjEajpodBCCGVIkqVS08SkScDOKaUOiwiJwO4DcD7lVJfsK5ToX3JFVJqTKQi1oMRo1jM\nsXFSajH4WhGBUoo3NCGkU8RYtjsdwB4RuQvA1wDcZhsn0hGY7B2HOTZOhBBCIizbKaXuBvD8CGMh\nbWU9JsZpG5KlN0agikHjRAghcw8rjJNZ7J1wTPaOA40TIYR0AponMotveY7J3oQQQgjNEzHIqsXE\niFNxGHUihJDOQPNEJnB5rhponAghpFPQPJFZuDwXDxonQgjpHDRPZBZGnOJA40QIIZ2E5okQQggh\nJAdRj2chhIARJ0II6TiMPJF42Lvz+giNEyGEdB6aJxIPHt9CCCGkB9A8kfJk1YfqC4w6EUJIL6B5\nIuVhfSgaJ0II6RE0TyQerA9FCCGkB9A8kXj0MeIEMOpECCE9g+aJNENX8qJonAghpHfQPJFmqHpn\nXh3mjMaJEEJ6Cc1T32g64lPXzryqzRmNEyGE9Baap77RdC2mqnfm1WHOaJwIIaTX0Dz1hbbVYqpq\nZ17V5ozGiRBCeg/NU1uJbW7KmIoqjFYVO/PMcVZhzmicCCGEIIJ5EpEzROROEblHRO4WkV+PMbDe\nU9XyWhFTkTWWmOaqTFvmOPtaNoEQQkjlxIg8PQbgKqXUeQB+DsCVIvLsCO32k6qX1/JGnELGEtPo\nbUjpxwfznAghhNSIKKXiNihyK4CPKaW+bL2vQvuSKyTqmGpnPcpHPrZhsszWNL6xrMe0cdqN/L9b\n/13FaKuqvzMapymUWgy+VkSglJrzG5oQQqaJmvMkImcDeC6Ar8Vsd+6IEYlp01EnvrGk5VFlRX9k\nIXnUf1dLVj9FIlDMcyKEEFIDy2M1JCKnAtgB4K1KqYdc1wyHwxPPB4MBBoNBrO7bgRk92YZi0RNN\nm3J2XGMxo2su07IBk4iS/f3TtgMXHgAGW5PX9t/VBhSLIHFnXeOMRiOMRqOmh0EIIZUSZdlORJYj\n+V/XF5VSv+O5pj/Ldm1acquK0KU8WNfJAnC64a0XZbadGMueMaB5csJlO0JI34m1bPdpAPt9xql3\ntGnJzUfRpOqs5GyX6TGvU0eAR25Knl+w2f13ReNECCGkxcQoVXABgF8B8CIR+YaIfF1ELiw/tDmm\nDf/zz6Jo+YGsPKdtxuvdnusObQHuOxV4yc52/l2ZxmlxT2PDIIQQ0k5Kmyel1FeVUo9TSj1XKfU8\npdTzlVJfijE4UgGxyg/4Ika2sXJdtx7AJ49kj6ENDPc2PQJCCCEtgxXG+0ZWpfFQc5UWMTINk+u6\nWEeoVFnPaXEPoMbP1ZARKEIIISegeeorRcoPaLJMS6gZKpsbFrsKu7lct3UdIOPXMkxeu1i5IvIg\nCCGEtB2aJ+ImzdjEMi3aZLWhorgvQXy41v+dzZuAa65JHgkhhPQGmqe+kmWAfBGnKo5ByWvGYi37\nadJ21qVFnM4/P3l+/vmMQBFCSI+geeobZQxQbNNS1ozFKAlRtCTB0WPAvn3J8337kteEEEJ6QfSz\n7bwd9alI5jygi1wWKUgZu4hlU0VFY9RyWrmid8aJRTIJIX2Hkae+oqM2RfKXiuYqZY0llCpznPLS\nM+NECCGE5imcNtciKkrZ/KXYieN19zsvMJ+KEEJaBc1TKG3+H3YR41Mmf6mqxPG6+s0bdWrSvHBH\nHyGEtA6apyyaMAp5+yhi7NqUOF5nv3mNU5PmhTv6CCGkldA8ZdGEUQg1Q3UYoKtS2mjqAOSi/RaJ\nODVpXrijjxBCWsnypgcwN9RhFNZjYpy2jftMM2tL4z9ldqtl/a7VGf03QR0RJ2BiXs4/vznzsmMn\nsGsXjRMhhLQIlipoI3nLCMQuHQAkESfTOB0E8OHIfcTA/O2+v4eyO+t6WI4gDZYqIIT0HS7btZG8\nZQSqiAB9GNPLeiHGKWvZsEi+WNZ3NljP7etjlCSgcSKEEGJA89RWiuYypV1bxLwczHGtbfbs/vLk\ncmV9x8730s9NAxWrlhMhhBBiQPPURvIkqecxKEV25YVGnFxmb0PG5z60AUr7jv13ZOZuuSJQeZK9\nuauNEEJICjRPbSYkST3EoJTZlee7Nq2Nc6z+gDAzaI5zAya/P+07+pol6/qNw8k1ecoN5C1NQKNF\nCCG9g+apzWRFnEINSplyC64ojn7f176dL2WO0WXsXO3onYZZBtL+nbsBnL528l6ecgN5SxOwgCUh\nhPQSmqd5xWeI0sxGnnILdhRoveN92wjZ7ZtRIZ95s9vW39PmLG8y/BKA4d7J6zy1kvJcm2W0GJEi\nhJDOwlIF804VZQrMts0Ik1l3KqQOlYksAK86Ml1WwG57A6aNk91n3vEO1wJb1yXP85QbCL1286ZJ\nDagdO7Pf7wgsVUAI6TtRzJOIfArJ/7ruV0r9pOcamqeYyAKgjlTfjzYkdhFO8/31AD5vjMc2dKdt\nB06+FHjkJuDQltk2bLSBylv4U++uU0PgpPeWKzEQaqDs61auSJbyNNdd17lSBzRPhJC+E2vZ7kYA\nL4/UFsnitO3A6Q8lj1Xjyju6CtOV0DcgGc/F49wf0xDJQmKcgORRFtxt28t6Zp++HCkTsyzBzW8s\nl4uUJ5fJNkY8UoUQQjpPFPOklPpTAIditEUySDMjQDUHF9vLZqsxGxVaFOAlO2fzodSRJOIEJI92\ntMw0Srut9zV2IUwb0zitXAEc+ETyvMh5dDHOs9uxM4k4hS7ZMT+KEELmCiaMzxtZZqRIIUr9OqRC\nuL3Dz4eZGH5oC3DfqdNLdiZL1qOvv6LlFvIQK3IU+j3u2COEkLmj1oOBh8PhieeDwQCDwaDO7rvD\noS3Ag5dNG6e8hwpvgDu64/qOzmGyDyLeBmCrAtYOgb3DJPqkc6Bs8uZnmX2aBx/bhyDbVcRjHOa7\nYyfwxS8CRx6efj/2GXd2lKsDBwCPRiOMRqOmh0EIIZUSbbediJwFYDcTxhvGNBeuQ3PtJO2DmD4A\nGJg1XrZhMdv6qpEMfsGWeDv/sn4HkH78iml0XEndaSbFtVsubQddGVM1hzvzmDBOCOk7Mc3T2UjM\n03M8n9M8VYG96840F2kRG9drYNYk2UbLPq7F3mnnI7SkQlp5BJPQc+tsc5JlVly75QD/DroY5id2\nRKtiaJ4IIX0nSs6TiHwOwJ8BOEdEviMir4/RLsnAtetOR4SycoW+b7W1G7O76uzilnaECkiMTsiS\nnGmIXMt6+r2QauihxsleFls4JTsZ3JXz5MuDipFcrvskhBAyN7BIZpOUKXApC4lx0tx36qyJyRN5\nSsNODtcmyxUhsn+Trxim3bcej31ci4tQ8wRMR4aAidnJihS5okGu9+Zw2a0sjDwRQvoOzVOT5DEw\nLnTxyQs2A7c4/sftyhW6CtMRJNdSnPl9e/eez/T4Xpvvu6qHw9GHqx8gzDT5TA9QXfHKOVt2KwvN\nEyGk79A8NUFoXk8IsgB88kh+E5Y38qSjQq5yAq5E9LSkc1ffrtIHZhshxsmOAtmmpmiUyGeOQqNT\nHYPmiRDSd2otVUDGuLbgF2E9gA3jpbqs8gS28TkY2P6Sca1rWc6s0WT+Jr38Zka69BhdBxTrsduJ\n6y7D5jMtZv7RsmXAmjXTRmrHzvzlAEzDdd6bk7PyVq4ANm7MtyMvlB6YL0IImXdYJLMJ7ITuogUf\nQ5KrNfbSmG+pTmNGksxlvqwinPZSnFmNXD+mjXM3po3dBkyiTot7/EUlzaTu/fsT4wQkZuaSiyff\nyWNMbEM23Dvp304Uj5E8zoKZhBAyF9A8/Y/27j9Wz/K+7/jn6+GD4LBYqaYRhJd1VQrFGC2JpihT\npfo4hUBS7LryD0inVe36R9RAFkLK0mDW8zgrhGItMDWQBm1t1EmZwbaU2eQnKD6umilZlYTEjg1h\n05bSqE20FU4ERNiLr/1xPxfP9Vznun89930/P87zfknoPD/v6z4Oir98r+/1/U5CKuhp0jE7lcnx\n7lQ6UCtaL55d539WDfjiLuSf0iAgyjt1FzbpvFrDAabrZYFT72RxgOLHojx+OD+QqhPU+IBsWy9r\nACpJh6/Lnnt5J/LqSgVfjG0BgKlE8DRJYdBTdaxKSlEmJ878+M/69VJjWsJMUzi4N84g5d1L3vtx\n8LVD+W0VdgRr97ZlgZOUBTHbevkBSpxZunCh2biVI0elGxck62XP957KuqmfPr12fp0P3qT6GaQ4\n+Nq5kywUAEwpCsYnrc3i8aLr+lN1RSfo8t5Lndorq9fasVH6xrks2DngBq0U8k7dhe0LfABlvSzj\ndGB79tz1stfyao68vEaXeYFT1Tojfy9Fn0+tXXerUOruZGALKBgHMO/IPE1anbqlPKkttPi6HdDt\n8QAAIABJREFUH0+87vlMUN578Tpl23e2KB0/n41sWVkeHmDsg8Pw3r56SPrf26S9u4fXc73h6/a2\nDR6ntu/Cba54Cy0ez+JVrTNa2Ch97J1rrxVrOli4qCknAGAqkHmaFk0aZhZlgfKue6eygCr13Ti7\n5AOkODuUWnPvbumrR7KA6YVb146Pidf5XL/ZZ5ih+uP+5/1WWUre2JXVVWnTpuz1Y8fWZqjC7x07\nVi3D478j5bdBiLVxam5KT96ReQIw78g8TYuqgdPN0eOyLFDedX1dU6rY/InovR0aLiC/Oed7tihd\n399Cu+SWdOB0c3CfT0j6lZcHxdjLJr3hn/VrnLapkK8v8oGMD242bcp+bt0qbSwZzyKVZ3jCa/vv\nhaf38rQR9Exh4AQAoM/T7Nmh/N5KVYR1Tb7+qEhqNEsqIAt7Ti2b9NRu6XAi4xSeqvPe21+nKNOU\n4oMLv80VZ55efmXweur5ufPlvZ/Ca0trT+/V7RsFAJh5bNvNiqLC8lG2/OJmlFU++38k7R/h/lLv\n+8/4rcArtg0Kw0cV1jLFNU5Fz+tc+9z5uZxnF2LbDsC8I3iadnFgVDfLFAcwVUappK4jpQf6xsqC\nMv9+fOKuVyF4Wrw0yx7lieuZuhydMqX1SONA8ARg3rFtN+3CbTpp7TZbUdYp/q5/Xne77yoNj1jJ\nGyZ8Z/A4HBcT3qO///Aewu26vKDkg3dk23Grq9KDD619P24yWXV0yqhB0KQCpzkO2gBgWhA8TatU\nbZIPOEJxgJT67rNaO19OKq938nygVHaqzzfkjD8X12mF3wkLw/ftHZ5H5y1eOigE37Qpe34+qHfy\nP8PaJGlQk+Qfh691vf3WRZAz59uFADAtOG03rcr6PxWdtEv1eIqvFQdd4Sm4POHMufjUXxysld2j\n+t/xW3W37Msfo/Kudw0er65mz+++e+2JN38CLz5Bl+qb1MYsujxdzKjr8n4BALUQPE27vOxQleaa\n8XePK3/WXdiOII/PQIXBUqp+KtWQM7zHeG7d4z+SrrlmcI0zZwaP41YBjz66NrsUF4qHbQy8+LVz\n56Vz5/qPz7VbC9VFkEPjTACYGgRP067sFF3R1lv8XT90N55RF7YjKBseHGeZ4nvw8+qK7vEbQR+n\nlWXpzMPD71955SBzc+78IJgK2w14eYFE2WuLl0oLC9njhYXsudQ82Am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"text/plain": "" }, "metadata": {} } ] }, { "metadata": {}, "cell_type": "markdown", "source": "The first performs **terribly**! But we could already predict this by looking at optCond in the right-hand column of the verbose output: the largest magnitude of the derivative not only stops shrinking beyond ~70 but actually bounces around between about 70 and 200.\n\nThe second performs much better. and looks like it's properly separated the points. Furthermore, fitting only required $28$ function evaluations compared to findMin's $26$. " }, { "metadata": {}, "cell_type": "markdown", "source": "# Setting alpha\n\n## From above/last time\n\nLast time, we used the approach with $\\alpha = \\alpha_0 = \\alpha_1 = \\ldots$ and we saw that the results obtained were highly dependent on the [sequence of] learning rates.\n\n## Lipschitz constant\n\nIf we can determine the Lipschitz constant of our objective function, then we get a very nice progress bound under certain assumptions on the objective function.\n\nNamely, assume that $f$ is strongly smooth. In particular, there exists $L$ such that for any $v, z$, \n$$\nv^T \\nabla^2 f(z) v \\leq L\\|v\\|_2^2\n$$\nFor some $z$, it then follows by Taylor's theorem that \n$$\nf(y) = f(x^t) + \\nabla f(x^t) (y-x^t) + \\frac{1}{2}(y-x^t)^T \\nabla^2 f(z) (y-x^t) %\n\\leq f(x^t) + \\nabla f(x^t) (y-x^t) + \\frac{L}{2}\\|y-x^t\\|_2^2\n$$\nSetting $y = x^{t+1}$ to be the minimizer of this equation yields\n$$\nx^{t+1} = x^t - \\frac{1}{L} \\|\\nabla f(x^t)\\|_2^2\n$$\nIn particular, gradient descent with $\\alpha_t := 1/L$ minimizes the upper bound (*i.e.*, maximizes the *worst-case* progress). Substituting this into the expression above, \n$$\nf(x^{t+1}) \\leq f(x^t) - \\frac{1}{2L} \\|\\nabla f(x^t)\\|_2^2\n$$\n*Caveat* *via* a direct quote from Mark Schmidt: \"In practice, you should never use $\\alpha = 1/L$.\"" }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def logisticLipschitz(X, lam=0):\n return .25 * np.max(np.linalg.eigvals(X.T @ X)) + lam", "execution_count": 17, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "objectiveParms_Lip = {'lam': 1}\nL_logist = logisticLipschitz(X, **objectiveParms_Lip)\n\nprint('Lipschitz constant is {:5.5g}, implying alpha is {:5.3e}'.format(L_logist, 1/L_logist))\n\nLR_Lip = LogisticRegressor(objectiveParms=objectiveParms_Lip, \n gdParms={'alpha':1/L_logist})", "execution_count": 18, "outputs": [ { "output_type": "stream", "text": "Lipschitz constant is 140.75, implying alpha is 7.105e-03\n", "name": "stdout" } ] }, { "metadata": { "scrolled": true, "trusted": true, "collapsed": false }, "cell_type": "code", "source": "%timeit LR_Lip.fit(X,y)", "execution_count": 19, "outputs": [ { "output_type": "stream", "text": "100 loops, best of 3: 3.39 ms per loop\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('LR_Lip.funEvals = {}'.format(LR_Lip.funEvals))", "execution_count": 20, "outputs": [ { "output_type": "stream", "text": "LR_Lip.funEvals = 42\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "Stops after only 42 function evaluations; largest partial derivative has magnitude .0090. This is *much* better than the step size of $1\\mathrm{e}-1$ but *worse* than taking a step size of $1\\mathrm{e}-2$. Why is that we can get away with a *larger* value of $\\alpha$? " }, { "metadata": {}, "cell_type": "markdown", "source": "## Adaptive step-size" }, { "metadata": {}, "cell_type": "markdown", "source": "This approach often gives bigger steps and faster progress, but step size never increases.\n\n1. Start with a small guess for $L$ \n$L \\leftarrow 1$\n2. Double $L$ if the *progress inequality* from above \n$f(x^{t+1}) \\leq f(x^t) - \\frac{1}{2L}\\|\\nabla f(x^t)\\|_2^2$\nis not satisfied." }, { "metadata": {}, "cell_type": "markdown", "source": "Notice that this \"backtracking\" approach makes sense when it is cheap to evaluate our function and expensive to compute the matrix product $Xw$. " }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "def adaptiveGradientDescent(funObj, w, X, y,\n objectiveParms={}, \n gdParms={}):\n # optimality tolerance\n optTol = gdParms.get('optTol')\n if optTol is None:\n optTol = 1e-2\n maxEvals = gdParms.get('maxEvals')\n if maxEvals is None:\n maxEvals = 500\n verbose = gdParms.get('verbose')\n if verbose is None:\n verbose = True\n # armijo?\n gamma = gdParms.get('gamma')\n if gamma is None:\n gamma = .5\n elif (gamma > .5) or (gamma <= 0):\n gamma = 1e-4\n # initial objective and gradient value\n f,g = funObj(w, X, y, **objectiveParms)\n # this took a single function evaluation\n funEvals = 1\n \n # initialize the step size\n L0 = gdParms.get('L0')\n if L0 is None:\n L0 = 1\n alpha = 1/L0\n \n # Gradient Descent loop\n while True:\n w_new = w - alpha*g\n f_new, g_new = funObj(w_new, X, y, **objectiveParms)\n funEvals += 1\n gg = g.T @ g\n while f_new > f - gamma*alpha*gg:\n if verbose:\n print('Backtracking...')\n alpha /= 2\n w_new = w - alpha*g\n f_new, g_new = funObj(w_new, X, y, **objectiveParms)\n funEvals += 1\n # Update parameters/function/gradient\n w = w_new\n f = f_new\n g = g_new\n # Test terminate conditions\n optCond = np.linalg.norm(g, np.inf)\n if verbose:\n print('%6d %15.5e %15.5e %15.5e' % (funEvals, alpha, f, optCond))\n # Reset alpha for next\n alpha = 1/L0\n if optCond < optTol:\n if verbose:\n print('Problem solved up to optimality tolerance')\n break\n if funEvals >= maxEvals:\n if verbose:\n print('At maximum number of function evaluations')\n break\n return (w, f, funEvals)", "execution_count": 21, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "LR_adap = LogisticRegressor(objectiveParms={'lam':1},\n gdParms={'L0':1},\n gd=adaptiveGradientDescent)", "execution_count": 22, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false, "scrolled": true }, "cell_type": "code", "source": "%timeit LR_adap.fit(X, y)", "execution_count": 23, "outputs": [ { "output_type": "stream", "text": "100 loops, best of 3: 4.08 ms per loop\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('LR_adap.funEvals = {}'.format(LR_adap.funEvals))", "execution_count": 24, "outputs": [ { "output_type": "stream", "text": "LR_adap.funEvals = 54\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "## Backtracking line-search\n\nChecks against the Armijo condition\n$$\nf(x^{t+1}) \\leq f(x^t) - \\alpha\\gamma \\|\\nabla f(x^t)\\|_2^2, \\quad \\gamma \\in (0, 1/2]\n$$\nSee Wikipedia for a more detailed discussion on the more technical [Wolfe conditions](https://en.wikipedia.org/wiki/Wolfe_conditions) (of which the Armijo rule is a special case)." }, { "metadata": {}, "cell_type": "markdown", "source": "Since this version was implemented above in adaptiveGradientDescent the only step required for this section is to choose gamma and fit the model." }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "LR_armijo = LogisticRegressor(objectiveParms={'lam':1},\n gdParms={'L0':1,'gamma':1e-4},\n gd=adaptiveGradientDescent)", "execution_count": 25, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "%timeit LR_armijo.fit(X,y)", "execution_count": 26, "outputs": [ { "output_type": "stream", "text": "100 loops, best of 3: 4.88 ms per loop\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('LR_armijo.funEvals = {}'.format(LR_armijo.funEvals))", "execution_count": 27, "outputs": [ { "output_type": "stream", "text": "LR_armijo.funEvals = 70\n", "name": "stdout" } ] }, { "metadata": {}, "cell_type": "markdown", "source": "## Comparison of the three" }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "plt.figure(figsize=(10,10))\nfor j, model, pTitle in zip(range(221,224), \n [LR_Lip, LR_adap, LR_armijo], \n ['Lipschitz', 'Adaptive', 'Armijo']):\n plt.subplot(j)\n binaryClassifier2DPlot(model)\n plt.title(pTitle)\n plt.axis('tight');", "execution_count": 28, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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iS8AZ1qR+WWaFpgl/fxSeVLq4bJenYb3SL6AZ72AdVNUvYFrDlgX4tSVAj0630YS/O2qY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\ntLJs0GU3ST95JKtDKrt2098rvJ/U2qnAIa4z8mNZrvug9M+25fVOUpZ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"text/plain": "" }, "metadata": {} } ] }, { "metadata": {}, "cell_type": "markdown", "source": "# Coordinate optimization\n\nIn this case, we must assume that our function $f(w; X,y)$ is the sum of a smooth function and a separable function\n$$\nf(w; X,y) = g(w; X,y) + \\sum_{j=1}^d h_j(w_i; X,y)\n$$\nThen at each iteration $t$ select an index $j_t$ and update that index via \n$$\nx^{t+1}_{j_t} = x^t_{j_t} + \\gamma_t \\quad\\iff \\quad x^{t+1} = x^t + \\gamma_t e_{j_t}\n$$\nfor some constant $\\gamma_t$ and where $e_{j_t}$ is the $j_t$-th unit vector.\n\nIn the case of logistic regression, one of the key tools here is that this allows us to track updates via \n$$\nXw^{t+1} = Xw^t + \\gamma_t X e_{j_t} = Xw^t + \\gamma_t x_{j_t}\n$$\nso that it is only necessary to keep track of the index $j_t$ that is selected and the update constant $\\gamma_t$ — no more than one expensive matrix multiplication is required [to compute $Xw^0$]. But how does one go about choosing $j_t$ on iteration $t$? We'll mention two ways. " }, { "metadata": {}, "cell_type": "markdown", "source": "## Lipschitz and Uniform Sampling\n\nIn the first selection method that we'll discuss, $j_t$ is chosen uniformly at random on each iteration from $[d] := \\{1, 2, \\ldots, d\\}$ – *i.e.*, $j_t \\sim \\mathrm{Unif}([d])$ and $j_t$ chosen according to $P(j_t = k) = L_k/Z$ for $k \\in [d]$, where $L_k$ denotes the $k$th coordinate-wise Lipschitz constant and $Z$ a normalizing factor so that $Z^{-1}\\sum_k L_k = 1$ gives a pmf on $[d]$." }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def weightedIndexSample(weights):\n \"\"\"\n weightedIndexSample returns a random integer in {1, 2, ..., weights.size}\n according to the probability mass function induced by weights\n \"\"\"\n return np.random.choice(range(weights.size), p=weights/weights.sum())", "execution_count": 29, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "def uniformIndexSample(v):\n \"\"\"\n uniformIndexSample returns a random integer in {1, 2, ..., v.size}\n uniformly at random. \n \"\"\"\n return np.random.choice(range(v.size))", "execution_count": 30, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "def lipschitzLogisticLipschitz(X, lam=0):\n \"\"\"\n lipschitzLogisticLipschitz returns a vector of the coordinate-wise\n Lipschitz constants for the logistic function given data X and \n regularizer lam. (i.e., each L_j, j in [d] satisfies the Lipschitz\n continuity inequality for coordinate j)\n \"\"\"\n return .25 * (X**2).sum(axis=0) + lam\ndef uniformLogisticLipschitz(X, lam=0):\n \"\"\"\n uniformLogisticLipschitz returns the uniform coordinate-wise Lipschitz \n constant for the logistic function given data X and regularizer lam \n (i.e., the smallest Lipschitz constant for which the Lipschitz continuity \n inequality is satisfied for each coordinate)\n \"\"\"\n return np.max(lipschitzLogisticLipschitz(X, lam))\n", "execution_count": 31, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "def randomizedCoordinateDescent(funObj, w, X, y, \n objectiveParms={}, \n gdParms={}):\n \"\"\"\n randomizedCoordinateDescent performs randomized coordinate \n descent on the convex function funObj to find the optimal \n parameter vector w given labeled training data [X, y]. \n \"\"\"\n # progress tolerance\n progTol = gdParms.get('progTol')\n if progTol is None:\n progTol = 1e-4\n # maximum number of passes through the data\n maxPasses = gdParms.get('maxPasses')\n if maxPasses is None:\n maxPasses = 500\n # whether to print output along the way\n verbose = gdParms.get('verbose')\n if verbose is None:\n verbose = False\n # Which Lipschitz method to use?\n # default: uniformLogisticLipschitz\n lipschitzMethod = gdParms.get('lipschitzMethod')\n if lipschitzMethod is None:\n L = uniformLogisticLipschitz(X, **objectiveParms)\n else:\n L = lipschitzMethod(X, **objectiveParms)\n # How to choose the coordinate/index jt on iteration t\n # default: weighted index sampling\n # default: if L is a uniform Lipschitz constant, then \n # uniform index sampling is performed.\n coordSample = gdParms.get('coordSample')\n if coordSample is None:\n coordSample = weightedIndexSample\n if isinstance(L, float):\n unif = np.ones(w.shape).ravel()\n funEvals = 0\n # For fast mulitiplication\n Xw = X @ w\n # For tracking progress\n w_old = w.copy()\n \n for t in range(maxPasses*d):\n # Choose index and Lipschitz coefficient\n if isinstance(L, float):\n Lt = L\n j = coordSample(unif)\n else:\n j = coordSample(L)\n Lt = L[j]\n # Compute objective value and partial derivative g_j\n Xj = X[:,j].reshape(-1,1)\n f, g_j = funObj(w, Xw, Xj, y, j, **objectiveParms)\n funEvals += 1\n # Variable update\n Xw -= g_j*Xj/Lt\n w[j] = w[j] - g_j/Lt\n # Check for lack of progress after each pass\n if np.mod(t,d) == 0:\n change = np.linalg.norm(w-w_old,2)*np.sqrt(d)\n if verbose:\n print('Passes = %d, function = %.4e, change = %15.4e' % (t/d, f, change))\n if change < progTol:\n if verbose:\n print('Parameters changed by less than progTol on pass %d.' % t)\n break\n w_old = w\n return (w, f, funEvals)\n \n \ndef coordinateObjective(w,Xw,Xj,y,j,lam):\n \"\"\"\n coordinateObjective computes the negative log-likelihood of the MAP estimate\n and the jth gradient for the logistic function using the matrix-product Xw,\n the labels y, an index j and regularizer lam\n \"\"\"\n yXw = y*Xw\n invsigmoid = 1+np.exp(-yXw)\n nll = np.sum(np.log(invsigmoid)) + .5*lam*np.linalg.norm(w,2)\n g_j = -Xj.T @ (y*(1-1/invsigmoid)) + lam*w[j, 0]\n return (nll, g_j)", "execution_count": 32, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "coordModels = {}\nfor name1, coordSamp in zip(['weighted', 'uniform'], \n [weightedIndexSample, uniformIndexSample]):\n for name2, lipMethod in zip(['coordLip', 'unifLip'], \n [lipschitzLogisticLipschitz, uniformLogisticLipschitz]):\n coordModels[(name1, name2)] = LogisticRegressor(objectiveParms={'lam':1},\n gdParms={'coordSample': coordSamp,\n 'lipschitzMethod': lipMethod},\n objective=coordinateObjective,\n gd=randomizedCoordinateDescent)", "execution_count": 33, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "for key, value in coordModels.items():\n print(key)\n %timeit value.fit(X,y)\n print('Number of Passes: {}\\n'.format(value.funEvals))", "execution_count": 34, "outputs": [ { "output_type": "stream", "text": "('uniform', 'coordLip')\n1000 loops, best of 3: 526 µs per loop\nNumber of Passes: 3\n\n('uniform', 'unifLip')\n1000 loops, best of 3: 541 µs per loop\nNumber of Passes: 3\n\n('weighted', 'coordLip')\nThe slowest run took 26.07 times longer than the fastest. This could mean that an intermediate result is being cached.\n1000 loops, best of 3: 649 µs per loop\nNumber of Passes: 3\n\n('weighted', 'unifLip')\n1000 loops, best of 3: 665 µs per loop\nNumber of Passes: 3\n\n", "name": "stdout" } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "plt.figure(figsize=(10,10))\nj = 221\nfor pTitle, model in coordModels.items():\n plt.subplot(j)\n j += 1\n binaryClassifier2DPlot(model)\n plt.title(pTitle)\n plt.axis('tight');", "execution_count": 35, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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3Jn11m9xTn8MejVkXTtYA735m9LhJfwGzDvMlntNfMzBhvMuEyB9ISuT2JXAX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"text/plain": "" }, "metadata": {} } ] }, { "metadata": {}, "cell_type": "markdown", "source": "## So which is better?" }, { "metadata": {}, "cell_type": "markdown", "source": "**Note:** For a more concrete/realistic example of which is better, see the exercises below. " }, { "metadata": {}, "cell_type": "markdown", "source": "Note that where uniform sampling is used - $\\forall j \\in [d], \\,\\, p(j) = d^{-1}$ - we achieve the guaranteed progress bound\n$$\n \\E (f(x^{t+1})) \\leq f(x^t) - \\frac{1}{2dL} \\|\\nabla f(x^t)\\|^2\n$$\nwhere $[d] := \\{1, \\ldots, d\\}$. Correspondingly, for Lipschitz sampling, the\npmf is given by\n$$\n\\forall j \\in [d],\\quad p(j) = \\frac{L_j}{\\sum_{j \\in [d]} L_j} =: \\frac{L_j}{d \\overline{L}}\n$$\nwhere $d\\overline{L}$ is the normalizing constant for the pmf, with\n$\\overline{L}$ the arithmetic mean of the coordinate-wise Lipschitz constants\n$L_j$. Hence, the guaranteed progress bound for Lipschitz sampling is\n$$\n\\E f(x^{t+1}) \\leq \\E \\big( f(x^t) - \\frac{1}{2L} |\\nabla_{j_t} f(x^t)|^2\\big) %\n= \\sum_{j=1}^d \\frac{L_{j_t}}{d\\overline{L}} \\big( f(x^t) - \\frac{1}{2L} |\\nabla_j f(x^t)|^2 \\big)\n$$\n$$\n= f(x^t) - \\frac{1}{2d L} \\sum_{j\\in d} \\frac{L_{j_t}}{\\overline{L}} |\\nabla_{j_t} f(x^t)|^2\n$$\nIn particular, by comparing the two equations above, uniform sampling is on average preferred to Lipschitz sampling if" }, { "metadata": {}, "cell_type": "markdown", "source": "$$\n\\ip{\\mathbf{1}, |\\nabla_{j_t} f(x^t)|^2} = \\|\\nabla f(x^t)\\|^2 > \\sum_{j_t\\in [d]} \\frac{L_{j_t}}{\\overline{L}}\n|\\nabla_{j_t} f(x^t)|^2 = \\ip{\\frac{L_{j_t}}{\\overline{L}}, |\\nabla_{j_t}\n f(x^t)|^2}.\n$$\nFirstly, by the Cauchy-Schwarz inequality, this certainly occurs if\n$\\|L_{j_t}\\|_2 < \\overline{L}$. Moreover, as an example, perhaps\nLipschitz sampling is sub-optimal on its second pass through the data, where\nthe gradient on the highly-preferred indices is already quite small. This would\ncause the two vectors to be closer to orthogonal, thereby being more likely to\nsatisfy the above inequality. Note: more generally, it is desirable to have a\nsampling scheme $p(j_t)$ which is well-aligned with the squared modulus of the $j_t$-gradient vector\n$\\big(|\\nabla_{j_t} f(x^t)|^2 \\big)$." }, { "metadata": {}, "cell_type": "markdown", "source": "# Exercises" }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "import pandas as pd\nimport requests", "execution_count": 36, "outputs": [] }, { "metadata": {}, "cell_type": "markdown", "source": "## Compare run time on larger data sets" }, { "metadata": {}, "cell_type": "markdown", "source": "This is a test data set that includes 500 observations of 100 features (*i.e.,* $X \\in \\reals^{n\\times d}$ with $n = 500$, $d = 100$; correspondingly, $y \\in \\{-1,1\\}^{500}$). The labels y are stored in the very last column. " }, { "metadata": { "trusted": true, "collapsed": true }, "cell_type": "code", "source": "bigData = pd.read_csv(filepath_or_buffer='./data/logisticData.csv',header=None)", "execution_count": 37, "outputs": [] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "print('bigData.shape = {}'.format(bigData.shape))\nbigData.head()", "execution_count": 38, "outputs": [ { "output_type": "stream", "text": "bigData.shape = (500, 101)\n", "name": "stdout" }, { "output_type": "execute_result", "data": { "text/html": "
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\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "text/plain": " 0 1 2 3 4 5 6 7 8 9 10 11 \\\n0 3.0 1.0 44.0 1735.14 21.0 1.0 1.0 179.5 70.4 120.0 67.0 2.0 \n1 4.0 1.0 43.0 1725.01 32.0 0.0 1.0 135.6 63.9 126.0 86.0 2.0 \n2 9.0 2.0 43.0 19451.83 48.0 0.0 1.0 149.7 61.8 131.0 73.0 1.0 \n3 10.0 1.0 6.0 27769.56 35.0 1.0 1.0 203.5 69.8 130.0 82.0 2.0 \n4 11.0 2.0 40.0 1245.52 48.0 1.0 1.0 155.3 66.2 120.0 70.0 1.0 \n\n 12 13 14 15 \n0 NaN 1.0 268.0 0 \n1 NaN 1.0 160.0 0 \n2 2.0 2.0 236.0 0 \n3 NaN 1.0 225.0 0 \n4 2.0 2.0 260.0 0 " }, "metadata": {}, "execution_count": 45 } ] }, { "metadata": { "trusted": true, "collapsed": false }, "cell_type": "code", "source": "smoking.tail()", "execution_count": 46, "outputs": [ { "output_type": "execute_result", "data": { "text/html": "
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\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", "text/plain": " 0 1 2 3 4 5 6 7 8 9 10 \\\n17026 53589.0 2.0 8.0 4667.32 81.0 0.0 1.0 190.2 61.8 129.0 72.0 \n17027 53593.0 2.0 17.0 18911.40 23.0 1.0 1.0 172.7 76.2 123.0 83.0 \n17028 53594.0 2.0 11.0 9323.23 26.0 0.0 1.0 234.9 67.1 124.0 83.0 \n17029 53616.0 2.0 21.0 2341.16 85.0 1.0 1.0 189.1 68.0 157.0 94.0 \n17030 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN \n\n 11 12 13 14 15 \n17026 1.0 2.0 2.0 168.0 0 \n17027 2.0 NaN 1.0 188.0 0 \n17028 2.0 NaN 1.0 137.0 0 \n17029 2.0 NaN 1.0 285.0 1 \n17030 NaN NaN NaN NaN 0 " }, "metadata": {}, "execution_count": 46 } ] }, { "metadata": {}, "cell_type": "markdown", "source": "**Note:** something's up with the last row there; should remove it. " }, { "metadata": {}, "cell_type": "markdown", "source": "# Stochastic Gradient Descent" }, { "metadata": {}, "cell_type": "markdown", "source": "So far what we've talked about has had a big effect on speed-up in the case where $d$ is large. When instead $n$ is very large we should look to Stochastic gradient descent. Note that Randomized coordinate descent is an example of stochastic gradient descent. Here, instead, we talk about \n\n* mini-batch stochast gradient descent (mini-batch SGD)\n* stochastic average gradient descent (SAG)\n* stochastic variance-reduced gradient methods (SVRG)" }, { "metadata": {}, "cell_type": "markdown", "source": "Generally, SGD operates as follows. We seek to minimize an objective function \n$$\nf(w; X,y) = \\frac{1}{n}\\sum_{i=1}^n f_i(w)\n$$\nwhere, typically, $f_i$ is associated to the $i$-th observation. In usual gradient descent, we march *via* the updates\n$$\nw \\leftarrow w - \\alpha \\nabla_w f(w; X,y)\n$$\nwhereas in SGD we approximate the gradient $\\nabla_w f(w; X,y)$ by $\\nabla_w f_i(w)$ so that the SGD algorithm is given by \n1. Let $w^0$ be an initial parameter vector; $(\\alpha_t)$ a sequence of learning rates\n2. Repeat until stopping condition satisfied:\n 1. Randomly shuffle $[n]$; \n 2. For each $i$ (in the new order):\n 1. $w^{t+1} \\leftarrow w^t - \\alpha_t \\nabla_w f_i(w)$\n\n3. Return $w^T$" }, { "metadata": {}, "cell_type": "markdown", "source": "## Mini-batch SGD" }, { "metadata": {}, "cell_type": "markdown", "source": "In mini-batch SGD we take *batches* of observations (say of size $m_t$ on iteration $t$) and approximate $\\nabla_w f(w; X,y)$ *via*\n$$\n\\nabla_w f(w; X,y) \\approx \\frac{1}{m_t} \\sum_{i=1}^{m_t} \\nabla_w f_i(w)\n$$" }, { "metadata": {}, "cell_type": "markdown", "source": "**Exercise:** Using the randomizedCoordinateDescent function as a starting point, write your own function that performs mini-batch SGD." }, { "metadata": {}, "cell_type": "markdown", "source": "## Stochastic Average Gradient\n\n$w$ is computed according to the updates\n$$\nw^{t+1} \\leftarrow w^t - \\frac{\\alpha_t}{n} \\sum_{i=1}^n g_i^t\n$$\nwhere $g_i^t$ is defined in the following way. On each iteration $t$, an index $i_t \\in [n]$ is chosen randomly and we define\n$$\ng_i^t := \\begin{cases}\n\\nabla_i f(w^t) & i = i_t\\\\\ng_i^{t-1} & \\text{otherwise}\n\\end{cases}\n$$\nwhere $\\nabla_i$ corresponds to the $i$-th element of the gradient (*i.e.*, $\\partial/\\partial w_i$)" }, { "metadata": {}, "cell_type": "markdown", "source": "### References\n1. [Original paper](https://arxiv.org/pdf/1202.6258.pdf)\n2. [Minimizing Finite Sums with SAG](https://arxiv.org/pdf/1309.2388.pdf)\n2. [SAG software](https://www.cs.ubc.ca/~schmidtm/Software/SAG.html)\n3. [SAG slides](https://www.cs.ubc.ca/~schmidtm/Documents/2014_Google_SAG.pdf)" }, { "metadata": {}, "cell_type": "markdown", "source": "## SVRG" }, { "metadata": {}, "cell_type": "markdown", "source": "We refer the interested reader to reference 3. below instead of repeating the derivation and intuition here." }, { "metadata": {}, "cell_type": "markdown", "source": "### References\n1. [Accelerating SGD using predictive variance reduction](https://papers.nips.cc/paper/4937-accelerating-stochastic-gradient-descent-using-predictive-variance-reduction.pdf)\n2. [Practical SVRG](https://arxiv.org/pdf/1511.01942.pdf)\n3. [SVRG intro from Stanford](http://cs.stanford.edu/~ppasupat/a9online/1321.html)" }, { "metadata": {}, "cell_type": "markdown", "source": "# Further reading\n\nWe've barely scratched the surface. Some things for which there was not enough time to write notes...\n\n## Regularization\n\n**Idea:** avoid *overfitting* the model to the data by constraining the size/magnitude/norm/behaviour of the parameters\n\n### $L^2$ regularization\n\n*cf.* Ridge regression, elastic net, *etc.*\n\n### $L^1$ regularization\n\nRobust linear regression is an excellent way of demonstrating the difference between optimizing the $L^1$ norm and optimizing the $L^2$ norm. \n\nAlso *cf.* Compressed Sensing; the $\\ell^1$ norm is *sparsity promoting*. That is, under certain conditions, the solution to the constrained $\\ell^1$ minimization problem \n$$\n\\hat x := \\argmin \\|x\\|_1 \\quad \\text{s.t.}\\quad \\|Ax - y\\|_2^2 \\leq \\eta\n$$\nis unique and $s$-sparse if elements of $A$ come from a certain distribution and $m \\geq C s \\log (N/s)$ for a known [and small] constant $C$.\n\n### Block sparsity\n\nThis constrains the structure of the parameters - *e.g.* if some parameters are zero then it forces other parameters to be zero as well. \n\n## Constrained Optimization\n### Projected Gradient\n\n### Proximal Gradient\n\n### Newton's method\n\nFrom a second-order Taylor expansion, one sees \n$$\nf(w^*) = f(w^j) + \\nabla f(w^j) p^j + \\frac{1}{2} (p^j)^T \\nabla^2 f(w^j) p_j + O (\\|p^j\\|^3)\n$$\nfor an unknown vector $p_j$. Since $\\nabla^2 f(w^j)$ is PSD, seek a particular vector that \"cancels\" this second-order term with the gradient term. Gets superlinear convergence. \n\n### References: \n1. Boyd & Vandenberghe's [Convex Optimization](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf)\n2. Bertsekas's Convex Optimization Algorithms\n3. [Ascher & Greif](http://gw2jh3xr2c.search.serialssolutions.com/?sid=sersol&SS_jc=TC0001261310&title=A%20first%20course%20in%20numerical%20methods) (pg. 261–265)\n\n## Dual methods\n### The Fenchel Dual, geometric multipliers and the KKT conditions\n### Dual coordinate ascent" } ], "metadata": { "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "name": "python", "nbconvert_exporter": "python", "mimetype": "text/x-python", "pygments_lexer": "ipython3", "version": "3.5.2" }, "kernelspec": { "name": "python3", "display_name": "Python 3", "language": "python" }, "toc": { "threshold": 4, "number_sections": true, "toc_cell": true, "toc_window_display": false, "toc_section_display": "block", "sideBar": true, "navigate_menu": true, "nav_menu": { "width": "252px", "height": "336px" } }, "gist": { "id": "f3ae3c29f1fb5ff53b329114296dbf99", "data": { "description": "Gradient Descent (master).ipynb", "public": true } }, "_draft": { "nbviewer_url": "https://gist.github.com/f3ae3c29f1fb5ff53b329114296dbf99" } }, "nbformat": 4, "nbformat_minor": 1 }