From 489c98a82f84f58d2c460adc049decf1b496ddec Mon Sep 17 00:00:00 2001 From: mhjensen Date: Sun, 8 Nov 2020 22:48:11 +0100 Subject: [PATCH] updating week 46 --- doc/pub/week46/html/._week46-bs000.html | 122 +++---- doc/pub/week46/html/._week46-bs001.html | 154 ++++----- doc/pub/week46/html/._week46-bs002.html | 218 +++++------- doc/pub/week46/html/._week46-bs003.html | 214 +++++++----- doc/pub/week46/html/._week46-bs004.html | 167 +++++----- doc/pub/week46/html/._week46-bs005.html | 169 +++++----- doc/pub/week46/html/._week46-bs006.html | 149 +++++---- doc/pub/week46/html/._week46-bs007.html | 141 ++++---- doc/pub/week46/html/._week46-bs008.html | 137 ++++---- doc/pub/week46/html/._week46-bs009.html | 138 ++++---- doc/pub/week46/html/._week46-bs010.html | 134 ++++---- doc/pub/week46/html/._week46-bs011.html | 160 ++++----- doc/pub/week46/html/._week46-bs012.html | 176 +++++----- doc/pub/week46/html/._week46-bs013.html | 163 ++++----- doc/pub/week46/html/._week46-bs014.html | 164 ++++----- doc/pub/week46/html/._week46-bs015.html | 157 +++++---- doc/pub/week46/html/._week46-bs016.html | 152 ++++----- doc/pub/week46/html/._week46-bs017.html | 157 ++++----- doc/pub/week46/html/._week46-bs018.html | 177 +++++----- doc/pub/week46/html/._week46-bs019.html | 216 ++++++------ doc/pub/week46/html/._week46-bs020.html | 211 +++++++----- doc/pub/week46/html/._week46-bs021.html | 180 +++++----- doc/pub/week46/html/._week46-bs022.html | 173 +++++----- doc/pub/week46/html/._week46-bs023.html | 329 +++++-------------- doc/pub/week46/html/._week46-bs024.html | 322 +++++++++++++----- doc/pub/week46/html/._week46-bs025.html | 148 +++++---- doc/pub/week46/html/._week46-bs026.html | 187 +++++------ doc/pub/week46/html/._week46-bs027.html | 188 +++++++---- doc/pub/week46/html/week46-bs.html | 122 +++---- doc/pub/week46/html/week46-reveal.html | 96 +++--- doc/pub/week46/html/week46-solarized.html | 150 +++++---- doc/pub/week46/html/week46.html | 156 +++++---- doc/pub/week46/ipynb/ipynb-week46-src.tar.gz | Bin 195 -> 191 bytes doc/pub/week46/ipynb/week46.ipynb | 20 +- 34 files changed, 2834 insertions(+), 2713 deletions(-) diff --git a/doc/pub/week46/html/._week46-bs000.html b/doc/pub/week46/html/._week46-bs000.html index 513efd003..8529411ec 100644 --- a/doc/pub/week46/html/._week46-bs000.html +++ b/doc/pub/week46/html/._week46-bs000.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -156,7 +158,7 @@ MathJax.Hub.Config({
    -

    Week 46: Support Vector Machines

    +

    Week 46: Gradient Boosting Summary and Support Vector Machines

    @@ -172,7 +174,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 16, 2020

    +

    Nov 8, 2020


    @@ -196,7 +198,7 @@ MathJax.Hub.Config({

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  • Contents
    • @@ -153,35 +155,17 @@ MathJax.Hub.Config({ -

    Support Vector Machines, overarching aims

    +

    Overview of week 46

    + +
      +
    • Thursday: Summary of Gradient Boosting and further examples of applications.
      • +
    • Friday: Support Vector Machines, classification and regression
      • +
    + +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.

    -A Support Vector Machine (SVM) is a very powerful and versatile -Machine Learning method, capable of performing linear or nonlinear -classification, regression, and even outlier detection. It is one of -the most popular models in Machine Learning, and anyone interested in -Machine Learning should have it in their toolbox. SVMs are -particularly well suited for classification of complex but small-sized or -medium-sized datasets. - -

    -The case with two well-separated classes only can be understood in an -intuitive way in terms of lines in a two-dimensional space separating -the two classes (see figure below). - -

    -The basic mathematics behind the SVM is however less familiar to most of us. -It relies on the definition of hyperplanes and the -definition of a margin which separates classes (in case of -classification problems) of variables. It is also used for regression -problems. - -

    -With SVMs we distinguish between hard margin and soft margins. The -latter introduces a so-called softening parameter to be discussed -below. We distinguish also between linear and non-linear -approaches. The latter are the most frequent ones since it is rather -unlikely that we can separate classes easily by say straight lines. +Overview of Support Vector Machines. see also this video.

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  • Contents
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    Hyperplanes and all that

    +

    Support Vector Machines, overarching aims

    -The theory behind support vector machines (SVM hereafter) is based on -the mathematical description of so-called hyperplanes. Let us start -with a two-dimensional case. This will also allow us to introduce our -first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here based on the usage of the petal data. +A Support Vector Machine (SVM) is a very powerful and versatile +Machine Learning method, capable of performing linear or nonlinear +classification, regression, and even outlier detection. It is one of +the most popular models in Machine Learning, and anyone interested in +Machine Learning should have it in their toolbox. SVMs are +particularly well suited for classification of complex but small-sized or +medium-sized datasets.

    -We assume here that our data set can be well separated into two -domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either squares or -circles. +The case with two well-separated classes only can be understood in an +intuitive way in terms of lines in a two-dimensional space separating +the two classes (see figure below). +

    +The basic mathematics behind the SVM is however less familiar to most of us. +It relies on the definition of hyperplanes and the +definition of a margin which separates classes (in case of +classification problems) of variables. It is also used for regression +problems. - -

    from sklearn import datasets
-from sklearn.svm import SVC, LinearSVC
-from sklearn.linear_model import SGDClassifier
-from sklearn.preprocessing import StandardScaler
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
+
    
+With SVMs we distinguish between hard margin and soft margins. The
+latter introduces a so-called softening parameter to be discussed
+below.  We distinguish also between linear and non-linear
+approaches. The latter are the most frequent ones since it is rather
+unlikely that we can separate classes easily by say straight lines.
 
-
-iris = datasets.load_iris()
-X = iris["data"][:, (2, 3)]  # petal length, petal width
-y = iris["target"]
-
-setosa_or_versicolor = (y == 0) | (y == 1)
-X = X[setosa_or_versicolor]
-y = y[setosa_or_versicolor]
-
-
-
-C = 5
-alpha = 1 / (C * len(X))
-
-lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
-svm_clf = SVC(kernel="linear", C=C)
-sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
-                        max_iter=100000, random_state=42)
-
-scaler = StandardScaler()
-X_scaled = scaler.fit_transform(X)
-
-lin_clf.fit(X_scaled, y)
-svm_clf.fit(X_scaled, y)
-sgd_clf.fit(X_scaled, y)
-
-print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
-print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
-print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
-
-# Compute the slope and bias of each decision boundary
-w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
-b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
-w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
-b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
-w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
-b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
-
-# Transform the decision boundary lines back to the original scale
-line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
-line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
-line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
-
-# Plot all three decision boundaries
-plt.figure(figsize=(11, 4))
-plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
-plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
-plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
-plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
-plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
-plt.xlabel("Petal length", fontsize=14)
-plt.ylabel("Petal width", fontsize=14)
-plt.legend(loc="upper center", fontsize=14)
-plt.axis([0, 5.5, 0, 2])
-
-plt.show()
-

    @@ -255,7 +203,7 @@ plt.show()

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  • Contents
    • @@ -153,34 +155,90 @@ MathJax.Hub.Config({ -

    What is a hyperplane?

    +

    Hyperplanes and all that

    -The aim of the SVM algorithm is to find a hyperplane in a -\( p \)-dimensional space, where \( p \) is the number of features that -distinctly classifies the data points. +The theory behind support vector machines (SVM hereafter) is based on +the mathematical description of so-called hyperplanes. Let us start +with a two-dimensional case. This will also allow us to introduce our +first SVM examples. These will be tailored to the case of two specific +classes, as displayed in the figure here based on the usage of the petal data.

    -In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \). -As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is -a two-dimensional subspace, or stated simply, a plane. - +We assume here that our data set can be well separated into two +domains, where a straight line does the job in the separating the two +classes. Here the two classes are represented by either squares or +circles.

    -In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as -$$ -b+w_1x_1+w_2x_2=0, -$$ -

    -where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line -\( b+w_1x_1+w_2x_2=0 \). -In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \). -We can then rewrite the above equation as + +

    from sklearn import datasets
+from sklearn.svm import SVC, LinearSVC
+from sklearn.linear_model import SGDClassifier
+from sklearn.preprocessing import StandardScaler
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
 
-$$
-\boldsymbol{x}^T\boldsymbol{w}+b=0.
-$$
 
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)]  # petal length, petal width
+y = iris["target"]
+
+setosa_or_versicolor = (y == 0) | (y == 1)
+X = X[setosa_or_versicolor]
+y = y[setosa_or_versicolor]
+
+
+
+C = 5
+alpha = 1 / (C * len(X))
+
+lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
+svm_clf = SVC(kernel="linear", C=C)
+sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
+                        max_iter=100000, random_state=42)
+
+scaler = StandardScaler()
+X_scaled = scaler.fit_transform(X)
+
+lin_clf.fit(X_scaled, y)
+svm_clf.fit(X_scaled, y)
+sgd_clf.fit(X_scaled, y)
+
+print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
+print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
+print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
+
+# Compute the slope and bias of each decision boundary
+w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
+b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
+w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
+b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
+w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
+b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
+
+# Transform the decision boundary lines back to the original scale
+line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
+line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
+line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
+
+# Plot all three decision boundaries
+plt.figure(figsize=(11, 4))
+plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
+plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
+plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
+plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
+plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
+plt.xlabel("Petal length", fontsize=14)
+plt.ylabel("Petal width", fontsize=14)
+plt.legend(loc="upper center", fontsize=14)
+plt.axis([0, 5.5, 0, 2])
+
+plt.show()
+

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    A \( p \)-dimensional space of features

    +

    What is a hyperplane?

    -We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \). -In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as -$$ -b+wx_1+w_2x_2+\dots +w_px_p=0. -$$ - -If we define a -matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \) -of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \), -$$ -\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}. -$$ - -If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have -$$ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0, -$$ - -if our output \( y_i=1 \). -In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if -$$ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0, -$$ - -for the class of observations \( y_i=-1 \), -then \( \boldsymbol{x}_i \) lies on the other side. +The aim of the SVM algorithm is to find a hyperplane in a +\( p \)-dimensional space, where \( p \) is the number of features that +distinctly classifies the data points.

    -Equivalently, for the two classes of observations we have +In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \). +As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is +a two-dimensional subspace, or stated simply, a plane. + +

    +In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as $$ -y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0. +b+w_1x_1+w_2x_2=0, $$

    -When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located. +where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line +\( b+w_1x_1+w_2x_2=0 \). +In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \). +We can then rewrite the above equation as + +$$ +\boldsymbol{x}^T\boldsymbol{w}+b=0. +$$

    @@ -212,7 +203,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a

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    • @@ -151,31 +153,46 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The two-dimensional case

    +

    A \( p \)-dimensional space of features

    -Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional -plane. To separate the two classes of data points, there are many -possible lines (hyperplanes if you prefer a more strict naming) -that could be chosen. Our objective is to find a -plane that has the maximum margin, i.e the maximum distance between -data points of both classes. Maximizing the margin distance provides -some reinforcement so that future data points can be classified with -more confidence. +We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \). +In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as +$$ +b+wx_1+w_2x_2+\dots +w_px_p=0. +$$ + +If we define a +matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \) +of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \), +$$ +\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}. +$$ + +If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have +$$ +b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0, +$$ + +if our output \( y_i=1 \). +In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if +$$ +b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0, +$$ + +for the class of observations \( y_i=-1 \), +then \( \boldsymbol{x}_i \) lies on the other side.

    -What a linear classifier attempts to accomplish is to split the -feature space into two half spaces by placing a hyperplane between the -data points. This hyperplane will be our decision boundary. All -points on one side of the plane will belong to class one and all points -on the other side of the plane will belong to the second class two. +Equivalently, for the two classes of observations we have +$$ +y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0. +$$

    -Unfortunately there are many ways in which we can place a hyperplane -to divide the data. Below is an example of two candidate hyperplanes -for our data sample. +When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.

    @@ -198,7 +215,7 @@ for our data sample.

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  • Contents
    • @@ -151,26 +153,31 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Getting into the details

    +

    The two-dimensional case

    -Let us define the function -$$ -f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, -$$ - -as the function that determines the line \( L \) that separates two classes (our two features), see the figure here. +Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional +plane. To separate the two classes of data points, there are many +possible lines (hyperplanes if you prefer a more strict naming) +that could be chosen. Our objective is to find a +plane that has the maximum margin, i.e the maximum distance between +data points of both classes. Maximizing the margin distance provides +some reinforcement so that future data points can be classified with +more confidence.

    -Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \). +What a linear classifier attempts to accomplish is to split the +feature space into two half spaces by placing a hyperplane between the +data points. This hyperplane will be our decision boundary. All +points on one side of the plane will belong to class one and all points +on the other side of the plane will belong to the second class two.

    -The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then -$$ -\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b). -$$ +Unfortunately there are many ways in which we can place a hyperplane +to divide the data. Below is an example of two candidate hyperplanes +for our data sample.

    @@ -194,7 +201,7 @@ $$

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    First attempt at a minimization approach

    +

    Getting into the details

    -How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could -do is to define a cost function which now contains the set of all -misclassified points \( M \) and attempt to minimize this function +Let us define the function +$$ +f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, +$$ -$$ -C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ +as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.

    -We could now for example define all values \( y_i =1 \) as misclassified in case we have \( \boldsymbol{w}^T\boldsymbol{x}_i+b < 0 \) and the opposite if we have \( y_i=-1 \). Taking the derivatives gives us -$$ -\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i, -$$ +Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \). -and +

    +The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then $$ -\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i. +\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b). $$

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    Solving the equations

    +

    First attempt at a minimization approach

    -We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations +How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could +do is to define a cost function which now contains the set of all +misclassified points \( M \) and attempt to minimize this function + $$ -b \leftarrow b +\eta \frac{\partial C}{\partial b}, +C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b). $$ -and +

    +We could now for example define all values \( y_i =1 \) as misclassified in case we have \( \boldsymbol{w}^T\boldsymbol{x}_i+b < 0 \) and the opposite if we have \( y_i=-1 \). Taking the derivatives gives us $$ -\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}}, +\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i, $$ -where \( \eta \) is our by now well-known learning rate. +and +$$ +\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i. +$$

    @@ -192,7 +201,7 @@ where \( \eta \) is our by now well-known learning rate.

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    Code Example

    +

    Solving the equations

    -The equations we discussed above can be coded rather easily (the -framework is similar to what we developed for logistic -regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way. -

    +We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations +$$ +b \leftarrow b +\eta \frac{\partial C}{\partial b}, +$$ + +and +$$ +\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}}, +$$ + +where \( \eta \) is our by now well-known learning rate. - -

    
-

    @@ -189,7 +195,7 @@ regression). We are going to set up a simple case with two classes only and we w

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    Problems with the Simpler Approach

    +

    Code Example

    -There are however problems with this approach, although it looks -pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes. - +The equations we discussed above can be coded rather easily (the +framework is similar to what we developed for logistic +regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.

    -For small -gaps between the entries, we may also end up needing many iterations -before the solutions converge and if the data cannot be separated -properly into two distinct classes, we may not experience a converge -at all. + +

    
+

    @@ -192,7 +192,7 @@ at all.

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    A better approach

    +

    Problems with the Simpler Approach

    -A better approach is rather to try to define a large margin between -the two classes (if they are well separated from the beginning). +There are however problems with this approach, although it looks +pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.

    -Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to -\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. -$$ - -All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. - -

    -We seek thus the largest value \( M \) defined by -$$ -\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, -$$ - -or just -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. -$$ - -If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of -\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. -$$ - -

    -We have thus defined our margin as the invers of the norm of -\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as -possible margin \( M \). Before we proceed, we need to remind ourselves -about Lagrangian multipliers. +For small +gaps between the entries, we may also end up needing many iterations +before the solutions converge and if the data cannot be separated +properly into two distinct classes, we may not experience a converge +at all.

    @@ -218,7 +194,7 @@ about Lagrangian multipliers.

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    A quick Reminder on Lagrangian Multipliers

    +

    A better approach

    -Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an -extreme we have -$$ -df=0. -$$ - -A necessary and sufficient condition is -$$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -$$ - -due to -$$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. -$$ - -In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) -so that they are no longer all independent. It is possible at least in principle to use each -constraint to eliminate one variable -and to proceed with a new and smaller set of independent varables. +A better approach is rather to try to define a large margin between +the two classes (if they are well separated from the beginning).

    -The use of so-called Lagrangian multipliers is an alternative technique when the elimination -of variables is incovenient or undesirable. Assume that we have an equation of constraint on -the variables \( x,y,z \) +Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to +\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition + $$ -\phi(x,y,z) = 0, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$ - resulting in +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. + +

    +We seek thus the largest value \( M \) defined by $$ -d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. +\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$ -Now we cannot set anymore +or just $$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. $$ -if \( df=0 \) is wanted -because there are now only two independent variables! Assume \( x \) and \( y \) are the independent -variables. -Then \( dz \) is no longer arbitrary. +If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of +\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. +$$ + +

    +We have thus defined our margin as the invers of the norm of +\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as +possible margin \( M \). Before we proceed, we need to remind ourselves +about Lagrangian multipliers.

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    Adding the Multiplier

    +

    A quick Reminder on Lagrangian Multipliers

    -However, we can add to +Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an +extreme we have $$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, +df=0. $$ -a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in +A necessary and sufficient condition is $$ -df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda -\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ -(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ -Our multiplier is chosen so that +due to $$ -\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. $$ +In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) +so that they are no longer all independent. It is possible at least in principle to use each +constraint to eliminate one variable +and to proceed with a new and smaller set of independent varables. +

    -We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have +The use of so-called Lagrangian multipliers is an alternative technique when the elimination +of variables is incovenient or undesirable. Assume that we have an equation of constraint on +the variables \( x,y,z \) $$ -\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, +\phi(x,y,z) = 0, $$ -and + resulting in $$ -\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. +d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. $$ -When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and -\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, -it is therefore often called -Lagrange's undetermined multiplier. -If we have a set of constraints \( \phi_k \) we have the equations +Now we cannot set anymore $$ -\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. +\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ +if \( df=0 \) is wanted +because there are now only two independent variables! Assume \( x \) and \( y \) are the independent +variables. +Then \( dz \) is no longer arbitrary. +

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    Setting up the Problem

    -In order to solve the above problem, we define the following Lagrangian function to be minimized -$$ -{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], -$$ - -where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). +

    Adding the Multiplier

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +However, we can add to $$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, $$ -and +a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in $$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda +\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ +(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. $$ -Inserting these constraints into the equation for \( {\cal L} \) we obtain +Our multiplier is chosen so that $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, +\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. $$ -subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). -We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition +

    +We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have $$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. +\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, $$ +and +$$ +\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. +$$ -

      -
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. -
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. -
    - -When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \). +When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and +\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined, +it is therefore often called +Lagrange's undetermined multiplier. +If we have a set of constraints \( \phi_k \) we have the equations +$$ +\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. +$$

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    The problem to solve

    +

    Setting up the Problem

    +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).

    -We can rewrite +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +$$ +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +$$ + +and +$$ +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +$$ + +Inserting these constraints into the equation for \( {\cal L} \) we obtain $$ {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ -y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). + +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.
      2. +
    2. If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
      3. +
    + +When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \).

    @@ -200,7 +219,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec

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    • diff --git a/doc/pub/week46/html/._week46-bs016.html b/doc/pub/week46/html/._week46-bs016.html index d450ffecd..c00ab470f 100644 --- a/doc/pub/week46/html/._week46-bs016.html +++ b/doc/pub/week46/html/._week46-bs016.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,36 +155,26 @@ MathJax.Hub.Config({ -

    The last steps

    +

    The problem to solve

    -Solving the above problem, yields the values of \( \lambda_i \). -To find the coefficients of your hyperplane we need simply to compute +We can rewrite $$ -\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, $$ -With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via +and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ +y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, $$ -resulting in -$$ -b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, -$$ - -or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have -$$ -b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). -$$ - -With our hyperplane coefficients we can use our classifier to assign any observation by simply using -$$ -y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ - -Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier. +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).

    @@ -210,7 +202,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr

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    • diff --git a/doc/pub/week46/html/._week46-bs017.html b/doc/pub/week46/html/._week46-bs017.html index 419283256..8b8e28c1d 100644 --- a/doc/pub/week46/html/._week46-bs017.html +++ b/doc/pub/week46/html/._week46-bs017.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,37 +155,36 @@ MathJax.Hub.Config({ -

    A soft classifier

    +

    The last steps

    -Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. +Solving the above problem, yields the values of \( \lambda_i \). +To find the coefficients of your hyperplane we need simply to compute +$$ +\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. +$$ -

    -Suppose now that classes overlap in feature space, as shown in the -figure here. One way to deal with this problem before we define the -so-called kernel approach, is to allow a kind of slack in the sense -that we allow some points to be on the wrong side of the margin. - -

    -We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and -modify our previous equation +With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via $$ y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, $$ -to +resulting in $$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, +b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, $$ -with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). -The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction -\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), -we bound the total amount by which predictions fall on the wrong side of their margins. +or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have +$$ +b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). +$$ -

    -Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of -misclassifications. +With our hyperplane coefficients we can use our classifier to assign any observation by simply using +$$ +y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). +$$ + +Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier.

    @@ -211,7 +212,7 @@ misclassifications.

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    • diff --git a/doc/pub/week46/html/._week46-bs018.html b/doc/pub/week46/html/._week46-bs018.html index 80fcb4b0a..2ab36a77a 100644 --- a/doc/pub/week46/html/._week46-bs018.html +++ b/doc/pub/week46/html/._week46-bs018.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,56 +155,37 @@ MathJax.Hub.Config({ -

    Soft optmization problem

    +

    A soft classifier

    -This has in turn the consequences that we change our optmization problem to finding the minimum of -$$ -{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, -$$ - -subject to -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, -$$ - -with the requirement \( \xi_i\geq 0 \). +Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.

    -Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +Suppose now that classes overlap in feature space, as shown in the +figure here. One way to deal with this problem before we define the +so-called kernel approach, is to allow a kind of slack in the sense +that we allow some points to be on the wrong side of the margin. + +

    +We introduce thus the so-called slack variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and +modify our previous equation $$ -\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, $$ -and +to $$ -\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, $$ -and -$$ -\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. -$$ +with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \). +The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction +\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \), +we bound the total amount by which predictions fall on the wrong side of their margins. -Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ - -but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). -We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads -$$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, -$$ - -$$ -\gamma_i\xi_i = 0, -$$ - -and -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. -$$ +

    +Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of +misclassifications.

    @@ -229,6 +212,8 @@ $$

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    • diff --git a/doc/pub/week46/html/._week46-bs019.html b/doc/pub/week46/html/._week46-bs019.html index 5c099d50e..2478e0739 100644 --- a/doc/pub/week46/html/._week46-bs019.html +++ b/doc/pub/week46/html/._week46-bs019.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,76 +155,57 @@ MathJax.Hub.Config({ -

    Kernels and non-linearity

    +

    Soft optmization problem

    -The cases we have studied till now, were all characterized by two classes -with a close to linear separability. The classifiers we have described -so far find linear boundaries in our input feature space. It is -possible to make our procedure more flexible by exploring the feature -space using other basis expansions such as higher-order polynomials, -wavelets, splines etc. +This has in turn the consequences that we change our optmization problem to finding the minimum of +$$ +{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, +$$ + +subject to +$$ +y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, +$$ + +with the requirement \( \xi_i\geq 0 \).

    -If our feature space is not easy to separate, as shown in the figure -here, we can achieve a better separation by introducing more complex -basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to -obtain a separation between the classes which is almost linear. +Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain +$$ +\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, +$$ -

    -The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that -we need to introduce for example a polynomial transformation to a two-dimensional training set. +and +$$ +\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, +$$ -

    +and +$$ +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. +$$ - -

    import numpy as np
-import os
+Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before
+$$
+{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
+$$
 
-np.random.seed(42)
+but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \). 
+We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads
+$$
+\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i,
+$$
 
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
+$$
+\gamma_i\xi_i = 0,
+$$
 
+and 
+$$
+y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i.
+$$
 
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
-X2D = np.c_[X1D, X1D**2]
-y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
-plt.gca().get_yaxis().set_ticks([])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.axis([-4.5, 4.5, -0.2, 0.2])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
-plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
-plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
-plt.axis([-4.5, 4.5, -1, 17])
-plt.subplots_adjust(right=1)
-plt.show()
-

    @@ -247,6 +230,7 @@ plt.show()

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    • diff --git a/doc/pub/week46/html/._week46-bs020.html b/doc/pub/week46/html/._week46-bs020.html index deecbf7ff..5cd032a42 100644 --- a/doc/pub/week46/html/._week46-bs020.html +++ b/doc/pub/week46/html/._week46-bs020.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,48 +155,76 @@ MathJax.Hub.Config({ -

    The equations

    +

    Kernels and non-linearity

    -Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) -$$ -z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). -$$ +The cases we have studied till now, were all characterized by two classes +with a close to linear separability. The classifiers we have described +so far find linear boundaries in our input feature space. It is +possible to make our procedure more flexible by exploring the feature +space using other basis expansions such as higher-order polynomials, +wavelets, splines etc.

    -With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors -$$ -y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, -$$ - -from which we also find \( b \). -To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -$$ - -For the above example, the kernel reads -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. -$$ +If our feature space is not easy to separate, as shown in the figure +here, we can achieve a better separation by introducing more complex +basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to +obtain a separation between the classes which is almost linear.

    -We note that this is nothing but the dot product of the two original -vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the -product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). +The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that +we need to introduce for example a polynomial transformation to a two-dimensional training set.

    -This leads to the so-called -kernel trick and the result leads to the same as if we went through -the trouble of performing the transformation -\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+

    @@ -218,6 +248,7 @@ the trouble of performing the transformation

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    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,38 +155,47 @@ MathJax.Hub.Config({ -

    The problem to solve

    -Using our definition of the kernel We can rewrite again the Lagrangian -$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -$$ - -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). -If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +

    The equations

    -We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type +Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) $$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} +z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). $$ -Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \). -Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into -\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \). +

    +With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) +$$ +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors +$$ +y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, +$$ + +from which we also find \( b \). +To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +$$ + +For the above example, the kernel reads +$$ +K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. +$$ + +

    +We note that this is nothing but the dot product of the two original +vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the +product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). + +

    +This leads to the so-called +kernel trick and the result leads to the same as if we went through +the trouble of performing the transformation +\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.

    @@ -208,6 +219,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.

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    • diff --git a/doc/pub/week46/html/._week46-bs022.html b/doc/pub/week46/html/._week46-bs022.html index a8fd3772b..d86d8b0f3 100644 --- a/doc/pub/week46/html/._week46-bs022.html +++ b/doc/pub/week46/html/._week46-bs022.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,40 +155,38 @@ MathJax.Hub.Config({ -

    Different kernels and Mercer's theorem

    - -

    -There are several popular kernels being used. These are - -

      -
    1. Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. -
    2. Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. -
    3. Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. -
    4. Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. -
    - -and many other ones. - -

    -An important theorem for us is Mercer's -theorem. The -theorem states that if a kernel function \( K \) is symmetric, continuous -and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there -exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into -another space (possibly with much higher dimensions) such that - +

    The problem to solve

    +Using our definition of the kernel We can rewrite again the Lagrangian $$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, $$ -

    -So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. +subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem +$$ +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ +y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, +$$ + +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).

    -Note that some frequently used kernels (such as the Sigmoid kernel) -don’t respect all of Mercer’s conditions, yet they generally work well -in practice. +We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type +$$ +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} +$$ + +Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \). +Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into +\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).

    @@ -209,6 +209,7 @@ in practice.

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    • diff --git a/doc/pub/week46/html/._week46-bs023.html b/doc/pub/week46/html/._week46-bs023.html index dffa845a2..041e9881f 100644 --- a/doc/pub/week46/html/._week46-bs023.html +++ b/doc/pub/week46/html/._week46-bs023.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,199 +155,41 @@ MathJax.Hub.Config({ -

    The moons example

    +

    Different kernels and Mercer's theorem

    +

    +There are several popular kernels being used. These are - -

    from __future__ import division, print_function, unicode_literals
+
      
+
    1.  Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),
      2. 
+
    2.  Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),
      3. 
+
    3.  Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),
      4. 
+
    4.  Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),
      5. 
+
    
 
-import numpy as np
-np.random.seed(42)
+and many other ones.
 
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
+
    
+An important theorem for us is Mercer's
+theorem.  The
+theorem states that if a kernel function \( K \) is symmetric, continuous
+and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there
+exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into
+another space (possibly with much higher dimensions) such that
 
+$$
+K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
+$$
 
-from sklearn.svm import SVC
-from sklearn import datasets
+
    
+So you can use \( K \) as a kernel since you know \( \phi \) exists, even if
+you don’t know what \( \phi \) is.
 
+
    
+Note that some frequently used kernels (such as the Sigmoid kernel)
+don’t respect all of Mercer’s conditions, yet they generally work well
+in practice.
 
-
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import StandardScaler
-from sklearn.svm import LinearSVC
-
-
-from sklearn.datasets import make_moons
-X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
-
-def plot_dataset(X, y, axes):
-    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
-    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
-    plt.axis(axes)
-    plt.grid(True, which='both')
-    plt.xlabel(r"$x_1$", fontsize=20)
-    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.show()
-
-from sklearn.datasets import make_moons
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-
-polynomial_svm_clf = Pipeline([
-        ("poly_features", PolynomialFeatures(degree=3)),
-        ("scaler", StandardScaler()),
-        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
-    ])
-
-polynomial_svm_clf.fit(X, y)
-
-def plot_predictions(clf, axes):
-    x0s = np.linspace(axes[0], axes[1], 100)
-    x1s = np.linspace(axes[2], axes[3], 100)
-    x0, x1 = np.meshgrid(x0s, x1s)
-    X = np.c_[x0.ravel(), x1.ravel()]
-    y_pred = clf.predict(X).reshape(x0.shape)
-    y_decision = clf.decision_function(X).reshape(x0.shape)
-    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
-    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
-
-plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-
-plt.show()
-
-
-from sklearn.svm import SVC
-
-poly_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
-    ])
-poly_kernel_svm_clf.fit(X, y)
-
-poly100_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
-    ])
-poly100_kernel_svm_clf.fit(X, y)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=3, r=1, C=5$", fontsize=18)
-
-plt.subplot(122)
-plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=10, r=100, C=5$", fontsize=18)
-
-plt.show()
-
-def gaussian_rbf(x, landmark, gamma):
-    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
-
-gamma = 0.3
-
-x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
-x2s = gaussian_rbf(x1s, -2, gamma)
-x3s = gaussian_rbf(x1s, 1, gamma)
-
-XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
-yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
-plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
-plt.plot(x1s, x2s, "g--")
-plt.plot(x1s, x3s, "b:")
-plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"Similarity", fontsize=14)
-plt.annotate(r'$\mathbf{x}$',
-             xy=(X1D[3, 0], 0),
-             xytext=(-0.5, 0.20),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
-plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
-plt.axis([-4.5, 4.5, -0.1, 1.1])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
-plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
-plt.xlabel(r"$x_2$", fontsize=20)
-plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
-plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
-             xy=(XK[3, 0], XK[3, 1]),
-             xytext=(0.65, 0.50),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
-plt.axis([-0.1, 1.1, -0.1, 1.1])
-    
-plt.subplots_adjust(right=1)
-
-plt.show()
-
-
-x1_example = X1D[3, 0]
-for landmark in (-2, 1):
-    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
-    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
-
-rbf_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
-    ])
-rbf_kernel_svm_clf.fit(X, y)
-
-
-from sklearn.svm import SVC
-
-gamma1, gamma2 = 0.1, 5
-C1, C2 = 0.001, 1000
-hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
-
-svm_clfs = []
-for gamma, C in hyperparams:
-    rbf_kernel_svm_clf = Pipeline([
-            ("scaler", StandardScaler()),
-            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
-        ])
-    rbf_kernel_svm_clf.fit(X, y)
-    svm_clfs.append(rbf_kernel_svm_clf)
-
-plt.figure(figsize=(11, 7))
-
-for i, svm_clf in enumerate(svm_clfs):
-    plt.subplot(221 + i)
-    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
-    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-    gamma, C = hyperparams[i]
-    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
-
-plt.show()
-

    @@ -366,6 +210,7 @@ plt.show()

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    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,28 +155,199 @@ MathJax.Hub.Config({ -

    Mathematical optimization of convex functions

    - +

    The moons example

    -A mathematical (quadratic) optimization problem, or just optimization problem, has the form -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ -subject to some constraints for say a selected set \( i=1,2,\dots, n \). -In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the -vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with. + +

    from __future__ import division, print_function, unicode_literals
 
-
    
-In our case we are particularly interested in a class of optimization problems called convex optmization problems. 
-In our discussion on gradient descent methods we discussed at length the definition of a convex function.
+import numpy as np
+np.random.seed(42)
 
-
    
-Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
 
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+

    @@ -194,6 +367,7 @@ Convex optimization problems play a central role in applied mathematics and we r

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    • diff --git a/doc/pub/week46/html/._week46-bs025.html b/doc/pub/week46/html/._week46-bs025.html index 9df02acb3..fe9b72791 100644 --- a/doc/pub/week46/html/._week46-bs025.html +++ b/doc/pub/week46/html/._week46-bs025.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,28 +155,27 @@ MathJax.Hub.Config({ -

    How do we solve these problems?

    +

    Mathematical optimization of convex functions

    -If we use Python as programming language and wish to venture beyond -scikit-learn, tensorflow and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming. +A mathematical (quadratic) optimization problem, or just optimization problem, has the form +$$ +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} +$$ + +subject to some constraints for say a selected set \( i=1,2,\dots, n \). +In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the +vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.

    -The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as +In our case we are particularly interested in a class of optimization problems called convex optmization problems. +In our discussion on gradient descent methods we discussed at length the definition of a convex function.

    - - -

    import numpy
-import cvxopt
-
    -

    -This will make our life much easier. You don't need t write your own optimizer. +Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.

    @@ -194,6 +195,7 @@ This will make our life much easier. You don't need t write your own optimizer.

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    • diff --git a/doc/pub/week46/html/._week46-bs026.html b/doc/pub/week46/html/._week46-bs026.html index 0d0eefb19..41e5739cc 100644 --- a/doc/pub/week46/html/._week46-bs026.html +++ b/doc/pub/week46/html/._week46-bs026.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source @@ -103,7 +104,7 @@ MathJax.Hub.Config({ - Week 46: Support Vector Machines + Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,71 +155,29 @@ MathJax.Hub.Config({ -

    A simple example

    +

    How do we solve these problems?

    -We remind ourselves about the general problem we want to solve -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber - &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. -\end{align*} -$$ +If we use Python as programming language and wish to venture beyond +scikit-learn, tensorflow and similar software which makes our +lives so much easier, we need to dive into the wonderful world of +quadratic programming. We can, if we wish, solve the minimization +problem using say standard gradient methods or conjugate gradient +methods. However, these methods tend to exhibit a rather slow +converge. So, welcome to the promised land of quadratic programming.

    -Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber - &\mathrm{subject to} \\ \nonumber - &x, y \geq 0 \\ \nonumber - &x+3y \geq 15 \\ \nonumber - &2x+5y \leq 100 \\ \nonumber - &3x+4y \leq 80. \\ \nonumber -\end{align*} -$$ +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as -The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) -$$ -\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. -$$ - -Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation -$$ -\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix -$$ -\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} -$$ - -is clearly positive semi-definite (all eigenvalues larger or equal zero). -Finally, the vector \( \boldsymbol{h} \) is defined as -$$ -\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -

    -Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero -The following code solves the equations for us

    -

    # Import the necessary packages
-import numpy
-from cvxopt import matrix
-from cvxopt import solvers
-P = matrix(numpy.diag([1,0]), tc=’d’)
-q = matrix(numpy.array([3,4]), tc=’d’)
-G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
-h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
-# Construct the QP, invoke solver
-sol = solvers.qp(P,q,G,h)
-# Extract optimal value and solution
-sol[’x’] 
-sol[’primal objective’]
+
    import numpy
+import cvxopt
 
    
+
    
+This will make our life much easier. You don't need t write your own optimizer.
+
 
    
 
    
 
@@ -235,6 +195,7 @@ sol[’primal objective’]
   
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diff --git a/doc/pub/week46/html/._week46-bs027.html b/doc/pub/week46/html/._week46-bs027.html
index 817d3168d..abdeff904 100644
--- a/doc/pub/week46/html/._week46-bs027.html
+++ b/doc/pub/week46/html/._week46-bs027.html
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
 
 
 
-
+
 
-Week 46: Support Vector Machines
+Week 46: Gradient Boosting Summary and Support Vector Machines
 
 
 
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
       
       
     
-    Week 46: Support Vector Machines
+    Week 46: Gradient Boosting Summary and Support Vector Machines
    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -153,28 +155,72 @@ MathJax.Hub.Config({ -

    Back to the more realistic cases

    +

    A simple example

    -We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have +We remind ourselves about the general problem we want to solve $$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber + &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. +\end{align*} $$ -subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and -\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). -With the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). +

    +Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem +$$ +\begin{align*} + &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber + &\mathrm{subject to} \\ \nonumber + &x, y \geq 0 \\ \nonumber + &x+3y \geq 15 \\ \nonumber + &2x+5y \leq 100 \\ \nonumber + &3x+4y \leq 80. \\ \nonumber +\end{align*} +$$ + +The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) +$$ +\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. +$$ + +Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation +$$ +\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$ + +We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix +$$ +\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} +$$ + +is clearly positive semi-definite (all eigenvalues larger or equal zero). +Finally, the vector \( \boldsymbol{h} \) is defined as +$$ +\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. +$$

    -code will be added - +Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero +The following code solves the equations for us

    + +

    # Import the necessary packages
+import numpy
+from cvxopt import matrix
+from cvxopt import solvers
+P = matrix(numpy.diag([1,0]), tc=’d’)
+q = matrix(numpy.array([3,4]), tc=’d’)
+G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
+h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
+# Construct the QP, invoke solver
+sol = solvers.qp(P,q,G,h)
+# Extract optimal value and solution
+sol[’x’] 
+sol[’primal objective’]
+
    +

      @@ -190,6 +236,8 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb
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    @@ -111,33 +112,34 @@ MathJax.Hub.Config({
  • Contents
    • @@ -156,7 +158,7 @@ MathJax.Hub.Config({
    -

    Week 46: Support Vector Machines

    +

    Week 46: Gradient Boosting Summary and Support Vector Machines

    @@ -172,7 +174,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 16, 2020

    +

    Nov 8, 2020


    @@ -196,7 +198,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/week46/html/week46-reveal.html b/doc/pub/week46/html/week46-reveal.html index 696e3908f..5fbc96d91 100644 --- a/doc/pub/week46/html/week46-reveal.html +++ b/doc/pub/week46/html/week46-reveal.html @@ -3,9 +3,9 @@ - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -132,7 +132,7 @@ MathJax.Hub.Config({ -

    Week 46: Support Vector Machines

    +

    Week 46: Gradient Boosting Summary and Support Vector Machines

    @@ -148,7 +148,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 16, 2020

    +

    Nov 8, 2020


    @@ -159,7 +159,23 @@ MathJax.Hub.Config({

    -

    Support Vector Machines, overarching aims

    +

    Overview of week 46

    + +
      +

    • Thursday: Summary of Gradient Boosting and further examples of applications.
      • +

    • Friday: Support Vector Machines, classification and regression
      • +
    +

    + +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion. + +

    +Overview of Support Vector Machines. see also this video. +

    + + +
    +

    Support Vector Machines, overarching aims

    A Support Vector Machine (SVM) is a very powerful and versatile @@ -192,7 +208,7 @@ unlikely that we can separate classes easily by say straight lines.

    -

    Hyperplanes and all that

    +

    Hyperplanes and all that

    The theory behind support vector machines (SVM hereafter) is based on @@ -245,9 +261,9 @@ lin_clf.fit(X_scaled, y) svm_clf.fit(X_scaled, y) sgd_clf.fit(X_scaled, y) -print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) -print("SVC: ", svm_clf.intercept_, svm_clf.coef_) -print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) +print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) +print("SVC: ", svm_clf.intercept_, svm_clf.coef_) +print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) # Compute the slope and bias of each decision boundary w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1] @@ -280,7 +296,7 @@ plt.show()

    -

    What is a hyperplane?

    +

    What is a hyperplane?

    The aim of the SVM algorithm is to find a hyperplane in a @@ -315,7 +331,7 @@ $$

    -

    A \( p \)-dimensional space of features

    +

    A \( p \)-dimensional space of features

    We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \). @@ -367,7 +383,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a

    -

    The two-dimensional case

    +

    The two-dimensional case

    Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional @@ -394,7 +410,7 @@ for our data sample.

    -

    Getting into the details

    +

    Getting into the details

    Let us define the function @@ -420,7 +436,7 @@ $$

    -

    First attempt at a minimization approach

    +

    First attempt at a minimization approach

    How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could @@ -451,7 +467,7 @@ $$

    -

    Solving the equations

    +

    Solving the equations

    We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations @@ -473,7 +489,7 @@ where \( \eta \) is our by now well-known learning rate.

    -

    Code Example

    +

    Code Example

    The equations we discussed above can be coded rather easily (the @@ -488,7 +504,7 @@ regression). We are going to set up a simple case with two classes only and we w

    -

    Problems with the Simpler Approach

    +

    Problems with the Simpler Approach

    There are however problems with this approach, although it looks @@ -504,7 +520,7 @@ at all.

    -

    A better approach

    +

    A better approach

    A better approach is rather to try to define a large margin between @@ -554,7 +570,7 @@ about Lagrangian multipliers.

    -

    A quick Reminder on Lagrangian Multipliers

    +

    A quick Reminder on Lagrangian Multipliers

    Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an @@ -616,7 +632,7 @@ Then \( dz \) is no longer arbitrary.

    -

    Adding the Multiplier

    +

    Adding the Multiplier

    However, we can add to @@ -671,7 +687,7 @@ $$

    -

    Setting up the Problem

    +

    Setting up the Problem

    In order to solve the above problem, we define the following Lagrangian function to be minimized

     
    $$ @@ -723,7 +739,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support

    -

    The problem to solve

    +

    The problem to solve

    We can rewrite @@ -751,7 +767,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec

    -

    The last steps

    +

    The last steps

    Solving the above problem, yields the values of \( \lambda_i \). @@ -795,7 +811,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr

    -

    A soft classifier

    +

    A soft classifier

    Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. @@ -834,7 +850,7 @@ misclassifications.

    -

    Soft optmization problem

    +

    Soft optmization problem

    This has in turn the consequences that we change our optmization problem to finding the minimum of @@ -906,7 +922,7 @@ $$

    -

    Kernels and non-linearity

    +

    Kernels and non-linearity

    The cases we have studied till now, were all characterized by two classes @@ -954,7 +970,7 @@ y = np.array([0, 11, 4)) plt.subplot(121) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs") plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^") @@ -963,7 +979,7 @@ plt.xlabel(r"$x_1$", fontsize=4.5, 4.5, -0.2, 0.2]) plt.subplot(122) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.axvline(x=0, color='k') plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs") @@ -980,7 +996,7 @@ plt.show()

    -

    The equations

    +

    The equations

    Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) @@ -1035,7 +1051,7 @@ the trouble of performing the transformation

    -

    The problem to solve

    +

    The problem to solve

    Using our definition of the kernel We can rewrite again the Lagrangian

     
    $$ @@ -1077,7 +1093,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.

    -

    Different kernels and Mercer's theorem

    +

    Different kernels and Mercer's theorem

    There are several popular kernels being used. These are @@ -1118,7 +1134,7 @@ in practice.

    -

    The moons example

    +

    The moons example

    @@ -1151,7 +1167,7 @@ X, y = make_moons(n_samples=100, noise=0][y==0], X[:, 1][y==0], "bs") plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^") plt.axis(axes) - plt.grid(True, which='both') + plt.grid(True, which='both') plt.xlabel(r"$x_1$", fontsize=20) plt.ylabel(r"$x_2$", fontsize=20, rotation=0) @@ -1229,7 +1245,7 @@ yk = np.array([0, 11, 4)) plt.subplot(121) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red") plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs") @@ -1251,7 +1267,7 @@ plt.text(1, 0.9 plt.axis([-4.5, 4.5, -0.1, 1.1]) plt.subplot(122) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.axvline(x=0, color='k') plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs") @@ -1276,7 +1292,7 @@ plt.show() x1_example = X1D[3, 0] for landmark in (-2, 1): k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma) - print("Phi({}, {}) = {}".format(x1_example, landmark, k)) + print("Phi({}, {}) = {}".format(x1_example, landmark, k)) rbf_kernel_svm_clf = Pipeline([ ("scaler", StandardScaler()), @@ -1315,7 +1331,7 @@ plt.show()

    -

    Mathematical optimization of convex functions

    +

    Mathematical optimization of convex functions

    A mathematical (quadratic) optimization problem, or just optimization problem, has the form @@ -1342,7 +1358,7 @@ Convex optimization problems play a central role in applied mathematics and we r

    -

    How do we solve these problems?

    +

    How do we solve these problems?

    If we use Python as programming language and wish to venture beyond @@ -1368,7 +1384,7 @@ This will make our life much easier. You don't need t write your own optimizer.

    -

    A simple example

    +

    A simple example

    We remind ourselves about the general problem we want to solve @@ -1449,7 +1465,7 @@ sol[’primal obj

    -

    Back to the more realistic cases

    +

    Back to the more realistic cases

    We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have diff --git a/doc/pub/week46/html/week46-solarized.html b/doc/pub/week46/html/week46-solarized.html index 35b8748a6..25201908f 100644 --- a/doc/pub/week46/html/week46-solarized.html +++ b/doc/pub/week46/html/week46-solarized.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines @@ -35,39 +35,40 @@ div { text-align: justify; text-justify: inter-word; } @@ -93,7 +94,7 @@ MathJax.Hub.Config({ -

    Week 46: Support Vector Machines

    +

    Week 46: Gradient Boosting Summary and Support Vector Machines

    @@ -109,12 +110,27 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 16, 2020

    +

    Nov 8, 2020












    -

    Support Vector Machines, overarching aims

    +

    Overview of week 46

    + +
      +
    • Thursday: Summary of Gradient Boosting and further examples of applications.
      • +
    • Friday: Support Vector Machines, classification and regression
      • +
    + +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion. + +

    +Overview of Support Vector Machines. see also this video. + +

    +









    + +

    Support Vector Machines, overarching aims

    A Support Vector Machine (SVM) is a very powerful and versatile @@ -147,7 +163,7 @@ unlikely that we can separate classes easily by say straight lines.











    -

    Hyperplanes and all that

    +

    Hyperplanes and all that

    The theory behind support vector machines (SVM hereafter) is based on @@ -200,9 +216,9 @@ lin_clf.fit(X_scaled, y) svm_clf.fit(X_scaled, y) sgd_clf.fit(X_scaled, y) -print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) -print("SVC: ", svm_clf.intercept_, svm_clf.coef_) -print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) +print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) +print("SVC: ", svm_clf.intercept_, svm_clf.coef_) +print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) # Compute the slope and bias of each decision boundary w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1] @@ -234,7 +250,7 @@ plt.show()











    -

    What is a hyperplane?

    +

    What is a hyperplane?

    The aim of the SVM algorithm is to find a hyperplane in a @@ -265,7 +281,7 @@ $$











    -

    A \( p \)-dimensional space of features

    +

    A \( p \)-dimensional space of features

    We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \). @@ -307,7 +323,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a

    -

    The two-dimensional case

    +

    The two-dimensional case

    Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional @@ -334,7 +350,7 @@ for our data sample.











    -

    Getting into the details

    +

    Getting into the details

    Let us define the function @@ -356,7 +372,7 @@ $$











    -

    First attempt at a minimization approach

    +

    First attempt at a minimization approach

    How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could @@ -381,7 +397,7 @@ $$











    -

    Solving the equations

    +

    Solving the equations

    We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations @@ -399,7 +415,7 @@ where \( \eta \) is our by now well-known learning rate.











    -

    Code Example

    +

    Code Example

    The equations we discussed above can be coded rather easily (the @@ -413,7 +429,7 @@ regression). We are going to set up a simple case with two classes only and we w











    -

    Problems with the Simpler Approach

    +

    Problems with the Simpler Approach

    There are however problems with this approach, although it looks @@ -429,7 +445,7 @@ at all.











    -

    A better approach

    +

    A better approach

    A better approach is rather to try to define a large margin between @@ -471,7 +487,7 @@ about Lagrangian multipliers.











    -

    A quick Reminder on Lagrangian Multipliers

    +

    A quick Reminder on Lagrangian Multipliers

    Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an @@ -521,7 +537,7 @@ Then \( dz \) is no longer arbitrary.











    -

    Adding the Multiplier

    +

    Adding the Multiplier

    However, we can add to @@ -564,7 +580,7 @@ $$











    -

    Setting up the Problem

    +

    Setting up the Problem

    In order to solve the above problem, we define the following Lagrangian function to be minimized $$ {\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], @@ -605,7 +621,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support











    -

    The problem to solve

    +

    The problem to solve

    We can rewrite @@ -629,7 +645,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec











    -

    The last steps

    +

    The last steps

    Solving the above problem, yields the values of \( \lambda_i \). @@ -663,7 +679,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr











    -

    A soft classifier

    +

    A soft classifier

    Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. @@ -698,7 +714,7 @@ misclassifications.











    -

    Soft optmization problem

    +

    Soft optmization problem

    This has in turn the consequences that we change our optmization problem to finding the minimum of @@ -752,7 +768,7 @@ $$











    -

    Kernels and non-linearity

    +

    Kernels and non-linearity

    The cases we have studied till now, were all characterized by two classes @@ -800,7 +816,7 @@ y = np.array([0, 11, 4)) plt.subplot(121) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs") plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^") @@ -809,7 +825,7 @@ plt.xlabel(r"$x_1$", fontsize=4.5, 4.5, -0.2, 0.2]) plt.subplot(122) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.axvline(x=0, color='k') plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs") @@ -825,7 +841,7 @@ plt.show()











    -

    The equations

    +

    The equations

    Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables) @@ -870,7 +886,7 @@ the trouble of performing the transformation











    -

    The problem to solve

    +

    The problem to solve

    Using our definition of the kernel We can rewrite again the Lagrangian $$ {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, @@ -906,7 +922,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.











    -

    Different kernels and Mercer's theorem

    +

    Different kernels and Mercer's theorem

    There are several popular kernels being used. These are @@ -944,7 +960,7 @@ in practice.











    -

    The moons example

    +

    The moons example

    @@ -977,7 +993,7 @@ X, y = make_moons(n_samples=100, noise=0][y==0], X[:, 1][y==0], "bs") plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^") plt.axis(axes) - plt.grid(True, which='both') + plt.grid(True, which='both') plt.xlabel(r"$x_1$", fontsize=20) plt.ylabel(r"$x_2$", fontsize=20, rotation=0) @@ -1055,7 +1071,7 @@ yk = np.array([0, 11, 4)) plt.subplot(121) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red") plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs") @@ -1077,7 +1093,7 @@ plt.text(1, 0.9 plt.axis([-4.5, 4.5, -0.1, 1.1]) plt.subplot(122) -plt.grid(True, which='both') +plt.grid(True, which='both') plt.axhline(y=0, color='k') plt.axvline(x=0, color='k') plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs") @@ -1102,7 +1118,7 @@ plt.show() x1_example = X1D[3, 0] for landmark in (-2, 1): k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma) - print("Phi({}, {}) = {}".format(x1_example, landmark, k)) + print("Phi({}, {}) = {}".format(x1_example, landmark, k)) rbf_kernel_svm_clf = Pipeline([ ("scaler", StandardScaler()), @@ -1140,7 +1156,7 @@ plt.show()











    -

    Mathematical optimization of convex functions

    +

    Mathematical optimization of convex functions

    A mathematical (quadratic) optimization problem, or just optimization problem, has the form @@ -1165,7 +1181,7 @@ Convex optimization problems play a central role in applied mathematics and we r











    -

    How do we solve these problems?

    +

    How do we solve these problems?

    If we use Python as programming language and wish to venture beyond @@ -1191,7 +1207,7 @@ This will make our life much easier. You don't need t write your own optimizer.











    -

    A simple example

    +

    A simple example

    We remind ourselves about the general problem we want to solve @@ -1259,7 +1275,7 @@ sol[’primal obj











    -

    Back to the more realistic cases

    +

    Back to the more realistic cases

    We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have diff --git a/doc/pub/week46/html/week46.html b/doc/pub/week46/html/week46.html index d8f773ff5..44693ef4c 100644 --- a/doc/pub/week46/html/week46.html +++ b/doc/pub/week46/html/week46.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Week 46: Support Vector Machines +Week 46: Gradient Boosting Summary and Support Vector Machines