We finalize our discussion on Support Vector Machines with an emphasis on kernel transformations and applications to regression. The following video attempts at giving an overview on this part. See also the follow-up video.
Friday's lecture is split in two parts. It starts with a summary of what we have done this semester and continues with perspectives for future studies and modern research projects in machine learning.
As discussed last week, 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.
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. I recommend you take a look at the lectures from last week on the binary classification problem.
The cases we studied last week 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.
If our feature space is not easy to separate, as shown in the figure generated by the code below, we can achieve a better separation by introducing a more complex basis functions. The ideal would be, as shown by the code example below, to, via a specific transformation to obtain a separation between the classes that is almost linear. See also Figures 12.2 and 12.3 of Hastie et al..
We can achieve a better separation by introducing more complex basis functions. The ideal would be (see Figures 12.2 and 12.3) to, via a specific transformation to obtain a separation between the classes which is almost linear.
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.
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()
Suppose we define a polynomial transformation of degree two only. We define a vector \( \boldsymbol{x}_i=[x_i,y_i] \) and have $$ \boldsymbol{z}_i^T = \phi(\boldsymbol{x}_i)^T =\left[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i\right]. $$
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 $$ y_i(\boldsymbol{z}_i^T\boldsymbol{w}+b)= 1 \hspace{0.1cm}\forall i, $$ 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 $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1, \sqrt{2}x_i, \sqrt{2}y_i, x_i^2, y_i^2, \sqrt{2}x_iy_i]\begin{bmatrix} 1\\ \sqrt{2}x_j \\ \sqrt{2}y_j \\ x_j^2\\ y_i^2\\ \sqrt{2}x_jy_j \end{bmatrix}=1+2x_ix_j+2y_iy_j+(x_ix_j)^2+(y_iy_j)^2+2x_ix_jy_iy_j. $$
We note that this dot product can be rewritten as $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[1+\boldsymbol{x}_i^T\boldsymbol{x}_j]^d, $$ where \( d=2 \) in our case and \( \boldsymbol{x}_i=[x_i,y_i] \) and \( \boldsymbol{x}_j=[x_j,y_j] \). To compute the last equation is however inefficient from a computational stand. Instead of computing the last equation for the kernel, we simply compute the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called kernel trick.
Using our definition of the kernel, we can rewrite again the Lagrangian $$ {\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, $$ 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 \( 0\leq \lambda_i \leq C \).
We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type $$ \begin{align*} &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber &\mathrm{s.t} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. \end{align*} $$ 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 \( \boldsymbol{h} \).
There are several popular kernels being used. These are
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 $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$
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 __future__ import division, print_function, unicode_literals
import numpy as np
np.random.seed(42)
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()
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.
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.
Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.
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 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.
The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as
import numpy
import cvxopt
This will make our life much easier. You don't need to write your own optimizer.
We remind ourselves about the general problem we want to solve $$ \begin{align*} &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber &\mathrm{s.t.} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. \end{align*} $$ Note: we use s.t. for subject to.
Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem $$ \begin{align*} &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber &\mathrm{s.t.} \\ \nonumber &x, y \geq 0 \\ \nonumber &x+3y \geq 15 \\ \nonumber &2x+5y \leq 100 \\ \nonumber &3x+4y \leq 80. \\ \nonumber \end{align*} $$ The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) $$ \frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}5\\ 3 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. $$ 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']
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 $$ \frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ \end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, $$ 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 \( 0\leq \lambda_i \leq C \).
We have the general problem $$ \begin{align*} &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber &\mathrm{s.t.} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. \end{align*} $$
We have two constraints, \( 0\le \lambda_i \) and \( \lambda_i \le C \). To do this we multiply the ones with the constraint \( \ge \) with \( -1 \) in order to get \( \le \). It means that the problem \( \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \) can be written as $$ \begin{bmatrix} -1& 0 & 0 & \dots & 0 \\ 0& -1 & 0 & \dots & 0 \\ 0& 0 & -1 & \dots & 0 \\ \dots& \dots & \dots & \dots & \dots \\ 0& 0 & 0 & \dots & -1 \\ 1 & 0 & 0 & \dots & 0 \\ 0& 1 & 0 & \dots & 0 \\ 0& 0 & 1 & \dots & 0 \\ \dots& \dots & \dots & \dots & \dots \\ 0& 0 & 0 & \dots & 1 \\ \end{bmatrix} \begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \lambda_3 \\ \dots \\ \lambda_n \\ \end{bmatrix} \preceq \begin{bmatrix} 0 \\ 0 \\ 0 \\ \dots \\ 0 \\ C \\ C \\ C \\ \dots \\ C \\ \end{bmatrix}. $$
And then we are ready to go.
More text will be added here. See also Hastie et al. section 12.3.
The course has two central parts
The following topics have been discussed:
The following topics will be covered
The course introduces a variety of central algorithms and methods essential for studies of data analysis and machine learning. The course is project based and through the various projects, normally three, you will be exposed to fundamental research problems in these fields, with the aim to reproduce state of the art scientific results. The students will learn to develop and structure large codes for studying these systems, get acquainted with computing facilities and learn to handle large scientific projects. A good scientific and ethical conduct is emphasized throughout the course.
Where to find recent results:
When do we resample?
The link here https://www.mn.uio.no/english/research/about/centre-focus/innovation/data-science/studies/ gives an excellent overview of courses on Machine learning at UiO.
Based on multi-layer nonlinear neural networks, deep learning can learn directly from raw data, automatically extract and abstract features from layer to layer, and then achieve the goal of regression, classification, or ranking. Deep learning has made breakthroughs in computer vision, speech processing and natural language, and reached or even surpassed human level. The success of deep learning is mainly due to the three factors: big data, big model, and big computing.
In the past few decades, many different architectures of deep neural networks have been proposed, such as
The approaches to machine learning are many, but are often split into two main categories. In supervised learning we know the answer to a problem, and let the computer deduce the logic behind it. On the other hand, unsupervised learning is a method for finding patterns and relationship in data sets without any prior knowledge of the system. Some authours also operate with a third category, namely reinforcement learning. This is a paradigm of learning inspired by behavioural psychology, where learning is achieved by trial-and-error, solely from rewards and punishment.
Another way to categorize machine learning tasks is to consider the desired output of a system. Some of the most common tasks are:
What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately. One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.
The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images.
Furthermore, they have been used to solve complicated quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models.
Why use a generative model rather than the more well known discriminative deep neural networks (DNN)?
A BM is what we would call an undirected probabilistic graphical model with stochastic continuous or discrete units.
It is interpreted as a stochastic recurrent neural network where the state of each unit(neurons/nodes) depends on the units it is connected to. The weights in the network represent thus the strength of the interaction between various units/nodes.
It turns into a Hopfield network if we choose deterministic rather than stochastic units. In contrast to a Hopfield network, a BM is a so-called generative model. It allows us to generate new samples from the learned distribution.
A standard BM network is divided into a set of observable and visible units \( \hat{x} \) and a set of unknown hidden units/nodes \( \hat{h} \).
Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to \( 1 \).
BMs are stackable, meaning they cwe can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning
However, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS. Here we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below.
The network layers:
The goal of the hidden layer is to increase the model's expressive power. We encode complex interactions between visible variables by introducing additional, hidden variables that interact with visible degrees of freedom in a simple manner, yet still reproduce the complex correlations between visible degrees in the data once marginalized over (integrated out).
The network parameters, to be optimized/learned:
The restricted Boltzmann machine is described by a Boltzmann distribution $$ \begin{align} P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}, \label{_auto1} \end{align} $$ where \( Z \) is the normalization constant or partition function, defined as $$ \begin{align} Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}. \label{_auto2} \end{align} $$ It is common to ignore \( T_0 \) by setting it to one.
The function \( E(\mathbf{x},\mathbf{h}) \) gives the energy of a configuration (pair of vectors) \( (\mathbf{x}, \mathbf{h}) \). The lower the energy of a configuration, the higher the probability of it. This function also depends on the parameters \( \mathbf{a} \), \( \mathbf{b} \) and \( W \). Thus, when we adjust them during the learning procedure, we are adjusting the energy function to best fit our problem.
An expression for the energy function is $$ E(\hat{x},\hat{h}) = -\sum_{ia}^{NA}b_i^a \alpha_i^a(x_i)-\sum_{jd}^{MD}c_j^d \beta_j^d(h_j)-\sum_{ijad}^{NAMD}b_i^a \alpha_i^a(x_i)c_j^d \beta_j^d(h_j)w_{ij}^{ad}. $$
Here \( \beta_j^d(h_j) \) and \( \alpha_i^a(x_j) \) are so-called transfer functions that map a given input value to a desired feature value. The labels \( a \) and \( d \) denote that there can be multiple transfer functions per variable. The first sum depends only on the visible units. The second on the hidden ones. Note that there is no connection between nodes in a layer.
The quantities \( b \) and \( c \) can be interpreted as the visible and hidden biases, respectively.
The connection between the nodes in the two layers is given by the weights \( w_{ij} \).
RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows: $$ \begin{align} E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j, \label{_auto3} \end{align} $$ where the binary values taken on by the nodes are most commonly 0 and 1.
Another varient is the RBM where the visible units are Gaussian while the hidden units remain binary: $$ \begin{align} E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}. \label{_auto4} \end{align} $$
Autoencoders are artificial neural networks capable of learning efficient representations of the input data (these representations are called codings) without any supervision (i.e., the training set is unlabeled). These codings typically have a much lower dimensionality than the input data, making autoencoders useful for dimensionality reduction.
More importantly, autoencoders act as powerful feature detectors, and they can be used for unsupervised pretraining of deep neural networks.
Lastly, they are capable of randomly generating new data that looks very similar to the training data; this is called a generative model. For example, you could train an autoencoder on pictures of faces, and it would then be able to generate new faces. Surprisingly, autoencoders work by simply learning to copy their inputs to their outputs. This may sound like a trivial task, but we will see that constraining the network in various ways can make it rather difficult. For example, you can limit the size of the internal representation, or you can add noise to the inputs and train the network to recover the original inputs. These constraints prevent the autoencoder from trivially copying the inputs directly to the outputs, which forces it to learn efficient ways of representing the data. In short, the codings are byproducts of the autoencoder’s attempt to learn the identity function under some constraints.
See also A. Geron's textbook, chapter 15.
This is an important topic if we aim at extracting a probability distribution. This gives us also a confidence interval and error estimates.
Bayesian machine learning allows us to encode our prior beliefs about what those models should look like, independent of what the data tells us. This is especially useful when we don’t have a ton of data to confidently learn our model.
Video on Bayesian deep learning
See also the slides here.
Reinforcement Learning (RL) is one of the most exciting fields of Machine Learning today, and also one of the oldest. It has been around since the 1950s, producing many interesting applications over the years.
It studies how agents take actions based on trial and error, so as to maximize some notion of cumulative reward in a dynamic system or environment. Due to its generality, the problem has also been studied in many other disciplines, such as game theory, control theory, operations research, information theory, multi-agent systems, swarm intelligence, statistics, and genetic algorithms.
In March 2016, AlphaGo, a computer program that plays the board game Go, beat Lee Sedol in a five-game match. This was the first time a computer Go program had beaten a 9-dan (highest rank) professional without handicaps. AlphaGo is based on deep convolutional neural networks and reinforcement learning. AlphaGo’s victory was a major milestone in artificial intelligence and it has also made reinforcement learning a hot research area in the field of machine learning.
Lecture on Reinforcement Learning.
See also A. Geron's textbook, chapter 16.
The goal of transfer learning is to transfer the model or knowledge obtained from a source task to the target task, in order to resolve the issues of insufficient training data in the target task. The rationality of doing so lies in that usually the source and target tasks have inter-correlations, and therefore either the features, samples, or models in the source task might provide useful information for us to better solve the target task. Transfer learning is a hot research topic in recent years, with many problems still waiting to be studied.
The conventional deep generative model has a potential problem: the model tends to generate extreme instances to maximize the probabilistic likelihood, which will hurt its performance. Adversarial learning utilizes the adversarial behaviors (e.g., generating adversarial instances or training an adversarial model) to enhance the robustness of the model and improve the quality of the generated data. In recent years, one of the most promising unsupervised learning technologies, generative adversarial networks (GAN), has already been successfully applied to image, speech, and text.
Lecture on adversial learning.
Dual learning is a new learning paradigm, the basic idea of which is to use the primal-dual structure between machine learning tasks to obtain effective feedback/regularization, and guide and strengthen the learning process, thus reducing the requirement of large-scale labeled data for deep learning. The idea of dual learning has been applied to many problems in machine learning, including machine translation, image style conversion, question answering and generation, image classification and generation, text classification and generation, image-to-text, and text-to-image.
Distributed computation will speed up machine learning algorithms, significantly improve their efficiency, and thus enlarge their application. When distributed meets machine learning, more than just implementing the machine learning algorithms in parallel is required.
Meta learning is an emerging research direction in machine learning. Roughly speaking, meta learning concerns learning how to learn, and focuses on the understanding and adaptation of the learning itself, instead of just completing a specific learning task. That is, a meta learner needs to be able to evaluate its own learning methods and adjust its own learning methods according to specific learning tasks.
While there has been much progress in machine learning, there are also challenges.
For example, the mainstream machine learning technologies are black-box approaches, making us concerned about their potential risks. To tackle this challenge, we may want to make machine learning more explainable and controllable. As another example, the computational complexity of machine learning algorithms is usually very high and we may want to invent lightweight algorithms or implementations. Furthermore, in many domains such as physics, chemistry, biology, and social sciences, people usually seek elegantly simple equations (e.g., the Schrödinger equation) to uncover the underlying laws behind various phenomena. In the field of machine learning, can we reveal simple laws instead of designing more complex models for data fitting? Although there are many challenges, we are still very optimistic about the future of machine learning. As we look forward to the future, here are what we think the research hotspots in the next ten years will be.
See the article on Discovery of Physics From Data: Universal Laws and Discrepancies
Machine learning, especially deep learning, evolves rapidly. The ability gap between machine and human on many complex cognitive tasks becomes narrower and narrower. However, we are still in the very early stage in terms of explaining why those effective models work and how they work.
What is missing: the gap between correlation and causation. Standard Machine Learning is based on what e have called a frequentist approach.
Most machine learning techniques, especially the statistical ones, depend highly on correlations in data sets to make predictions and analyses. In contrast, rational humans tend to reply on clear and trustworthy causality relations obtained via logical reasoning on real and clear facts. It is one of the core goals of explainable machine learning to transition from solving problems by data correlation to solving problems by logical reasoning.
Bayesian Machine Learning is one of the exciting research directions in this field.
Quantum machine learning is an emerging interdisciplinary research area at the intersection of quantum computing and machine learning.
Quantum computers use effects such as quantum coherence and quantum entanglement to process information, which is fundamentally different from classical computers. Quantum algorithms have surpassed the best classical algorithms in several problems (e.g., searching for an unsorted database, inverting a sparse matrix), which we call quantum acceleration.
When quantum computing meets machine learning, it can be a mutually beneficial and reinforcing process, as it allows us to take advantage of quantum computing to improve the performance of classical machine learning algorithms. In addition, we can also use the machine learning algorithms (on classic computers) to analyze and improve quantum computing systems.
Read interview with Maria Schuld on her work on Quantum Machine Learning. See also her recent textbook.
Many quantum machine learning algorithms are based on variants of quantum algorithms for solving linear equations, which can efficiently solve N-variable linear equations with complexity of O(log2 N) under certain conditions. The quantum matrix inversion algorithm can accelerate many machine learning methods, such as least square linear regression, least square version of support vector machine, Gaussian process, and more. The training of these algorithms can be simplified to solve linear equations. The key bottleneck of this type of quantum machine learning algorithms is data input—that is, how to initialize the quantum system with the entire data set. Although efficient data-input algorithms exist for certain situations, how to efficiently input data into a quantum system is as yet unknown for most cases.
In quantum reinforcement learning, a quantum agent interacts with the classical environment to obtain rewards from the environment, so as to adjust and improve its behavioral strategies. In some cases, it achieves quantum acceleration by the quantum processing capabilities of the agent or the possibility of exploring the environment through quantum superposition. Such algorithms have been proposed in superconducting circuits and systems of trapped ions.
Dedicated quantum information processors, such as quantum annealers and programmable photonic circuits, are well suited for building deep quantum networks. The simplest deep quantum network is the Boltzmann machine. The classical Boltzmann machine consists of bits with tunable interactions and is trained by adjusting the interaction of these bits so that the distribution of its expression conforms to the statistics of the data. To quantize the Boltzmann machine, the neural network can simply be represented as a set of interacting quantum spins that correspond to an adjustable Ising model. Then, by initializing the input neurons in the Boltzmann machine to a fixed state and allowing the system to heat up, we can read out the output qubits to get the result.
Machine learning aims to imitate how humans learn. While we have developed successful machine learning algorithms, until now we have ignored one important fact: humans are social. Each of us is one part of the total society and it is difficult for us to live, learn, and improve ourselves, alone and isolated. Therefore, we should design machines with social properties. Can we let machines evolve by imitating human society so as to achieve more effective, intelligent, interpretable “social machine learning”?
And much more.
Early computer scientist Alan Kay said, The best way to predict the future is to create it. Therefore, all machine learning practitioners, whether scholars or engineers, professors or students, need to work together to advance these important research topics. Together, we will not just predict the future, but create it.
