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updating week 46

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mhjensen
2020-11-09 23:21:39 +01:00
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@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
<!-- tocinfo
{'highest level': 2,
'sections': [('Overview of week 46', 2, None, '___sec0'),
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
('Hyperplanes and all that', 2, None, '___sec2'),
('What is a hyperplane?', 2, None, '___sec3'),
('A $p$-dimensional space of features', 2, None, '___sec4'),
('The two-dimensional case', 2, None, '___sec5'),
('Getting into the details', 2, None, '___sec6'),
('First attempt at a minimization approach', 2, None, '___sec7'),
('Solving the equations', 2, None, '___sec8'),
('Code Example', 2, None, '___sec9'),
('Problems with the Simpler Approach', 2, None, '___sec10'),
('A better approach', 2, None, '___sec11'),
('Thursday', 2, None, '___sec1'),
('Friday', 2, None, '___sec2'),
('Support Vector Machines, overarching aims', 2, None, '___sec3'),
('Hyperplanes and all that', 2, None, '___sec4'),
('What is a hyperplane?', 2, None, '___sec5'),
('A $p$-dimensional space of features', 2, None, '___sec6'),
('The two-dimensional case', 2, None, '___sec7'),
('Getting into the details', 2, None, '___sec8'),
('First attempt at a minimization approach', 2, None, '___sec9'),
('Solving the equations', 2, None, '___sec10'),
('Code Example', 2, None, '___sec11'),
('Problems with the Simpler Approach', 2, None, '___sec12'),
('A better approach', 2, None, '___sec13'),
('A quick Reminder on Lagrangian Multipliers',
2,
None,
'___sec12'),
('Adding the Multiplier', 2, None, '___sec13'),
('Setting up the Problem', 2, None, '___sec14'),
('The problem to solve', 2, None, '___sec15'),
('The last steps', 2, None, '___sec16'),
('A soft classifier', 2, None, '___sec17'),
('Soft optmization problem', 2, None, '___sec18'),
('Kernels and non-linearity', 2, None, '___sec19'),
('The equations', 2, None, '___sec20'),
('The problem to solve', 2, None, '___sec21'),
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
('The moons example', 2, None, '___sec23'),
'___sec14'),
('Adding the Multiplier', 2, None, '___sec15'),
('Setting up the Problem', 2, None, '___sec16'),
('The problem to solve', 2, None, '___sec17'),
('The last steps', 2, None, '___sec18'),
('A soft classifier', 2, None, '___sec19'),
('Soft optmization problem', 2, None, '___sec20'),
('Kernels and non-linearity', 2, None, '___sec21'),
('The equations', 2, None, '___sec22'),
('The problem to solve', 2, None, '___sec23'),
("Different kernels and Mercer's theorem", 2, None, '___sec24'),
('The moons example', 2, None, '___sec25'),
('Mathematical optimization of convex functions',
2,
None,
'___sec24'),
('How do we solve these problems?', 2, None, '___sec25'),
('A simple example', 2, None, '___sec26'),
('Back to the more realistic cases', 2, None, '___sec27')]}
'___sec26'),
('How do we solve these problems?', 2, None, '___sec27'),
('A simple example', 2, None, '___sec28'),
('Back to the more realistic cases', 2, None, '___sec29')]}
end of tocinfo -->
<body>
@@ -113,33 +115,35 @@ MathJax.Hub.Config({
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">A simple example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Getting into the details</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">How do we solve these problems?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" style="font-size: 80%;">Back to the more realistic cases</a></li>
</ul>
</li>
@@ -155,71 +159,28 @@ MathJax.Hub.Config({
<a name="part0027"></a>
<!-- !split -->
<h2 id="___sec26" class="anchor">A simple example </h2>
<h2 id="___sec26" class="anchor">Mathematical optimization of convex functions </h2>
<p>
We remind ourselves about the general problem we want to solve
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
$$
\begin{align*}
&\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber
&\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f.
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
\end{align*}
$$
<p>
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
$$
\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
$$
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.
<p>
Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero
The following code solves the equations for us
<p>
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.
<p>
Convex optimization problems play a central role in applied mathematics and we recommend strongly <a href="http://web.stanford.edu/~boyd/cvxbook/" target="_self">Boyd and Vandenberghe's text on the topics</a>.
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Import the necessary packages</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> matrix
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> solvers
P <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>diag([<span style="color: #666666">1</span>,<span style="color: #666666">0</span>]), tc<span style="color: #666666">=</span>’d’)
q <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]), tc<span style="color: #666666">=</span>’d’)
G <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([[<span style="color: #666666">-1</span>,<span style="color: #666666">0</span>],[<span style="color: #666666">0</span>,<span style="color: #666666">-1</span>],[<span style="color: #666666">-1</span>,<span style="color: #666666">-3</span>],[<span style="color: #666666">2</span>,<span style="color: #666666">5</span>],[<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]]), tc<span style="color: #666666">=</span>’d’)
h <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>,<span style="color: #666666">0</span>,<span style="color: #666666">-15</span>,<span style="color: #666666">100</span>,<span style="color: #666666">80</span>]), tc<span style="color: #666666">=</span>’d’)
<span style="color: #408080; font-style: italic"># Construct the QP, invoke solver</span>
sol <span style="color: #666666">=</span> solvers<span style="color: #666666">.</span>qp(P,q,G,h)
<span style="color: #408080; font-style: italic"># Extract optimal value and solution</span>
sol[’x’]
sol[’primal objective’]
</pre></div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
@@ -237,6 +198,8 @@ sol[’primal objective’]
<li><a href="._week46-bs026.html">27</a></li>
<li class="active"><a href="._week46-bs027.html">28</a></li>
<li><a href="._week46-bs028.html">29</a></li>
<li><a href="._week46-bs029.html">30</a></li>
<li><a href="._week46-bs030.html">31</a></li>
<li><a href="._week46-bs028.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->