\documentclass[12pt,a4paper,pdftex]{exam}
\newcommand{\exercisetopic}{Gradient descent}
\newcommand{\exercisenum}{9}
\newcommand{\exercisedate}{December 22th, 2020}
\input{../../exercisesheader}
\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}
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\begin{document}
\input{../../exercisestitle}
\begin{questions}
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\question \qt{Read sections 8.1 -- 8.5 of chapter 8 on ``optimization
and gradient descent!}\vspace{-3ex}
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\question \qt{Fitting the time constant of an exponential function}
Let's assume we record the membrane potential from a photoreceptor
neuron. We define the resting potential of the neuron to be at
0\,mV. By means of a brief current injection we increase the
membrane potential by exactly 1\,mV. We then record how the membrane
potential decays exponentially down to the resting potential. We are
interested in the membrane time constant and therefore want to fit
an exponential function to the recorded time course of the membrane
potential.
\begin{parts}
\part Implement (and document!) the exponential function
\begin{equation}
\label{expfunc}
x(t) = e^{-t/\tau} \quad , \qquad \tau \in \reZ
\end{equation}
with the membrane time constant $\tau$ as a matlab function
\code{expdecay(t, tau)} that takes as arguments a vector of time
points and the value of the membrane time constant. The function
returns \eqnref{expfunc} computed for each time point as a vector.
\begin{solution}
\lstinputlisting{expdecay.m}
\end{solution}
\part Let's first generate the data. Set the membrane time
constant to 10\,ms. Generate a time vector with sample times
between zero and six times the membrane time constant and a
sampling interval of 0.05\,ms. Then compute a vector containing
the corresponding measurements of the membrane potential using the
\code{expdecay()} function and adding some measurement noise with
a standard deviation of 0.05\.mV (\code{randn()} function).
Plot the data.
\begin{solution}
\lstinputlisting{expdecaydata.m}
\end{solution}
\part Write a function that returns the mean squared error. The
function takes as arguments the measured data (time points and
corresponding voltage measurements) and a value for the membrane
time constant at which the mean squred error should be
evaluated. Plot the mean squared error as a function of $\tau$.
\begin{solution}
\lstinputlisting{expdecaymse.m}
\lstinputlisting{expdecaymseplot.m}
\end{solution}
\part Write a function that returns the gradient of the mean
squared error, i.e. its derivative with respect to the $\tau$
parameter. The function takes as arguments the measured data
(time points and corresponding voltage measurements) and a value
for the membrane time constant at which the gradient should be
evaluated. Plot the gradient as a function of $\tau$.
\begin{solution}
\lstinputlisting{expdecaygradient.m}
\newsolutionpage
\lstinputlisting{expdecaygradientplot.m}
\includegraphics{expdecaygradientplot}
\end{solution}
\part Implement the gradient descent algorithm for finding the
least squares for the exponential function \eqref{expfunc}. The
function takes as arguments the measured data, an initial value
for the estimation of the membrane time constant, the $\epsilon$
factor, and the threshold for the length of the gradient where to
terminate the algorithm. The function should return the estimated
membrane time constant at the minimum of the mean squared error,
and a vector with the time constants as well as a vector with the
mean squared errors for each step of the algorithm.
\begin{solution}
\lstinputlisting{expdecaydescent.m}
\end{solution}
\part Call the gradient descent function with the generated data.
Watch the value of the gradient and of tau and adapt $\epsilon$
and the threshold accordingly (they differ quite dramatically from
the ones in the script for the cubic fit).
\part Generate three plots: (i) the values of the time constant
for each iteration step, (ii) the mean squared error for each
iteration step, and (iii) the measured data and the fitted
exponential function.
\begin{solution}
\lstinputlisting{expdecayplot.m}
\includegraphics{expdecayplot}
\end{solution}
\end{parts}
\continuepage
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\question \qt{Read sections 8.6 -- 8.8 of chapter 8 on ``optimization
and gradient descent!}\vspace{-3ex}
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\question \qt{Fitting an exponential function with two parameters}
Usually we do not know the exact value of the voltage step. So let's
make this a paramter, too:
\begin{equation}
\label{exp2func}
x(t) = c e^{-t/\tau} \quad , \qquad c, \tau \in \reZ
\end{equation}
\begin{parts}
\part Implement the exponential function \eqref{exp2func} as a
matlab function \code{exp2func(t, p)} that takes as arguments a
vector of time points and a vector with the two parameter values
$c$ and $\tau$. The function returns \eqnref{exp2func} computed for
each time point as a vector.
\begin{solution}
\lstinputlisting{exp2func.m}
\end{solution}
\part Plot the error surface, the mean squared error, for a range
of values for the two parameters $c$ and $\tau$. Try the
\code{surface()}, the \code{contour()} as well as the
\code{contourf()} functions to visualize the error surface for the
data from the previous exercise. You may also want to try to plot
the logarithm of the mean squared error.
\begin{solution}
\lstinputlisting{exp2errorsurface.m}
\end{solution}
\newsolutionpage
\part Implement the gradient descent algorithm for finding the
least squares of an arbitrary function. The function takes as
arguments the measured data, a vector holding the initial values
of the function parameters, the $\epsilon$ factor, and the
threshold for the length of the gradient where to terminate the
algorithm. The function should return the estimated parameter
values in a vector, and a 2D vector with the parameter values as
well as a vector with the mean squared errors for each step of the
algorithm.
\begin{solution}
\lstinputlisting{gradientdescent.m}
\end{solution}
\part Call the gradient descent function with the generated data.
Set $\epsilon=1.0$ and the threshold value initially to 0.001.
\part Plot the path of the gradient descent algorithm into the
error surface.
Why does (potentially) the path of the gradient descent algorithm
appear to be not perpendicular to the contour lines of the error
surface?
\begin{solution}
Because on the plot the $c$ and $\tau$ axes are (in general) not
equally scaled. That is a unit step in $c$ direction covers a
physical distance on paper or the screen than a unit stpe in
$\tau$ direction. Use \code{axis equal} to force the plot to
have equally scaled $c$ and $\tau$ axes.
\end{solution}
\part Plot the data together with the fitted exponential function.
\part Then adapt $\epsilon$ and the threshold value to improve the
convergence of the algortihm to the minimum of the cost function.
\begin{solution}
\lstinputlisting{exp2plot.m}
\includegraphics{exp2plot}
\end{solution}
\part Use the function \code{lsqcurvefit()} provided by matlab to
find $c$ and $\tau$ for the exponential function describing the
data. Compare the resulting fit parameters of this function with
the ones of your gradient descent algorithm. Plot the data
together with the fitted exponential function.
\begin{solution}
\lstinputlisting{exp2fit.m}
\includegraphics{exp2fit}\\
Resulting values for $c$ and $\tau$ are similar but not identical.
\end{solution}
\end{parts}
\end{questions}
\end{document}