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