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\chapter{Optimization and gradient descent}
\exercisechapter{Optimization and gradient descent}
Optimization problems arise in many different contexts. For example,
to understand the behavior of a given neuronal system, the system is
probed with a range of input signals and then the resulting responses
are measured. This input-output relation can be described by a
model. Such a model can be a simple function that maps the input
signals to corresponding responses, it can be a filter, or a system of
differential equations. In any case, the model has some parameters
that specify how input and output signals are related. Which
combination of parameter values are best suited to describe the
input-output relation? The process of finding the best parameter
values is an optimization problem. For a simple parameterized function
that maps input to output values, this is the special case of a
\enterm{curve fitting} problem, where the average distance between the
curve and the response values is minimized. One basic numerical method
used for such optimization problems is the so called gradient descent,
which is introduced in this chapter.
\begin{figure}[t]
\includegraphics{cubicfunc}
\titlecaption{Example data suggesting a cubic relation.}{The length
$x$ and weight $y$ of $n=34$ male tigers (blue, left). Assuming a
cubic relation between size and weight leaves us with a single
free parameters, a scaling factor. The cubic relation is shown for
a few values of this scaling factor (orange and red,
right).}\label{cubicdatafig}
\end{figure}
For demonstrating the curve-fitting problem let's take the simple
example of weights and sizes measured for a number of male tigers
(\figref{cubicdatafig}). Weight $y$ is proportional to volume
$V$ via the density $\rho$. The volume $V$ of any object is
proportional to its length $x$ cubed. The factor $\alpha$ relating
volume and size cubed depends on the shape of the object and we do not
know this factor for tigers. For the data set we thus expect a cubic
relation between weight and length
\begin{equation}
\label{cubicfunc}
y = f(x; c) = c\cdot x^3
\end{equation}
where $c=\rho\alpha$, the product of a tiger's density and form
factor, is the only free parameter in the relation. We would like to
find out which value of $c$ best describes the measured data. In the
following we use this example to illustrate the gradient descent as a
basic method for finding such an optimal parameter.
\section{The error function --- mean squared error}
Before we go ahead finding the optimal parameter value we need to
specify what exactly we consider as an optimal fit. In our example we
search the parameter that describes the relation of $x$ and $y$
best. What is meant by this? The length $x_i$ of each tiger is
associated with a weight $y_i$ and for each $x_i$ we have a
\emph{prediction} or \emph{estimation} $y^{est}(x_i)$ of the weight by
the model \eqnref{cubicfunc} for a specific value of the parameter
$c$. Prediction and actual data value ideally match (in a perfect
noise-free world), but in general the estimate and measurement are
separated by some distance or error $y_i - y^{est}(x_i)$. In our
example the estimate of the weight for the length $x_i$ is given by
equation \eqref{cubicfunc} $y^{est}(x_i) = f(x_i;c)$. The best fitting
model with parameter $c$ is the one that somehow minimizes the
distances between observations $y_i$ and corresponding estimations
$y^{est}(x_i)$ (\figref{cubicerrorsfig}).
As a first guess we could simply minimize the sum of the distances,
$\sum_{i=1}^N y_i - y^{est}(x_i)$. This, however, does not work
because positive and negative errors would cancel out, no matter how
large they are, and sum up to values close to zero. Better is to sum
over absolute distances: $\sum_{i=1}^N |y_i - y^{est}(x_i)|$. This sum
can only be small if all deviations are indeed small no matter if they
are above or below the prediction. The sum of the squared distances,
$\sum_{i=1}^N (y_i - y^{est}(x_i))^2$, turns out to be an even better
choice. Instead of the sum we could also minimize the average distance
\begin{equation}
\label{meansquarederror}
f_{mse}(\{(x_i, y_i)\}|\{y^{est}(x_i)\}) = \frac{1}{N} \sum_{i=1}^N (y_i - y^{est}(x_i))^2
\end{equation}
This is known as the \enterm[mean squared error]{mean squared error}
(\determ[quadratischer Fehler!mittlerer]{mittlerer quadratischer
Fehler}). Similar to the absolute distance, the square of the errors
is always positive and thus positive and negative error values do not
cancel each other out. In addition, the square punishes large
deviations over small deviations. In
chapter~\ref{maximumlikelihoodchapter} we show that minimizing the
mean squared error is equivalent to maximizing the likelihood that the
observations originate from the model, if the data are normally
distributed around the model prediction.
\begin{exercise}{meansquarederrorline.m}{}\label{mseexercise}
Simulate $n=40$ tigers ranging from 2.2 to 3.9\,m in size and store
these sizes in a vector \varcode{x}. Compute the corresponding
predicted weights \varcode{yest} for each tiger according to
\eqnref{cubicfunc} with $c=6$\,\kilo\gram\per\meter\cubed. From the
predictions generate simulated measurements of the tiger's weights
\varcode{y}, by adding normally distributed random numbers to the
predictions scaled to a standard deviation of 50\,\kilo\gram.
Compute the \emph{mean squared error} between \varcode{y} and
\varcode{yest} in a single line of code.
\end{exercise}
\section{Cost function}
The mean squared error is a so called \enterm{objective function} or
\enterm{cost function} (\determ{Kostenfunktion}). A cost function
assigns to a model prediction $\{y^{est}(x_i)\}$ for a given data set
$\{(x_i, y_i)\}$ a single scalar value that we want to minimize. Here
we aim to adapt the model parameter to minimize the mean squared error
\eqref{meansquarederror}. In general, the \enterm{cost function} can
be any function that describes the quality of a fit by mapping the
data and the predictions to a single scalar value.
\begin{figure}[t]
\includegraphics{cubicerrors}
\titlecaption{Estimating the \emph{mean squared error}.} {The
deviation error (orange) between the prediction (red line) and the
observations (blue dots) is calculated for each data point
(left). Then the deviations are squared and the average is
calculated (right).}
\label{cubicerrorsfig}
\end{figure}
Replacing $y^{est}$ in the mean squared error \eqref{meansquarederror}
with our cubic model \eqref{cubicfunc}, the cost function reads
\begin{eqnarray}
f_{cost}(c|\{(x_i, y_i)\}) & = & \frac{1}{N} \sum_{i=1}^N (y_i - f(x_i;c))^2 \label{msefunc} \\
& = & \frac{1}{N} \sum_{i=1}^N (y_i - c x_i^3)^2 \label{msecube}
\end{eqnarray}
The optimization process tries to find a value for the factor $c$ such
that the cost function is minimized. With the mean squared error as
the cost function this optimization process is also called method of
\enterm{least squares} (\determ[quadratischer
Fehler!kleinster]{Methode der kleinsten Quadrate}).
\begin{exercise}{meanSquaredErrorCubic.m}{}
Implement the objective function \eqref{msecube} as a function
\varcode{meanSquaredErrorCubic()}. The function takes three
arguments. The first is a vector of $x$-values and the second
contains the measurements $y$ for each value of $x$. The third
argument is the value of the factor $c$. The function returns the
mean squared error.
\end{exercise}
\section{Graph of the cost function}
For each value of the parameter $c$ of the model we can use
\eqnref{msecube} to calculate the corresponding value of the cost
function. The cost function $f_{cost}(c|\{(x_i, y_i)\}|)$ is a
function $f_{cost}(c)$ that maps the parameter value $c$ to a scalar
error value. For a given data set we thus can simply plot the cost
function as a function of the parameter $c$ (\figref{cubiccostfig}).
\begin{exercise}{plotcubiccosts.m}{}
Calculate the mean squared error between the data and the cubic
function for a range of values of the factor $c$ using the
\varcode{meanSquaredErrorCubic()} function from the previous
exercise. Plot the graph of the cost function.
\end{exercise}
\begin{figure}[t]
\includegraphics{cubiccost}
\titlecaption{Minimum of the cost function.}{For a given data set
the cost function, the mean squared error \eqnref{msecube}, as a
function of the unknown parameter $c$ has a minimum close to the
true value of $c$ that was used to generate the data
(left). Simply taking the absolute minimum of the cost function
computed for a pre-set range of values for the parameter $c$, has
the disadvantage to be limited in precision (right) and the
possibility to entirely miss the global minimum if it is outside
the computed range.}\label{cubiccostfig}
\end{figure}
By looking at the plot of the cost function we can visually identify
the position of the minimum and thus estimate the optimal value for
the parameter $c$. How can we use the cost function to guide an
automatic optimization process?
The obvious approach would be to calculate the mean squared error for
a range of parameter values and then find the position of the minimum
using the \code{min()} function. This approach, however has several
disadvantages: (i) The accuracy of the estimation of the best
parameter is limited by the resolution used to sample the parameter
space. The coarser the parameters are sampled the less precise is the
obtained position of the minimum (\figref{cubiccostfig}, right). (ii)
The range of parameter values might not include the absolute minimum.
(iii) In particular for functions with more than a single free
parameter it is computationally expensive to calculate the cost
function for each parameter combination at a sufficient
resolution. The number of combinations increases exponentially with
the number of free parameters. This is known as the \enterm{curse of
dimensionality}.
So we need a different approach. We want a procedure that finds the
minimum of the cost function with a minimal number of computations and
to arbitrary precision.
\section{Gradient}
\begin{figure}[t]
\includegraphics{cubicgradient}
\titlecaption{Derivative of the cost function.}{The gradient, the
derivative \eqref{costderivative} of the cost function, is
negative to the left of the minimum (vertical line) of the cost
function, zero (horizontal line) at, and positive to the right of
the minimum (left). For each value of the parameter $c$ the
negative gradient (arrows) points towards the minimum of the cost
function (right).} \label{gradientcubicfig}
\end{figure}
\begin{ibox}[b]{\label{differentialquotientbox}Difference quotient and derivative}
\includegraphics[width=0.33\textwidth]{derivative}
\hfill
\begin{minipage}[b]{0.63\textwidth}
The difference quotient
\begin{equation}
\label{difffrac}
m = \frac{f(x + \Delta x) - f(x)}{\Delta x}
\end{equation}
of a function $y = f(x)$ is the slope of the secant (red) defined
by the points $(x,f(x))$ and $(x+\Delta x,f(x+\Delta x))$ at
distance $\Delta x$.
The slope of the function $y=f(x)$ at the position $x$ (orange) is
given by the derivative $f'(x)$ of the function at that position.
It is defined by the difference quotient in the limit of
infinitesimally (red and yellow) small distances $\Delta x$:
\begin{equation}
\label{derivative}
f'(x) = \frac{{\rm d} f(x)}{{\rm d}x} = \lim\limits_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{equation}
\end{minipage}\vspace{2ex}
It is not possible to calculate the exact value of the derivative
\eqref{derivative} numerically. The derivative can only be estimated
by computing the difference quotient \eqref{difffrac} using
sufficiently small $\Delta x$.
\end{ibox}
Imagine to place a ball at some point on the cost function
\figref{cubiccostfig}. Naturally, it would roll down the slope and
eventually stop at the minimum of the error surface (if it had no
inertia). We will use this analogy to develop an algorithm to find our
way to the minimum of the cost function. The ball always follows the
steepest slope. Thus we need to figure out the direction of the slope
at the position of the ball.
In our one-dimensional example of a single free parameter the slope is
simply the derivative of the cost function with respect to the
parameter $c$ (\figref{gradientcubicfig}, left). This derivative is called
the \entermde{Gradient}{gradient} of the cost function:
\begin{equation}
\label{costderivative}
\nabla f_{cost}(c) = \frac{{\rm d} f_{cost}(c)}{{\rm d} c}
\end{equation}
There is no need to calculate this derivative analytically, because it
can be approximated numerically by the difference quotient
(Box~\ref{differentialquotientbox}) for small steps $\Delta c$:
\begin{equation}
\frac{{\rm d} f_{cost}(c)}{{\rm d} c} =
\lim\limits_{\Delta c \to 0} \frac{f_{cost}(c + \Delta c) - f_{cost}(c)}{\Delta c}
\approx \frac{f_{cost}(c + \Delta c) - f_{cost}(c)}{\Delta c}
\end{equation}
Choose, for example, $\Delta c = 10^{-7}$. The derivative is positive
for positive slopes. Since want to go down the hill, we choose the
opposite direction (\figref{gradientcubicfig}, right).
\begin{exercise}{meanSquaredGradientCubic.m}{}\label{gradientcubic}
Implement a function \varcode{meanSquaredGradientCubic()}, that
takes the $x$- and $y$-data and the parameter $c$ as input
arguments. The function should return the derivative of the mean
squared error $f_{cost}(c)$ with respect to $c$ at the position
$c$.
\end{exercise}
\begin{exercise}{plotcubicgradient.m}{}
Using the \varcode{meanSquaredGradientCubic()} function from the
previous exercise, plot the derivative of the cost function as a
function of $c$.
\end{exercise}
\section{Gradient descent}
\begin{figure}[t]
\includegraphics{cubicmse}
\titlecaption{Gradient descent.}{The algorithm starts at an
arbitrary position. At each point the gradient is estimated and
the position is updated as long as the length of the gradient is
sufficiently large. The dots show the positions after each
iteration of the algorithm.} \label{gradientdescentcubicfig}
\end{figure}
Finally, we are able to implement the optimization itself. By now it
should be obvious why it is called the gradient descent method. From a
starting position on we iteratively walk down the slope of the cost
function against its gradient. All ingredients necessary for this
algorithm are already there. We need: (i) the cost function
(\varcode{meanSquaredErrorCubic()}), and (ii) the gradient
(\varcode{meanSquaredGradientCubic()}). The algorithm of the gradient
descent works as follows:
\begin{enumerate}
\item Start with some arbitrary value $p_0$ for the parameter $c$.
\item \label{computegradient} Compute the gradient
\eqnref{costderivative} at the current position $p_i$.
\item If the length of the gradient, the absolute value of the
derivative \eqnref{costderivative}, is smaller than some threshold
value, we assume to have reached the minimum and stop the search.
We return the current parameter value $p_i$ as our best estimate of
the parameter $c$ that minimizes the cost function.
We are actually looking for the point at which the derivative of the
cost function equals zero, but this is impossible because of
numerical imprecision. We thus apply a threshold below which we are
sufficiently close to zero.
\item \label{gradientstep} If the length of the gradient exceeds the
threshold we take a small step into the opposite direction:
\begin{equation}
\label{gradientdescent}
p_{i+1} = p_i - \epsilon \cdot \nabla f_{cost}(p_i)
\end{equation}
where $\epsilon$ is a factor linking the gradient to
appropriate steps in the parameter space.
\item Repeat steps \ref{computegradient} -- \ref{gradientstep}.
\end{enumerate}
\Figref{gradientdescentcubicfig} illustrates the gradient descent ---
the path the imaginary ball has chosen to reach the minimum. We walk
along the parameter axis against the gradient as long as the gradient
differs sufficiently from zero. At steep slopes we take large steps
(the distance between the red dots in \figref{gradientdescentcubicfig}
is large).
\begin{exercise}{gradientDescentCubic.m}{}\label{gradientdescentexercise}
Implement the gradient descent algorithm for the problem of fitting
a cubic function \eqref{cubicfunc} to some measured data pairs $x$
and $y$ as a function \varcode{gradientDescentCubic()} that returns
the estimated best fitting parameter value $c$ as well as two
vectors with all the parameter values and the corresponding values
of the cost function that the algorithm iterated trough. As
additional arguments that function takes the initial value for the
parameter $c$, the factor $\epsilon$ connecting the gradient with
iteration steps in \eqnref{gradientdescent}, and the threshold value
for the absolute value of the gradient terminating the algorithm.
\end{exercise}
\begin{exercise}{plotgradientdescentcubic.m}{}\label{plotgradientdescentexercise}
Use the function \varcode{gradientDescentCubic()} to fit the
simulated data from exercise~\ref{mseexercise}. Plot the returned
values of the parameter $c$ as as well as the corresponding mean
squared errors as a function of iteration step (two plots). Compare
the result of the gradient descent method with the true value of $c$
used to simulate the data. Inspect the plots and adapt $\epsilon$
and the threshold to make the algorithm behave as intended. Finally
plot the data together with the best fitting cubic relation
\eqref{cubicfunc}.
\end{exercise}
The $\epsilon$ parameter in \eqnref{gradientdescent} is critical. If
too large, the algorithm does not converge to the minimum of the cost
function (try it!). At medium values it oscillates around the minimum
but might nevertheless converge. Only for sufficiently small values
(here $\epsilon = 0.00001$) does the algorithm follow the slope
downwards towards the minimum.
The terminating condition on the absolute value of the gradient
influences how often the cost function is evaluated. The smaller the
threshold value the more often the cost is computed and the more
precisely the fit parameter is estimated. If it is too small, however,
the increase in precision is negligible, in particular in comparison
to the increased computational effort. Have a look at the derivatives
that we plotted in exercise~\ref{gradientcubic} and decide on a
sensible value for the threshold. Run the gradient descent algorithm
and check how the resulting $c$ parameter values converge and how many
iterations were needed. Then reduce the threshold (by factors of ten)
and check how this changes the results.
Many modern algorithms for finding the minimum of a function are based
on the basic idea of the gradient descent. Luckily these algorithms
choose $\epsilon$ in a smart adaptive way and they also come up with
sensible default values for the termination condition. On the other
hand, these algorithm often take optional arguments that let you
control how they behave. Now you know what this is all about.
\section{N-dimensional minimization problems}
So far we were concerned about finding the right value for a single
parameter that minimizes a cost function. Often we deal with
functions that have more than a single parameter, in general $n$
parameters. We then need to find the minimum in an $n$ dimensional
parameter space.
For our tiger problem, we could have also fitted the exponent $a$ of
the power-law relation between size and weight, instead of assuming a
cubic relation with $a=3$:
\begin{equation}
\label{powerfunc}
y = f(x; c, a) = f(x; \vec p) = c\cdot x^a
\end{equation}
We then could check whether the resulting estimate of the exponent
$a$ indeed is close to the expected power of three. The
power-law \eqref{powerfunc} has two free parameters $c$ and $a$.
Instead of a single parameter we are now dealing with a vector $\vec
p$ containing $n$ parameter values. Here, $\vec p = (c, a)$. All
the concepts we introduced on the example of the one dimensional
problem of tiger weights generalize to $n$-dimensional problems. We
only need to adapt a few things. The cost function for the mean
squared error reads
\begin{equation}
\label{ndimcostfunc}
f_{cost}(\vec p|\{(x_i, y_i)\}) = \frac{1}{N} \sum_{i=1}^N (y_i - f(x_i;\vec p))^2
\end{equation}
\begin{figure}[t]
\includegraphics{powergradientdescent}
\titlecaption{Gradient descent on an error surface.}{Contour plot of
the cost function \eqref{ndimcostfunc} for the fit of a power law
\eqref{powerfunc} to some data. Here the cost function is a long
and narrow valley on the plane spanned by the two parameters $c$
and $a$ of the power law. The red line marks the path of the
gradient descent algorithm. The gradient is always perpendicular
to the contour lines. The algorithm quickly descends into the
valley and then slowly creeps on the shallow bottom of the valley
towards the global minimum where it terminates (yellow circle).
} \label{powergradientdescentfig}
\end{figure}
For two-dimensional problems the graph of the cost function is an
\enterm{error surface} (\determ{{Fehlerfl\"ache}}). The two parameters
span a two-dimensional plane. The cost function assigns to each
parameter combination on this plane a single value. This results in a
landscape over the parameter plane with mountains and valleys and we
are searching for the position of the bottom of the deepest valley
(\figref{powergradientdescentfig}).
\begin{ibox}[t]{\label{partialderivativebox}Partial derivatives and gradient}
Some functions depend on more than a single variable. For example, the function
\[ z = f(x,y) \]
depends on both $x$ and $y$. Using the partial derivatives
\[ \frac{\partial f(x,y)}{\partial x} = \lim\limits_{\Delta x \to 0} \frac{f(x + \Delta x,y) - f(x,y)}{\Delta x} \]
and
\[ \frac{\partial f(x,y)}{\partial y} = \lim\limits_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x,y)}{\Delta y} \]
one can calculate the slopes in the directions of each of the
variables by means of the respective difference quotient
(see box~\ref{differentialquotientbox}). \vspace{1ex}
\begin{minipage}[t]{0.5\textwidth}
\mbox{}\\[-2ex]
\includegraphics[width=1\textwidth]{gradient}
\end{minipage}
\hfill
\begin{minipage}[t]{0.46\textwidth}
For example, the partial derivatives of
\[ f(x,y) = x^2+y^2 \vspace{-1ex} \] are
\[ \frac{\partial f(x,y)}{\partial x} = 2x \; , \quad \frac{\partial f(x,y)}{\partial y} = 2y \; .\]
The gradient is a vector with the partial derivatives as coordinates:
\[ \nabla f(x,y) = \left( \begin{array}{c} \frac{\partial f(x,y)}{\partial x} \\[1ex] \frac{\partial f(x,y)}{\partial y} \end{array} \right) \]
This vector points into the direction of the strongest ascend of
$f(x,y)$.
\end{minipage}
\vspace{0.5ex} The figure shows the contour lines of a bivariate
Gaussian $f(x,y) = \exp(-(x^2+y^2)/2)$ and the gradient (thick
arrows) together with the corresponding partial derivatives (thin
arrows) for three different locations.
\end{ibox}
When we place a ball somewhere on the slopes of a hill, it rolls
downwards and eventually stops at the bottom. The ball always rolls in
the direction of the steepest slope. For a two-dimensional parameter
space of our example, the \entermde{Gradient}{gradient}
(box~\ref{partialderivativebox}) of the cost function is a vector
\begin{equation}
\label{gradientpowerlaw}
\nabla f_{cost}(c, a) = \left( \frac{\partial f_{cost}(c, a)}{\partial c},
\frac{\partial f_{cost}(c, a)}{\partial a} \right)
\end{equation}
that points into the direction of the strongest ascend of the cost
function. The gradient is given by the partial derivatives
(box~\ref{partialderivativebox}) of the mean squared error with
respect to the parameters $c$ and $a$ of the power law
relation. In general, for $n$-dimensional problems, the gradient is an
$n-$ dimensional vector containing for each of the $n$ parameters
$p_j$ the respective partial derivatives as coordinates:
\begin{equation}
\label{gradient}
\nabla f_{cost}(\vec p) = \left( \frac{\partial f_{cost}(\vec p)}{\partial p_j} \right)
\end{equation}
The iterative equation \eqref{gradientdescent} of the gradient descent
stays exactly the same, with the only difference that the current
parameter value $p_i$ becomes a vector $\vec p_i$ of parameter values:
\begin{equation}
\label{ndimgradientdescent}
\vec p_{i+1} = \vec p_i - \epsilon \cdot \nabla f_{cost}(\vec p_i)
\end{equation}
The algorithm proceeds along the negative gradient
(\figref{powergradientdescentfig}).
For the termination condition we need the length of the gradient. In
one dimension it was just the absolute value of the derivative. For
$n$ dimensions this is according to the \enterm{Pythagorean theorem}
(\determ[Pythagoras!Satz des]{Satz des Pythagoras}) the square root of
the sum of the squared partial derivatives:
\begin{equation}
\label{ndimabsgradient}
|\nabla f_{cost}(\vec p_i)| = \sqrt{\sum_{i=1}^n \left(\frac{\partial f_{cost}(\vec p)}{\partial p_i}\right)^2}
\end{equation}
The \code{norm()} function implements this given a vector with the
partial derivatives.
\subsection{Passing a function as an argument to another function}
So far, all our code for the gradient descent algorithm was tailored
to a specific function, the cubic relation \eqref{cubicfunc}. It would
be much better if we could pass an arbitrary function to our gradient
algorithm. Then we would not need to rewrite it every time anew.
This is possible. We can indeed pass a function as an argument to
another function. For this use the \code{@}-operator. As an example
let's define a function that produces a standard plot for a function:
\pageinputlisting[caption={Example function taking a function as argument.}]{funcPlotter.m}
This function can then be used as follows for plotting a sine wave. We
pass the built in \varcode{sin()} function as \varcode{@sin} as an
argument to our function:
\pageinputlisting[caption={Passing a function handle as an argument to a function.}]{funcplotterexamples.m}
\subsection{Gradient descent algorithm for arbitrary functions}
Now we are ready to adapt the gradient descent algorithm from
exercise~\ref{gradientdescentexercise} to arbitrary functions with $n$
parameters that we want to fit to some data.
\begin{exercise}{gradientDescent.m}{}
Adapt the function \varcode{gradientDescentCubic()} from
exercise~\ref{gradientdescentexercise} to implement the gradient
descent algorithm for any function \varcode{func(x, p)} that takes
as first argument the $x$-data values and as second argument a
vector with parameter values. The new function takes a vector $\vec
p_0$ for the initial parameter values and also returns the best
parameter combination as a vector. Use a \varcode{for} loop over the
two dimensions for computing the gradient.
\end{exercise}
For testing our new function we need to implement the power law
\eqref{powerfunc}:
\begin{exercise}{powerLaw.m}{}
Write a function that implements \eqref{powerfunc}. The function
gets as arguments a vector $x$ containing the $x$-data values and
another vector containing as elements the parameters for the power
law, i.e. the factor $c$ and the power $a$. It returns a vector
with the computed $y$ values for each $x$ value.
\end{exercise}
Now let's use the new gradient descent function to fit a power law to
our tiger data-set (\figref{powergradientdescentfig}):
\begin{exercise}{plotgradientdescentpower.m}{}
Use the function \varcode{gradientDescent()} to fit the
\varcode{powerLaw()} function to the simulated data from
exercise~\ref{mseexercise}. Plot the returned values of the two
parameters against each other. Compare the result of the gradient
descent method with the true values of $c$ and $a$ used to simulate
the data. Observe the norm of the gradient and inspect the plots to
adapt $\epsilon$ and the threshold if necessary. Finally plot the
data together with the best fitting power-law \eqref{powerfunc}.
\end{exercise}
\section{Fitting non-linear functions to data}
The gradient descent is a basic numerical method for solving
optimization problems. It is used to find the global minimum of an
objective function.
Curve fitting is a specific optimization problem and a common
application for the gradient descent method. For the case of fitting
straight lines to data pairs, the error surface (using the mean
squared error) has exactly one clearly defined global minimum. In
fact, the position of the minimum can be analytically calculated as
shown in the next chapter. For linear fitting problems numerical
methods like the gradient descent are not needed.
Fitting problems that involve nonlinear functions of the parameters,
e.g. the power law \eqref{powerfunc} or the exponential function
$f(t;\tau) = e^{-t/\tau}$, do in general not have an analytical
solution for the least squares. To find the least squares for such
functions numerical methods such as the gradient descent have to be
applied.
The suggested gradient descent algorithm requires manually tuned
values for $\epsilon$ and the threshold for terminating the iteration.
The algorithm can be improved in multiple ways to converge more
robustly and faster. Most importantly, $\epsilon$ is made dependent
on the changes of the gradient from one iteration to the next. These
and other numerical tricks have already been implemented in
pre-defined functions. Generic optimization functions such as
\mcode{fminsearch()} have been implemented for arbitrary objective
functions, while the more specialized function \mcode{lsqcurvefit()}
is specifically designed for optimizations in the least square error
sense.
\begin{exercise}{plotlsqcurvefitpower.m}{}
Use the \matlab-function \varcode{lsqcurvefit()} instead of
\varcode{gradientDescent()} to fit the \varcode{powerLaw()} function
to the simulated data from exercise~\ref{mseexercise}. Plot the data
and the resulting best fitting power law function.
\end{exercise}
\begin{important}[Beware of secondary minima!]
Finding the absolute minimum is not always as easy as in the case of
fitting a straight line. Often, the cost function has secondary or
local minima in which the gradient descent stops even though there
is a more optimal solution, a global minimum that is lower than the
local minimum. Starting from good initial positions is a good
approach to avoid getting stuck in local minima. Also keep in mind
that cost functions tend to be simpler (less local minima) the fewer
parameters are fitted from the data. Each additional parameter
increases complexity and is computationally more expensive.
\end{important}
\section{Evolution as an optimization problem}
Evolution is a biological implementation of an optimization
algorithm. The objective function is an organism's fitness. This needs
to be maximized (this is the same as minimizing the negative fitness).
The parameters of this optimization problem are all the many genes on
the DNA. This is a very high-dimensional optimization problem. By
cross-over and mutations a population of a species moves along the
high-dimensional parameter space. Selection processes make sure that
only organisms with higher fitness pass on their genes to the next
generations. In this way the algorithm is not directed towards higher
fitness, as the gradient descent method would be. Rather, some
neighborhood of the parameter space is randomly probed. That way it is
even possible to escape a local maximum and find a potentially better
maximum. For this reason, \enterm[genetic algorithm]{genetic
algorithms} try to mimic evolution in the context of
high-dimensional optimization problems, in particular with discrete
parameter values. In biological evolution, the objective function,
however, is not a fixed function. It may change in time by changing
abiotic and biotic environmental conditions, making this a very
complex but also interesting optimization problem.
How should a neuron or neural network be designed? As a particular
aspect of the general evolution of a species, this is a fundamental
question in the neurosciences. Maintaining a neural system is
costly. By their simple presence neurons incur costs. They need to be
built and maintained, they occupy space and consume
resources. Equipping a neuron with more ion channels also costs. And
neural activity makes it more costly to maintain concentration
gradients of ions. This all boils down to the consumption of more ATP,
the currency of metabolism. On the other hand each neuron provides
some useful function. In the end, neurons make the organism to behave
in some sensible way to increase the overall fitness of the
organism. On the level of neurons that means that they should
faithfully represent and process behaviorally relevant sensory
stimuli, make a sensible decision, store some important memory, or
initiate and control movements in a directed way. Unfortunately there
is a tradeoff. Better neural function usually involves higher
costs. More ion channels reduce intrinsic noise which usually favors
the precision of neural responses. Higher neuronal activity improves
the quality of the encoding of sensory stimuli. More neurons are
required for more complex computations. And so on.
Understanding why a neuronal system is designed in some specific way
requires to also understand these tradeoffs. The number of neurons,
the number and types of ion channels, the length of axons, the number
of synapses, the way neurons are connected, etc. are all parameters
that could be optimized. For the objective function the function of
the neurons needs to be quantified, for example by measures from
information or detection theory, and their dependence on the
parameters. From these benefits the costs need to be subtracted. And
then one is interested in finding the maximum of the resulting
objective function. Maximization (or minimization) problems are not
only a tool for data analysis, rather they are at the core of many ---
not only biological or neuroscientific --- problems.
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\printsolutions
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