some fixes

regression/lecture/regression.tex

@@ -362,7 +362,8 @@ 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.
+downwards towards the minimum. Change $\epsilon$ by factors of ten to
+adapt it to a specific problem.
 
 The terminating condition on the absolute value of the gradient
 influences how often the cost function is evaluated. The smaller the
@@ -560,7 +561,7 @@ For testing our new function we need to implement the power law
 \end{exercise}
 
 Now let's use the new gradient descent function to fit a power law to
-our tiger data-set (\figref{powergradientdescentfig}):
+our tiger data-set:
 
 \begin{exercise}{plotgradientdescentpower.m}{}
   Use the function \varcode{gradientDescent()} to fit the
@@ -573,12 +574,21 @@ our tiger data-set (\figref{powergradientdescentfig}):
   data together with the best fitting power-law \eqref{powerfunc}.
 \end{exercise}
 
+Note that in our specific example on tiger sizes and weights the
+simulated data look on a first glance like being linearly related
+(\figref{cubicdatafig}). The true cubic relation between weights and
+sizes is not that obvious, because of the limited range of tiger
+sizes. Nevertheless, the cost function has a minimum at the bottom of
+a valley that is very narrow in the direction of the expontent
+(\figref{powergradientdescentfig}). The exponent of about three is
+thus clearly defined by the data.
+
 
 \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.
+objective function.  
 
 Curve fitting is a specific optimization problem and a common
 application for the gradient descent method.  For the case of fitting
@@ -650,6 +660,8 @@ 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.
 
+\subsection{Optimal design of neural systems}
+
 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