Jason, the scientist, tries to figure out the relationship between the
temperature and the reaction rate for H2 and 02. He knows the relationship
(a.k.a, reaction function) is
K = a * T<sup>b</sup> * e<sup>-c/T</sup>, K is the reaction rate, T is the temperature
He decides to use gradient descent to find out the value for a, b and c. Jason
gathers all past lab results, 10,000 records in total. Each record has the format
of (T, K', Ku) where K' is the reaction rate measured in the lab test and Ku is
the uncertainty of K'. Jason uses the loss function below.
\begin{equation}
L (T, a, b, c) = \frac{1}{20000} \sum \left(\frac{K(T, a, b, c) - K'}{K_u}\right)^2
\end{equation}
K(T,a,b,c) is the value calculated by the reaction function. K' is the reaction
rate measured in a lab result. Ku is the uncertainty of K'
1.1 (3 pts) You can assume the learning rates for a, b and c are La, Lb
and Lc respectively. Write the update functions for parameter a, b and c
for the process of gradient descent. Please explain why.
1.2 (3 pts) Jason knows that the approximate values of a, b and c,
which are 10<sup>-10</sup>, -0.1 and 60. The range of temperature for the lab
results is [300, 2000]. $\frac{K_u}{K}$ is approximately 10%. Please choose the
initial values for the learning rates La, Lb and Lc. Explain why.
1.3 (1pt) Jason finds that if he choose different initial values of a, b and
c, he often gets different final results. Please explain why.
