代写CO-496作业、代做Python编程设计作业、代写Ptensorflow, pytorch作业、代做Python语言作业
Coursework 1
Mathematics for Machine Learning (CO-496)
Instructions
This coursework has both writing and coding components. The coding part must
be done in python. The code will be run on the standard CSG installation and will
be tested on the labTS 1
system. On starting the course you will be given a gitlab
repository. Every commit you make to this repository can be marked automatically.
You are not permitted to use any symbolic manipulation libraries (e.g. sympy) or
automatic differentiation tools (e.g. tensorflow, pytorch) for your submitted code
(though, of course, you may find these useful for checking your answers). You
should not need to import anything other than numpy for the submitted code for this
assignment.
The writing assignment requires plots, which you can create using any method of
your choice. You should not submit the code used to create these plots.
No aspect of your submission may be hand-drawn. You are strongly encouraged to
use LATEX to create the written component.
You are required to submit the following:
A .pdf file for your written answers, submitted through CATe
A .py file which implements all the methods for the coding exercises, submitted
through gitlab.
1https://teaching.doc.ic.ac.uk/labts
1
1 Differentiation
In this question, we define the following constants:
We define also the following functions, which are all R
2 → R
f1(x) = x
T x + x
TBx a
T x + b
T x
f2(x) = sin(x a)T(x a)+ (x b)
TB(x b)
f3(x) = 1 exp
(x a)T（x a)+ exp
a) Write f1(x) in the completed square form (x ? c)
>C(x c) + c0.
3 marks for
correct C
correct c
correct c0
lose a mark for any incorrect notation
b) Explain how you can tell that f1 has a minimum point. State the minimum value
of f1 and find the input which achieves this minimum.
2 marks for
the appropriate eigenvalues and accompanying explanation
the minimum value stated and the input which achieves this value
c) Write a python function grad_f1(x) that return the gradient for f1.
4 marks
d) Write a python function grad_f2(x) that return the gradient for f2.
6 marks
e) Optional: Write a python function grad_f3(x) that return the gradient for f3. If
you don’t do this question you still need the gradient for the next part, but you
can use an automatic differentiation package. autograd is recommended as a
good lightweight option, but you could also use e.g. pytorch, tensorflow, mxnet
or chainer.
There are no marks for this question, but you can still submit the code to labts to
verify your answer.
2
f) Use your gradients to implement a gradient descent algorithm with 50 iterations
to find a local minimum for both f2 and f3. Show the steps of your algorithm
on a contour plot of the function. Start from the point (1, 1) and state the step
size you used. Produce separate contour plots for the two functions, using first
component of x on the x axis and the second on the y.
4 marks for:
sensible scales, plots labeled, contours visible (use at least 20)
correct functions shown
gradient descent for first plot with 50 points, converging to minimum
gradient descent for second plot, converging to one of the minima
g) For the two functions f2 and f3, discuss the qualitative differences you observe
when performing gradient descent with step sizes varying between 0.1 and 1,
again starting the point (1, ?1). Briefly describe also what happens in the two
cases with grossly mis-specified step-sizes (i.e. greater than 1), with a reason to
explain the difference in behaviour.
6 marks for:
observation about number of minima for f2
observation about large stepsizes for f2
observation about number of minima for f3
observation about different behaviour for different moderate stepsizes for f3
observation about large stepsizes for f3
explanation for the difference in behaviour
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