IN4320 Exercise 1 Some Optima & Some Geometry solution

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We are going to consider a regularized version of the nearest mean classifier (NMC) for twoclass data. In the general setting, training data consists of pairs of d-dimensional feature
vectors xi and their label yi
, the latter of which we agree to encode with an element from the
set {+, −}. The two means for the two classes are also indexed + and −, i.e., m− belongs
to the class with labels −, while m+ belongs to the + class. Now, consider the following loss
function (or objective function) L:
L(m−, m+) := X
N
i
kxi − myi
k
2 + λkm− − m+k1 , (1)
with λ the regularization parameter, N the number of samples in the training set, and k · k1
the L1-norm. Minimizing L for both m− and m+ gives us the solution to the regularized loss
(for that λ). These optimal means, in turn, define the regularized NMC, which classifies new
data to the mean nearest to that data point.
Before you start: my assessment is that if your report goes significantly over 4 pages, you
are probably on the wrong track.
Some Optima & Some Geometry
1 Assume d = 1 and we are in a situation where we know m− is fixed to 1 and we only have
to optimize for m+. The only observations that we have for that + class are x1 = −1
and x2 = 1.
a Draw the loss function as a function of m+ for all λ ∈ {0, 2, 4, 6}. Be precise. A
rough sketch or artist impression is not enough.
b Derive for every of the four functions the minimizer and their minimum values.
Also determine for every loss the point where the derivative equals 0.
2 We now consider the setting in which both means have to be determined through a
minimization of the loss. In this case, we have L : R
2 → R.
a Generally, what does the regularizer in Equation (1) actually try to enforce and
what will, therefore, eventually happen to the means if λ gets larger and larger
(i.e., what is the limiting behavior of the two solution means)?
b Precisely describe how the contour lines for the general function L typically look
like when we are trying to find two 1-dimensional class means.
Hints: 1) the loss is convex, so the contours are convex as well and 2) the contours
consist of the concatenation of two basic geometric shapes.
c Let us consider a handful of data points. For the + class we have the same
observations as in Exercise 1. For the − class we now have observations x3 = 3 and
x4 = −1. Clearly, if we set λ to 0, we would find as optimal solution (m−, m+) =
(1, 0). Assume we have a large enough λ as under Exercise 2a: determine the exact
solution (m−, m+) in that case.
Some Programming & Some Experimenting
Through BlackBoard you can find a two-class digit classification task in 64 dimensions, i.e.,
d = 64. It is named optdigitsubset and consists of the pixel values of small 8 × 8 images,
which are ordered in N = 1125 rows of 64 columns wide. The first 554 rows contains the
values of 8 × 8 images of zeros, while the remaining block of 571 rows contains the 64 pixel
values of 571 ones. The actual feature vectors are obtained by running through the rows
of the images of the digits from top to bottom, concatenating all 8 rows of 8 pixels into a
64-dimensional vector.
3 Implement an optimizer for the regularized NMC from Equation (1) and convince yourself that it indeed optimizes what it should optimize. You can use gradient descent or
any other approach of your liking. You are even allowed to use optimization toolboxes
and the like. You can either implement a general version of the regularized NMC or
one that is completely dedicated to the data set that is given. (Note, however, that
the latter is probably not necessarily easier to implement and it is probably harder to
debug.)
a Describe in no more than 200 words the main ingredients of and/or considerations
behind your optimization routine. In particular, sketch the search strategy that
you employ and explain how you decide when you have reached the sought-after
minimum.
b Similar to all 64-dimensional samples in the data sets, you can restructure the
64-dimensional solution means that you find into an 8 × 8 images again. (For
instance, in Matlab you can just use something like reshape(m,[8 8]).) Plot the
two solution mean images that you find for λ = 0 and plot the two solution images
for a large λ (i.e., one for which the solution does not change anymore with an
even further increase of λ).