I wanted to confirm my answer with what others are getting.

I coded up the solution in MATLAB. I seem to have gotten somewhat lucky when I answered [d] for this question, since the ratio from one of the runs was 0.61, which is closer to 70% than 50%.

However, I was re-running the code after submission, and it seems that the result fluctuates around 0.6, sometimes dropping below that.

My code follows. I would appreciate if anyone has insights as to where I am going wrong. It is most likely in the % Quadratic Programming % section, since PLA is lifted straight from the previous assignments I coded.

Code:

clear all
clc
N = 100;
Ntest = 1000;
TRIALS = 1000;
better = 0;
sv_count = 0;
for ii = 1:TRIALS
% Generate random line
p1 = 2*rand(2, 1) - 1;
p2 = 2*rand(2, 1) - 1;
% Possibility of divide by 0. Better to just use w0 + x1*w1 + x2*w2 form.
slope = (p2(2) - p1(2)) / (p2(1) - p1(1));
intercept = p1(2) - slope*p1(1);
line = [slope intercept];
% Generate random points
% points are (x, y) coords
points = 2*rand(N, 2) - 1;
y = sign(points(:, 2) - polyval(line, points(:, 1)));
X = [ones(size(points, 1), 1) points]; % Add extra dimension
% PLA
wpla = zeros(3, 1);
while true
% Classify
yfit = sign(X*wpla);
combined = [X y];
bad = combined(yfit ~= y, :);
if size(bad, 1) == 0
break;
end
i = randi(size(bad, 1), 1);
wpla = wpla + bad(i, 4)*bad(i, [1:3])';
end
% Quadratic programming
X_svm = X(:, [2:3]); % Discard the constant term
H = (y*y').*(X_svm*X_svm');
f = ones(N, 1)*-1;
A = y';
b = 0;
lb = zeros(N, 1);
ub = 1e6*ones(N, 1);
options = optimset('Algorithm','interior-point-convex','MaxIter',1500, 'Display', 'off');
alpha = quadprog(H, f, [], [], A, b, lb, ub, [], options);
wquad = zeros(3, 1);
wquad(2) = sum(X_svm(:, 1).*(alpha.*y));
wquad(3) = sum(X_svm(:, 2).*(alpha.*y));
% Get index of maximum alpha
[maxval, maxind] = max(alpha);
wquad(1) = 1/y(maxind) - X_svm(maxind, :)*wquad(2:3);
% Eout
points = 2*rand(Ntest, 2) - 1;
X = [ones(size(points, 1), 1) points];
y = 2*[points(:, 2) > polyval(line, points(:, 1))] - 1;
e_pla = sum(sign(X*wpla) ~= y)/size(X, 1);
e_svm = sum(sign(X*wquad) ~= y)/size(X, 1);
better = better + (e_svm < e_pla);
% SVM
sv_count = sv_count + sum(alpha > 0.001); % Floating point tolerance
end
(better/TRIALS) * 100
sv_count/TRIALS