def perce(X, y, w_init, rho, max_iter=1000):
? ??
? ? (N, nfeatures) = X.shape

? ? # Augment the feature vectors by adding a 1 to each one. (see lecture notes)
? ? X = np.hstack((X, np.ones((N, 1))))
? ? nfeatures += 1

? ? w = w_init ?# initialise weights
? ? iter = 0
? ? mis_class = N ?# start by assuming all samples are misclassified

? ? while mis_class > 0 and iter < max_iter:
? ? ? ? iter += 1
? ? ? ? mis_class = 0
? ? ? ? gradient = np.zeros(nfeatures) ?# initaliase the gradients to 0

? ? ? ? # loop over every training sample.
? ? ? ? for i in range(N):
? ? ? ? ? ? # each misclassified point will cause the gradient to change
? ? ? ? ? ? if np.inner(X[i, :], w) * y[i] <= 0:
? ? ? ? ? ? ? ? mis_class += 1
? ? ? ? ? ? ? ? gradient += -y[i] * X[i, :]
? ? ? ? # update the weight vector ready for the next iteration
? ? ? ? # Note, also that the learning rate decays over time (rho/iter)
? ? ? ? w -= rho / iter * gradient

? ? return w, iter, mis_class

def perce_fast(X, y, w_init, rho, max_iter=10000):
??
? ? (N, nfeatures) = X.shape
? ? X = np.hstack((X, np.ones((N, 1))))
? ? nfeatures += 1
? ? w = w_init
? ? iter = 0
? ? mis_class = N
? ? yy = np.tile(y, (nfeatures, 1)).T
? ? while mis_class > 0 and iter < max_iter:
? ? ? ? iter += 1
? ? ? ? # Compute set of misclassified points
? ? ? ? mc = (np.dot(X, w.transpose()) * y) <= 0
? ? ? ? mis_class = np.sum(mc)
? ? ? ? # Update weights. Note, the learning rate decays over time (rho/iter)
? ? ? ? w -= rho / iter * (np.sum(-yy[mc, :] * X[mc, :], axis=0))
? ? return w, iter, np.sum(mc)