SVM.md

import matplotlib.pyplot as plt
import numpy as np
import scipy.optimize, numpy.random
import time

def f(x):
    return x**2

def df(x):
    return 2*x

from random import choice

from functools import partial

from random import randint

begin = time.time()
print(scipy.optimize.minimize(f, numpy.random.randint(-1000, 1000), jac=df))
end = time.time()
print(f"Running time: { end - begin }")

      fun: 7.703719777548943e-34
 hess_inv: array([[0.5]])
      jac: array([5.55111512e-17])
  message: 'Optimization terminated successfully.'
     nfev: 7
      nit: 3
     njev: 7
   status: 0
  success: True
        x: array([2.77555756e-17])
Running time: 0.002002239227294922

begin = time.time()
print(scipy.optimize.minimize(f, numpy.random.randint(-1000, 1000), jac=False))
end = time.time()
print(f"Running time: { end - begin }")

      fun: 5.5357882549173466e-17
 hess_inv: array([[0.49999993]])
      jac: array([-2.97817368e-08])
  message: 'Optimization terminated successfully.'
     nfev: 14
      nit: 3
     njev: 7
   status: 0
  success: True
        x: array([-7.4402878e-09])
Running time: 0.0029959678649902344

The function with a custom jac parameter takes less time to run and makes fewer evaluations.

Calculating the hessian is also more precise in the case with a pre-determined jacobian parameter.

The developer takes part in the optimization only if they decide to pass a custom derivative function (jac).

v = np.array((randint(-100, 100), randint(-100, 100)))
xs = np.array([np.array((randint(-100, 100), randint(-100, 100))) for i in range(500)])
ys = np.array([np.dot(v, x) * randint(-1, 9) for x in xs])

def to_train_data(a:np.ndarray, b:np.ndarray) -> np.ndarray:
    return np.array([np.array([x, y]) for x, y in zip(a, b)])

tr_data:np.ndarray = to_train_data(xs, ys)

#Write in python the loss function for support vector machines from equation (7.48) of Daumé. You can use the following hinge loss surrogate:
def hinge_loss_surrogate(y_gold, y_pred):
    return np.max([0, 1 - y_gold * y_pred])

def svm_loss(wb:np.ndarray,
             C:float,
             D:np.ndarray):
    w = wb[:-1]
    b = wb[-1]
    # replace with your implementation, must call hinge_loss_surrogate
    cumul_loss = 0
    for item in D:
        cumul_loss += hinge_loss_surrogate(item[1], np.dot(w, item[0]) + b)    
    loss = np.dot(w, w) / 2 + C * cumul_loss
    return loss

#Use scipy.optimize.minimize with jac=False to implement support vector machines.
def svm(D:np.ndarray,
        C:float=1):
    # compute w and b with scipy.optimize.minimize and return them
    wb = np.zeros(D[0][0].shape[0] + 1)
    minfunc = partial(svm_loss, C=C, D=D)
    res = scipy.optimize.minimize(minfunc, x0=wb, jac=False).x
    return res[:-1], res[-1]

#Implement the gradient of svm_loss, and add an optional flag to svm to use it:
def gradient_hinge_loss_surrogate(y_gold, y_pred):
    if hinge_loss_surrogate(y_gold, y_pred) == 0:
        return 0
    else:
        return -y_gold

def gradient_svm_loss(wb:np.ndarray,
                      D:np.ndarray,
                      C:float=1):
    
    w, b = wb[:-1], wb[-1]
    der_by_w = 0
    der_by_b = 0
    for item in D:
        der_by_w += item[0] * gradient_hinge_loss_surrogate(
            item[1],
            np.dot(w, item[0]) + b
        )       
        der_by_b += gradient_hinge_loss_surrogate(
            item[1],
            np.dot(w, item[0]) + b
        )
    der_w = w + C * der_by_w
    der_b = C * der_by_b
    return np.hstack([der_w, der_b])

def svm(D:np.ndarray,
        C:float=1,
        use_gradient=False):

    wb = np.zeros(D[0][0].shape[0] + 1)
    minfunc = partial(svm_loss, C=C, D=D)
    minder = partial(gradient_svm_loss, C=C, D=D)
    if use_gradient:
        res = scipy.optimize.minimize(minfunc, x0=wb, jac=minder).x
        new_w, new_b = res[:-1], res[-1]
    else:
        res = scipy.optimize.minimize(minfunc, x0=wb, jac=False).x
        new_w, new_b = res[:-1], res[-1]
    return new_w, new_b

#Use numpy.random.normal to generate two isolated clusters of points in 2 dimensions, one x_plus and one x_minus, and 

#graph the three hyperplanes found by training:
#an averaged perceptron
#support vector machine without gradient
#support vector machine with gradient.
#How do they compare?
x_plus = np.random.normal(loc=[1,1], scale=0.5, size=(20,2))
x_minus = np.random.normal(loc=[-1,-1], scale=0.5, size=(20,2))

xes = np.vstack([x_plus, x_minus])

pos_labels = np.ones((x_plus.shape[0], 1))
neg_labels = pos_labels.copy() * -1

labels = np.vstack([pos_labels, neg_labels])

totrain = to_train_data(xes, labels)

def averaged_perceptron_train(D:np.ndarray,
                              maxiter: int,
                              weights:np.ndarray=None,
                              bias:np.float=None) -> tuple:
    if weights is None:
        weights = np.zeros(D[0][0].shape[0])
    cached_w: Vector = np.zeros(D[0][0].shape[0])
    if bias is None:
        bias = 0
    cached_b = 0
    iteration: int = 1
    counter = 1
    while iteration <= maxiter:
        for item in D:
            x, y = item[0], item[1]
            act = np.dot(x, weights) + bias
            if y * act <= 0:
                weights = weights + ( y * x )
                bias = bias + y
                cached_w = cached_w + counter * y * x
                cached_b = cached_b + counter * y
            counter += 1
        iteration += 1
    weigth_correction = cached_w / counter
    bias_correction = cached_b / counter
    return (weights - weigth_correction,
            bias - bias_correction)

w1, b1 = svm(totrain)
w2, b2 = svm(totrain, use_gradient=True)

w3, b3 = averaged_perceptron_train(totrain, 250)

def build_hyperplane(w:np.ndarray, b:np.float):
    x = [-2, 2]
    y = [-(w[0] * x[0] + b) / w[1], -(w[0] * x[1] + b) / w[1]]
    dots = choice(['k--', 'k:', 'k'])
    return x, y, dots

# plot the hyperplanes they find
fig, ax = plt.subplots()
ax.scatter(
	x_plus[:,0], x_plus[:,1],
	marker='+',
	color='g'
)
ax.scatter(
	x_minus[:,0], x_minus[:,1],
	marker='x',
	color='y'
)
x1, y1, d1 = build_hyperplane(w1, b1)
ax.plot(x1, y1, d1, label="Autograd", alpha=0.3, color="r")
x2, y2, d2 = build_hyperplane(w2, b2)
ax.plot(x2, y2, d2, label="Manual gradient", alpha=0.3, color="g")
x3, y3, d3 = build_hyperplane(w3, b3)
ax.plot(x3, y3, d3, label="Perceptron", alpha=0.3, color="b")
legend = ax.legend(shadow=True, fontsize='x-large')
# save as svm-svm-perceptron.pdf
plt.savefig("svm-svm-perceptron.pdf") 

Separation by the support vector machine is more precise as long as the data is linearly separable.

As can be seen from the picture, the average perceptron is adjusting the hyperplane more freely to the point where it starts misclassifying objects