Bonjour à tous !

Je suis en train de coder un SVM non linéaire pour de la reconnaissance de chiffres manuscrits. J'utilise la base de donné MNIST. Seulement voilà, j'ai un problème. J'ai choisis l'algorithme SMO pour l'apprentissage, et celui-ci ne fonctionne pas correctement. Je me suis basé sur le papier original de Platt et d'autres ouvrages pour le coder, mais j'ai du manquer quelque chose à un moment... J'ai pourtant vérifié, revérifié et comparé avec le pseudo-code donné dans les différentes ressources sur le sujet. Tout à pourtant l'air en ordre...
La problème a l'air de venir du faire que la valeur du biais crois de façon déraisonnable au fur et à mesure que l'algorithme regarde les exemples, parfois même jusqu'à finalement atteindre la valeur "Inf". Je n'ai pas réussi à trouver pourquoi... Et cette énorme valeur conduit évidemment le SVM à classer tous les exemples dans la même catégorie...

Si vous pouviez m'indiquer mon erreur, je vous en serai très reconnaissant
Mon code est en python, le voici:

Code Python : Sélectionner tout - Visualiser dans une fenêtre à part
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import numpy
import gzip
import struct
import matplotlib
from sklearn import datasets
from copy import copy
 
class SVM:
 
    def __init__(self, constant, data_set, label_set):
        self._N = len(data_set)
        if self._N != len(label_set):
            raise Exception("Data size and label size don't match.")
 
 
        self._C = constant
        self._epsilon = 0.001
        self._tol = 0.001
 
        self._data = [numpy.ndarray.flatten((1/255)*elt) for elt in data_set]
        self._dimension = len(self._data[0])
        self._label = label_set
        self._alphas = numpy.zeros((1, self._N))
        self._b = 0
        self._errors = numpy.ndarray((2, 0))
 
 
 
 
 
 
 
    def kernel(self, x1, x2):
        x1 = x1.reshape(1,self._dimension)
        result = numpy.power(numpy.dot(x1, x2), 3)
 
        return result
 
 
 
 
 
 
    def evaluate(self, x):
        result = 0
        i = 0
        while i < self._N:
            result +=  self._alphas[0, i]*self._label[i]*self.kernel(x, self._data[i])
            i += 1
 
        result += self._b        
        return result
 
 
 
 
 
 
    def update(self, i1, i2, E2):
        i1 = int(i1)
        i2 = int(i2)
        if i1 == i2:
            return 0
 
        y1 = self._label[i1]
        y2 = self._label[i2]
        alpha1 = self._alphas[0, i1]
        alpha2 = self._alphas[0, i2]
 
        #If alpha1 is non-bound, its error is in the cache.
        #So we check its position to extract its error.
        #Else, we compute it.
        if alpha1 > 0 and alpha1 < self._C :
            position = 0
            for i, elt in enumerate(self._errors[0, :]):
                if elt == i1:
                    position = i
 
            E1 = self._errors[1, position]
        else:
            E1 = self.evaluate(self._data[i1]) - y1
 
 
        s = y1*y2
        H = L = 0
 
        if y1 != y2:
            L = max(0, alpha2 - alpha1)
            H = min(self._C, self._C + alpha2 - alpha1)
        else:
            L = max(0, alpha2 + alpha1 - self._C)
            H = min(self._C, alpha2 + alpha1)
 
 
        if H == L:
            return 0
 
        K11 = self.kernel(self._data[i1], self._data[i1])
        K12 = self.kernel(self._data[i1], self._data[i2])
        K22 = self.kernel(self._data[i2], self._data[i2])
 
        eta = K11 + K22 - 2*K12
        if eta > 0:
            alpha2_new = alpha2 + (y2*(E1 - E2)/eta)
            if alpha2_new < L:
                alpha2_new = L
            elif alpha2_new > H:
                alpha2_new = H
 
        else:
            f1 = y1*(E1 + self._b) - alpha1*K11 - s*alpha2*K12
            f2 = y2*(E2 + self._b) - alpha2*K22 - s*alpha1*K12
 
            L1 = alpha1 + s*(alpha2 - L)
            H1 = alpha1 + s*(alpha2 - H)
 
            FuncL = L1*f1 + L*f2 + (1/2)*numpy.square(L1)*K11 + (1/2)*numpy.square(L)*K22 + s*L1*L*K12
            FuncH = H1*f1 + H*f2 + (1/2)*numpy.square(H1)*K11 + (1/2)*numpy.square(H)*K22 + s*H1*H*K12
 
            if FuncL < FuncH - self._epsilon:
                alpha2_new = L
            elif FuncL > FuncH + self._epsilon:
                alpha2_new = H
            else:
                alpha2_new = alpha2
 
 
 
        if numpy.abs(alpha2_new - alpha2) < self._epsilon*(alpha2_new+alpha2+ self._epsilon):
            return 0
 
        alpha1_new = alpha1 + s*(alpha2 - alpha2_new)
 
        #Update of the threshold.
        b1 = E1 + y1*(alpha1_new - alpha1)*K11 + y2*(alpha2_new - alpha2)*K12 + self._b
        b2 = E2 + y1*(alpha1_new - alpha1)*K12 + y2*(alpha2_new - alpha2)*K22 + self._b
 
        if L < alpha1_new < H:
            b_new = b1
        elif L < alpha2_new < H:
            b_new = b2
        else:
            b_new = (b1+b2)/2
 
 
#Update the cache error
 
        #If alpha2 was bound and its new value is non-bound, we add its index and its error to the cache.
        #If alpha2 was unbound and its new value is bound, we delete it from the cache.
        if (alpha2 == 0 or alpha2 == self._C) and (alpha2_new > 0 and alpha2_new < self._C):
            vector_alpha2_new = numpy.array([i2, E2])
            vector_alpha2_new = vector_alpha2_new.reshape((2, 1))
            self._errors = numpy.concatenate((self._errors, vector_alpha2_new), 1)
 
 
        if (alpha2 > 0 and alpha2 < self._C) and (alpha2_new == 0 or alpha2_new == self._C):
            l = 0
            position = 0
            while l < len(self._errors[0, :]):
                if self._errors[0, l] == i2:
                    position = l
                l += 1
 
            self._errors = numpy.delete(self._errors, position, 1)
 
 
        #We do the exact same thing with alpha1.
        if (alpha1 == 0 or alpha1 == self._C) and (alpha1_new > 0 and alpha1_new < self._C):
            vector_alpha1_new = numpy.array([i1, E1])
            vector_alpha1_new = vector_alpha1_new.reshape((2, 1))
            self._errors = numpy.concatenate((self._errors, vector_alpha1_new), 1)
 
 
        if (alpha1 > 0 and alpha1 < self._C) and (alpha1_new == 0 or alpha1_new == self._C):
            l = 0
            position = 0
            while l < len(self._errors[0, :]):
                if self._errors[0, l] == i1:
                    position = l
                l += 1
 
            self._errors = numpy.delete(self._errors, position, 1)        
 
 
 
        #Then we update the error for each non bound point using the new values for alpha1 and alpha2.
        for i,error in enumerate(self._errors[1, :]):
            self._errors[1, i] = error + (alpha2_new - alpha2)*y2*self.kernel(self._data[i2], self._data[int(self._errors[0, i])]) + (alpha1_new - alpha1)*y1*self.kernel(self._data[i1], self._data[int(self._errors[0, i])]) - self._b + b_new
 
 
        #Storing the new values of alpha1 and alpha2:
 
        self._alphas[0, i1] = alpha1_new
        self._alphas[0, i2] = alpha2_new
        self._b = b_new
 
        print(self._errors)
        return 1
 
 
 
 
    def examineExample(self, i2):
        i2 = int(i2)
        y2 = self._label[i2]
        alpha2 = self._alphas[0, i2]
 
        if alpha2 > 0 and alpha2 < self._C:
            position = 0
            for i, elt in enumerate(self._errors[0, :]):
                if elt == i2:
                    position = i
 
            E2 = self._errors[1, position]
        else:
            E2 = self.evaluate(self._data[i2]) - y2
 
        r2 = E2*y2
 
        if (r2< -self._tol and alpha2 < self._C) or (r2 > self._tol and alpha2 > 0):
 
            n = numpy.shape(self._errors)[1]            
            if n > 1:    
                i1 = 0
 
                if E2 > 0:
                    min = self._errors[1, 0]
                    position = 0
                    for l, elt in enumerate(self._errors[1, :]):
                        if elt < min:
                            min = elt
                            position = l
 
                    i1 = self._errors[0, position]
 
                else:
                    max = self._errors[1, 0]
                    position = 0
                    for l, elt in enumerate(self._errors[1, :]):
                        if elt > max:
                            max = elt
                            position = l
 
                    i1 = self._errors[0, position]
 
                if self.update(i1, i2, E2):
                    return 1
 
 
 
            #loop over all non bound examples starting at a random point.
            list_index = [i for i in range(n)]
            numpy.random.shuffle(list_index)
 
            for i in list_index:
                i1 = self._errors[0, i]
                if self.update(i1, i2, E2):
                    return 1
 
 
            #Loop over all the training examples, starting at a random point.
            list_bound = [i for i in range(self._N) if not numpy.any(self._errors[0, :] == i)]
            numpy.random.shuffle(list_bound)
 
            for i in list_bound:
                i1 = i
                if self.update(i1, i2, E2):
                    return 1
 
 
        return 0
 
 
 
 
    def SMO(self):
        numChanged = 0
        examineAll = 1
        cpt = 1
        while(numChanged > 0 or examineAll):
            numChanged = 0
 
            if examineAll == 1:
                for i in range(self._N):
                    numChanged += self.examineExample(i)
 
            else:
                for i in self._errors[0, :]:
                    numChanged += self.examineExample(i)
 
            if examineAll == 1:
                examineAll = 0
            elif numChanged == 0:
                examineAll = 1
 
            cpt += 1    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
def load_training_data(a, b):
    train = gzip.open("train-images-idx3-ubyte.gz", "rb")
    labels = gzip.open("train-labels-idx1-ubyte.gz", "rb")
 
    train.read(4)
    labels.read(4)
 
    number_images = train.read(4)
    number_images = struct.unpack(">I", number_images)[0]
 
    rows = train.read(4)
    rows = struct.unpack(">I", rows)[0]
 
    cols = train.read(4)
    cols = struct.unpack(">I", cols)[0]
 
    number_labels = labels.read(4)
    number_labels = struct.unpack(">I", number_labels)[0]
 
    image_list = []
    label_list = []
    if number_images != number_labels:
        raise Exception("The number of labels doesn't match with the number of images")
    else:
        for l in range(number_labels):
            if l % 1000 == 0:
                print("l:{}".format(l))
 
            mat = numpy.zeros((rows, cols), dtype = numpy.uint8)
            for i in range(rows):
                for j in range(cols):
                    pixel = train.read(1)
                    pixel = struct.unpack(">B", pixel)[0]
                    mat[i][j] = pixel
 
 
            image_list += [mat]
            lab = labels.read(1)
            lab = struct.unpack(">B", lab)[0]
            label_list += [lab]
 
 
    train.close()
    labels.close()
 
 
    i = 0
    index_a = []
    index_b = []
    while i < number_labels:
        if label_list[i] == a:
            index_a += [i]
        elif label_list[i] == b:
            index_b += [i]
 
        i += 1
 
    image_list = [m for i,m in enumerate(image_list) if (i in index_a) | (i in index_b)]
    mean = (a+b)/2
    label_list = [ numpy.sign(m - mean) for l,m in enumerate(label_list) if l in index_a+index_b]
 
    return ([image_list, label_list])
 
 
 
 
 
def load_test_data():
    test = gzip.open("t10k-images-idx3-ubyte.gz", "rb")
    labels = gzip.open("t10k-labels-idx1-ubyte.gz", "rb")
 
    test.read(4)
    labels.read(4)
 
    number_images = test.read(4)
    number_images = struct.unpack(">I", number_images)[0]
 
    rows = test.read(4)
    rows = struct.unpack(">I", rows)[0]
 
    cols = test.read(4)
    cols = struct.unpack(">I", cols)[0]
 
    number_labels = labels.read(4)
    number_labels = struct.unpack(">I", number_labels)[0]
 
    image_list = []
    label_list = []
    if number_images != number_labels:
        raise Exception("The number of labels doesn't match with the number of images")
    else:
        for l in range(number_labels):
            if l % 1000 == 0:
                print("l:{}".format(l))
 
            mat = numpy.zeros((rows, cols), dtype = numpy.uint8)
            for i in range(rows):
                for j in range(cols):
                    pixel = test.read(1)
                    pixel = struct.unpack(">B", pixel)[0]
                    mat[i][j] = pixel
 
 
            image_list += [mat]
            lab = labels.read(1)
            lab = struct.unpack(">B", lab)[0]
            label_list += [lab]
 
 
    test.close()
    labels.close()
 
    return ([image_list, label_list])    
 
data = load_training_data(0, 7)
images_training = data[0]
labels_training = data[1]
 
svm = SVM(0.1, images_training[0:200], labels_training[0:200])
 
svm.SMO()
 
 
 
 
def view(image, label=""):
    print("Number : {}".format(label))
    pylab.imshow(image, cmap = pylab.cm.gray)
    pylab.show()


Je ne sais vraiment plus quoi, et ça m'aiderai beaucoup que quelqu'un d'un peu plus expérimenté que moi m'aide là dessus.
Merci d'avance