多层感知器无法收敛

Question

This is a simple MLP I am writing for binary image classification, with backpropagation:

class MLP:
    def __init__(self, size, epochs = 1000, learning_rate = 1):
        self.l1weights = numpy.random.random((size + 1, 3))
        self.l2weights = numpy.random.random(3)
        self.epochs = epochs
        self.learning_rate = learning_rate

    def predict(self, _input_):
        #Append bias at the beginning of input
        l1output = self.sigmoid(numpy.dot(numpy.append([1], _input_), self.l1weights))
        l2output = self.sigmoid(numpy.dot(l1output, self.l2weights))
        return l1output, l2output

    def train(self, training_set, training_goal):
        for epoch in range(self.epochs):
            l1squared_error = 0
            l2squarederror = 0
            for set_index in range(training_goal.shape[0]):
                set = training_set[set_index]
                l1output, l2output = self.predict(set)

                l2error = training_goal[set_index] - l2output
                l1error = l2error * self.dsigmoid(l2output) * self.l2weights

                self.l1weights[0] = self.l1weights[0] + self.learning_rate * l1error
                for index in range(len(self.l1weights) - 1):
                    self.l1weights[index + 1] += self.learning_rate * l1error * self.dsigmoid(l1output)
                for index in range(len(self.l2weights)):
                    self.l2weights[index] += self.learning_rate * l2error * self.dsigmoid(l2output)

                l1squared_error += sum(l1error ** 2)
                l2squarederror += l2error ** 2
            print("Squared error at epoch " + str(epoch) + " : " + str(l1squared_error) + ", " + str(l2squarederror))

    def sigmoid(self, _input_):
        #Sigmoid sigmoid function
        return 1 / (1 + numpy.exp(-_input_))

    def dsigmoid(self, _input_):
        return _input_ * (1 - _input_)

When run sometimes all output converges into 1 but for some reason the predictions for 0 converge into 0.5 while predictions for 1 stay near 0.75, with error from layer 2 staying the same after ~1000 epochs, if it does relatively more successfully. This is from testing with 2x2 image classification with the code below:

def image_class(input):
    return 1 if input >= 2 else 0

training_set = ((numpy.arange(2**4)[:,None] & (1 << numpy.arange(4))) != 0)
training_goals = numpy.array([image_class(sum(i)) for i in training_set])

mlp = MLP(size=4)
mlp.train(training_set, training_goals)

Answer 1

我可以通过以下步骤解决此问题：在输出层之后立即添加逐步激活而不是Sigmoid的层，并将其与初始网络分开进行训练，至少使用2x2识别。