公式推导

\(L_{i}=(\sigma(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j})-t_{i})^{2}\).

\(\nabla_{i}=\frac{dL}{d\omega_{i}}=2(\sigma(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j})-t_{i})\sigma(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j})(1-\sigma(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}))x_{i}^{j}\).

其中这里取激活函数\(\sigma(x)=\frac{1}{1+e^{-x}}\)，\(\frac{\partial\sigma(x)}{\partial x}=\frac{-1}{(1-e^{-x})^{2}}=\sigma(x)(1-\sigma(x))\)

将\(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}\)带入激活函数\(\sigma(x)\)可以得到\(\sigma(\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j})=\frac{1}{1+e^{-\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}}}\)

所以\(\nabla_{i}=\frac{dL}{d\omega_{i}}=2(\frac{1}{1+e^{-\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}}})-t_{i})(\frac{1}{1+e^{-\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}}})(1-\frac{1}{1+e^{-\Sigma_{j=0}^{2}x^{i}_{j}\omega_{j}}})x_{i}^{j}\)

代码实现

import math
import random

class Perceptron:
    def __init__(self) -> None:
        self.theta = 0.5
        self.eps = 1e-4
        self.tot = 0
        self.delta = [10.0, 10.0, 10.0]
        self.W = [random.random(), random.random(), random.random()]
        self.X = [[1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]
        self.T = [0, 0, 0, 1]

    def __calSigmod(self, x:float):
        return 1.0 / (1.0 + math.pow(math.e, -x))

    def __calSegma(self, x:list):
        res = 0.0
        for i in range(3):
            res = res + x[i] * self.W[i]
        # res = (res + 1.0) / 2.0
        return res

    def check(self):
        for p in range(4):
            for i in range(3):
                s = self.__calSigmod(self.__calSegma(self.X[p]))
                t = (s - self.T[p]) * s * (1.0 - s) * self.X[p][i]
                if t > self.eps:
                    return True
        return False

    def iterate(self):
        self.tot = self.tot + 1
        p = random.randint(0, 3)
        for i in range(3):
            s = self.__calSigmod(self.__calSegma(self.X[p]))
            self.delta[i] = (s - self.T[p]) * s * (1 - s) * self.X[p][i]
        for i in range(3):
            self.W[i] = self.W[i] - self.delta[i] * self.theta

    def printResult(self):
        print('Iteration time: ', self.tot)
        print('Coe: ', self.W)
        print('Result:')
        for i in range(4):
            x = self.X[i]
            print(x[1:], ': ', end='')
            res = 0.0
            for j in range(3):
                res = res + x[j] * self.W[j]
            res = self.__calSigmod(res)
            print('%.0f(%f)' % (res, res))


def main():
    perceptron = Perceptron()
    while perceptron.check():
        perceptron.iterate()
    perceptron.printResult()


if __name__ == '__main__':
    main()

运行结果

Iteration time:  201751
Coe:  [-13.56814252989061, 8.978083107458904, 8.976992168326456]
Result:
[0, 0] : 0(0.000001)
[0, 1] : 0(0.010039)
[1, 0] : 0(0.010050)
[1, 1] : 1(0.987714)