Question

这就是我为线性回归生成训练数据的方式。

!pip install grapher, numpy

from grapher import Grapher
import matplotlib.pyplot as plt
import numpy as np

# Secret: y = 3x + 4

# x, y = [float(row[0]) for row in rows], [float(row[5]) for row in rows]

x, y = [a for a in range(-20, 20)], [3*a + 4 for a in range(-20, 20)]


g = Grapher(['3*x + 4'], title="y = 3x+4")
plt.scatter(x, y)
g.plot()

然后，我在一个简单的二次函数 (x - 7)^2 上尝试了梯度下降

def n(x):
    return (x-7)**2

cur_x = 0
lr = 0.001
ittr = 10000
n = 0
prev_x = -1
max_precision = 0.0000001
precision = 1

while n < ittr and precision > max_precision:
    prev_x = cur_x

    cur_x = cur_x - lr * (2*(cur_x - 7))
    precision = abs(prev_x - cur_x)

    n+=1
    if n%100 == 0:
        print(n, ':')
        print(cur_x)
        print()

print(cur_x)

这很完美。

然后我做了一个线性回归类来实现同样的事情。

class LinearRegression:
    def __init__(self, X, Y):
        self.X = X
        self.Y = Y
        self.m = 1
        self.c = 0
        self.learning_rate = 0.01
        self.max_precision = 0.000001
        self.itter = 10000

    def h(self, x, m, c):
        return m * x + c

    def J(self, m, c):
        loss = 0
        for x in self.X:
            loss += (self.h(x, m, c) - self.Y[self.X.index(x)])**2

        return loss/2

    def calc_loss(self):
        return self.J(self.m, self.c)

    def guess_answer(self, step=1):
        losses = []
        mcvalues = []
        for m in np.arange(-10, 10, step):
            for c in np.arange(-10, 10, step):
                mcvalues.append((m, c))
                losses.append(self.J(m, c))

        minloss = sorted(losses)[0]
        return mcvalues[losses.index(minloss)]

    def gradient_decent(self):
        print('Orignal: ', self.m, self.c)

        nm = 0
        nc = 0

        prev_m = 0
        perv_c = -1

        mprecision = 1
        cprecision = 1

        while nm < self.itter and mprecision > self.max_precision:
            prev_m = self.m
            nm += 1

            self.m = self.m - self.learning_rate * sum([(self.h(x, self.m, self.c) - self.Y[self.X.index(x)])*x for x in self.X])
            mprecision = abs(self.m - prev_m)

        return self.m, self.c

    def graph_loss(self):
        plt.scatter(0, self.J(0))
        print(self.J(0))
        plt.plot(self.X, [self.J(x) for x in self.X])

    def check_loss(self):
        plt.plot([m for m in range(-20, 20)], [self.J(m, 0) for m in range(-20, 20)])
        x1 = 10
        y1 = self.J(x1, 0)
        l = sum([(self.h(x, x1, self.c) - self.Y[self.X.index(x)])*x for x in self.X])
        print(l)
        plt.plot([m for m in range(-20, 20)], [(l*(m - x1)) + y1 for m in range(-20, 20)])
        plt.scatter([x1], [y1])

LinearRegression(x, y).gradient_decent()

输出是