machine-learning - 线性回归爆炸的梯度下降

Question

我正在尝试使用此资源实现线性回归的梯度下降:https://spin.atomicobject.com/2014/06/24/gradient-descent-linear-regression/

我的问题是我的权重正在爆炸(呈指数级增长)并且基本上与预期相反。

首先我创建了一个数据集:

def y(x, a):
    return 2*x + a*np.random.random_sample(len(x)) - a/2

x = np.arange(20)
y_true = y(x,10)

看起来像这样:

以及待优化的线性函数:

def y_predict(x, m, b):
    return m*x + b

因此，对于一些随机选择的参数，结果如下:

m0 = 1
b0 = 1

a = y_predict(x, m0, b0)

plt.scatter(x, y_true)
plt.plot(x, a)
plt.show()

现在成本如下:

cost = (1/2)* np.sum((y_true - a) ** 2)

成本相对于预测的偏导数 (dc_da):

dc_da = (a - y_true) # still a vector

成本相对于斜率参数(dc_dm)的偏导数:

dc_dm = dc_da.dot(x) # now a constant 

以及成本相对于 y 轴截距参数 (dc_db) 的偏导数:

dc_db = np.sum(dc_da) # also a constant

最后是梯度下降的实现:

iterations = 10

m0 = 1
b0 = 1

learning_rate = 0.1

N = len(x)

for i in range(iterations):

    a = y_predict(x, m0, b0)

    cost = (1/2) * np.sum((y_true - a) ** 2)

    dc_da = (a - y_true)

    mgrad = dc_da.dot(x)
    bgrad = np.sum(dc_da)

    m0 -= learning_rate * (2 / N) * mgrad
    b0 -= learning_rate * (2 / N) * bgrad

    if (i % 2 == 0):
        print("Iteration {}".format(i))
        print("Cost: {}, m: {}, b: {}\n".format(cost, m0, b0))

其结果是:

Iteration 0
Cost: 1341.5241150881411, m: 26.02473879743261, b: 2.8683883457327797

Iteration 2
Cost: 409781757.38124645, m: 13657.166910552878, b: 1053.5831308528543

Iteration 4
Cost: 132510115599264.75, m: 7765058.4350503925, b: 598610.1166795876

Iteration 6
Cost: 4.284947676217907e+19, m: 4415631880.089208, b: 340401694.5610262

Iteration 8
Cost: 1.3856132043127762e+25, m: 2510967578365.3584, b: 193570850213.62192

显然，出了什么问题。但我不知道我的实现有什么问题。

感谢您的阅读

Answer 1

问题在于学习率。

学习率为 0.1 时，步长太大，导致它们逃脱了下坡梯度。

学习率为 0.001 时，结果如下:

Iteration 0
Cost: 1341.5241150881411, m: 1.250247387974326, b: 1.0186838834573277

Iteration 20
Cost: 74.23350734398517, m: 2.0054600094398487, b: 1.0648169455682297

Iteration 40
Cost: 74.14854910310204, m: 2.00886824141609, b: 1.0531220375231194

Iteration 60
Cost: 74.07892801481468, m: 2.0097830838155835, b: 1.0413622803654885

Iteration 80
Cost: 74.01078231057598, m: 2.0106800645568503, b: 1.0297271562539492

看起来像:

plt.scatter(x,y_true)
plt.plot(x, a)
plt.show()