Gradient Descent for Linear Regression Exploding

Question

I am trying to implement gradient descent for linear regression using this resource: https://spin.atomicobject.com/2014/06/24/gradient-descent-linear-regression/

My problem is that my weights are exploding (increasing exponentially) and essentially doing the opposite of what is intended.

First I created a data set:

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)

Which looks like this:

And the linear function to be optimized:

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

So for some randomly chosen parameters, this is the result:

m0 = 1
b0 = 1

a = y_predict(x, m0, b0)

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

Now the cost would look like this:

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

The partial derivative of the cost with respect to the prediction (dc_da):

dc_da = (a - y_true) # still a vector

The partial derivative of the cost with respect to the slope parameter (dc_dm):

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

And the partial derivative of the cost with respect to the y-intercept parameter (dc_db):

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

And finally the implementation of gradient descent:

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))

For which the result is:

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

What is wrong with my implementation?

elMentat · Accepted Answer · 2018-05-07 17:22:59Z

3

The problem is the learning rate.

At a learning rate of 0.1, the step sizes were too large, causing them to escape the downhill gradient.

At a learning rate of 0.001, this is the result:

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

Which looks like:

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

answered May 7, 2018 at 17:22

elMentat

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