Pytorch Linear Regression with squared features

Question

I am new to PyTorch and I would like to implement linear regression partly with PyTorch and partly on my own. I want to use squared features for my regression:

import torch

# init
x = torch.tensor([1,2,3,4,5])
y = torch.tensor([[1],[4],[9],[16],[25]])
w = torch.tensor([[0.5], [0.5], [0.5]], requires_grad=True)

iterations = 30
alpha = 0.01

def forward(X):
    # feature transformation [1, x, x^2]
    psi = torch.tensor([[1.0, x[0], x[0]**2]])
    for i in range(1, len(X)):
        psi = torch.cat((psi, torch.tensor([[1.0, x[i], x[i]**2]])), 0)
    return torch.matmul(psi, w)
    
def loss(y, y_hat):
    return ((y-y_hat)**2).mean()

for i in range(iterations):
    
    y_hat = forward(x)

    l = loss(y, y_hat)
    l.backward()
    
    with torch.no_grad():
        w -= alpha * w.grad 
    w.grad.zero_()

    if i%10 == 0:
        print(f'Iteration {i}: The weight is:\n{w.detach().numpy()}\nThe loss is:{l}\n')

When I execute my code, the regression doesn't learn the correct features and the loss increases permanently. The output is the following:

Iteration 0: The weight is:
[[0.57 ]
 [0.81 ]
 [1.898]]
The loss is:25.450000762939453

Iteration 10: The weight is:
[[ 5529.5835]
 [22452.398 ]
 [97326.12  ]]
The loss is:210414632960.0

Iteration 20: The weight is:
[[5.0884394e+08]
 [2.0662339e+09]
 [8.9567642e+09]]
The loss is:1.7820802835250162e+21

Does somebody know, why my model is not learning?

UPDATE

Is there a reason why it performs so poorly? I thought it's because of the low number of training data. But also with 10 data points, it is not performing well :

Answer 1

You should normalize your data. Also, since you're trying to fit x -> ax² + bx + c, c is essentially the bias. It should be wiser to remove it from the training data (I'm referring to psi here) and use a separate parameter for the bias.

What could be done:

• normalize your input data and targets with mean and standard deviation.

• separate the parameters into w (a two-component weight tensor) and b (the bias).

• you don't need to construct psi on every inference since x is identical.

• you can build psi with torch.stack([torch.ones_like(x), x, x**2], 1), but here we won't need the ones, as we've essentially detached the bias from the weight tensor.

Here's how it would look like:

x = torch.tensor([1,2,3,4,5]).float()
psi = torch.stack([x, x**2], 1).float()
psi = (psi - psi.mean(0)) / psi.std(0)

y = torch.tensor([[1],[4],[9],[16],[25]]).float()
y = (y - y.mean(0)) / y.std(0)

w = torch.tensor([[0.5], [0.5]], requires_grad=True)
b = torch.tensor([0.5], requires_grad=True)

iterations = 30
alpha = 0.02
def loss(y, y_hat):
    return ((y-y_hat)**2).mean()

for i in range(iterations):
    y_hat = torch.matmul(psi, w) + b
    l = loss(y, y_hat)
    l.backward()
    
    with torch.no_grad():
        w -= alpha * w.grad 
        b -= alpha * b.grad 
    w.grad.zero_()
    b.grad.zero_()

    if i%10 == 0:
        print(f'Iteration {i}: The weight is:\n{w.detach().numpy()}\nThe loss is:{l}\n')

And the results:

Iteration 0: The weight is:
[[0.49954653]
 [0.5004535 ]]
The loss is:0.25755801796913147

Iteration 10: The weight is:
[[0.49503425]
 [0.5049657 ]]
The loss is:0.07994867861270905

Iteration 20: The weight is:
[[0.49056274]
 [0.50943726]]
The loss is:0.028329044580459595