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A third order polynomial, trained to predict $$y=\sin(x)$$ from $$-\pi$$ to $$\pi$$ by minimizing squared Euclidean distance. Instead of writing the polynomial as $$y=a+bx+cx^2+dx^3$$, we write the polynomial as $$y=a+b P_3(c+dx)$$ where $$P_3(x)=\frac{1}{2}\left(5x^3-3x\right)$$ is the Legendre polynomial of degree three.

This implementation computes the forward pass using operations on PyTorch Tensors, and uses PyTorch autograd to compute gradients.

In this implementation we implement our own custom autograd function to perform $$P_3'(x)$$. By mathematics, $$P_3'(x)=\frac{3}{2}\left(5x^2-1\right)$$

import torch
import math

"""
We can implement our own custom autograd Functions by subclassing
torch.autograd.Function and implementing the forward and backward passes
which operate on Tensors.
"""

@staticmethod
def forward(ctx, input):
"""
In the forward pass we receive a Tensor containing the input and return
a Tensor containing the output. ctx is a context object that can be used
to stash information for backward computation. You can cache arbitrary
objects for use in the backward pass using the ctx.save_for_backward method.
"""
ctx.save_for_backward(input)
return 0.5 * (5 * input ** 3 - 3 * input)

@staticmethod
"""
In the backward pass we receive a Tensor containing the gradient of the loss
with respect to the output, and we need to compute the gradient of the loss
with respect to the input.
"""
input, = ctx.saved_tensors
return grad_output * 1.5 * (5 * input ** 2 - 1)

dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0")  # Uncomment this to run on GPU

# Create Tensors to hold input and outputs.
# By default, requires_grad=False, which indicates that we do not need to
# compute gradients with respect to these Tensors during the backward pass.
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)

# Create random Tensors for weights. For this example, we need
# 4 weights: y = a + b * P3(c + d * x), these weights need to be initialized
# not too far from the correct result to ensure convergence.
# respect to these Tensors during the backward pass.
a = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
b = torch.full((), -1.0, device=device, dtype=dtype, requires_grad=True)
c = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
d = torch.full((), 0.3, device=device, dtype=dtype, requires_grad=True)

learning_rate = 5e-6
for t in range(2000):
# To apply our Function, we use Function.apply method. We alias this as 'P3'.
P3 = LegendrePolynomial3.apply

# Forward pass: compute predicted y using operations; we compute
# P3 using our custom autograd operation.
y_pred = a + b * P3(c + d * x)

# Compute and print loss
loss = (y_pred - y).pow(2).sum()
if t % 100 == 99:
print(t, loss.item())

# Use autograd to compute the backward pass.
loss.backward()

# Update weights using gradient descent

# Manually zero the gradients after updating weights

print(f'Result: y = {a.item()} + {b.item()} * P3({c.item()} + {d.item()} x)')


Total running time of the script: ( 0 minutes 0.000 seconds)

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