Replace "Learning PyTorch with Examples" with fitting sine function w…

…ith a third order polynomial (#1265)

* Replace tutorial with fitting sine function with third order polynomial

* more

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* no tensor.data

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* P3

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beginner_source/examples_autograd/polynomial_autograd.py

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+# -*- coding: utf-8 -*-
+"""
+PyTorch: Tensors and autograd
+-------------------------------
+
+A third order polynomial, trained to predict :math:`y=\sin(x)` from :math:`-\pi`
+to :math:`pi` by minimizing squared Euclidean distance.
+
+This implementation computes the forward pass using operations on PyTorch
+Tensors, and uses PyTorch autograd to compute gradients.
+
+
+A PyTorch Tensor represents a node in a computational graph. If ``x`` is a
+Tensor that has ``x.requires_grad=True`` then ``x.grad`` is another Tensor
+holding the gradient of ``x`` with respect to some scalar value.
+"""
+import torch
+import math
+
+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 a third order polynomial, we need
+# 4 weights: y = a + b x + c x^2 + d x^3
+# Setting requires_grad=True indicates that we want to compute gradients with
+# respect to these Tensors during the backward pass.
+a = torch.randn((), device=device, dtype=dtype, requires_grad=True)
+b = torch.randn((), device=device, dtype=dtype, requires_grad=True)
+c = torch.randn((), device=device, dtype=dtype, requires_grad=True)
+d = torch.randn((), device=device, dtype=dtype, requires_grad=True)
+
+learning_rate = 1e-6
+for t in range(2000):
+    # Forward pass: compute predicted y using operations on Tensors.
+    y_pred = a + b * x + c * x ** 2 + d * x ** 3
+
+    # Compute and print loss using operations on Tensors.
+    # Now loss is a Tensor of shape (1,)
+    # loss.item() gets the scalar value held in the loss.
+    loss = (y_pred - y).pow(2).sum()
+    if t % 100 == 99:
+        print(t, loss.item())
+
+    # Use autograd to compute the backward pass. This call will compute the
+    # gradient of loss with respect to all Tensors with requires_grad=True.
+    # After this call a.grad, b.grad. c.grad and d.grad will be Tensors holding
+    # the gradient of the loss with respect to a, b, c, d respectively.
+    loss.backward()
+
+    # Manually update weights using gradient descent. Wrap in torch.no_grad()
+    # because weights have requires_grad=True, but we don't need to track this
+    # in autograd.
+    with torch.no_grad():
+        a -= learning_rate * a.grad
+        b -= learning_rate * b.grad
+        c -= learning_rate * c.grad
+        d -= learning_rate * d.grad
+
+        # Manually zero the gradients after updating weights
+        a.grad = None
+        b.grad = None
+        c.grad = None
+        d.grad = None
+
+print(f'Result: y = {a.item()} + {b.item()} x + {c.item()} x^2 + {d.item()} x^3')

beginner_source/examples_autograd/polynomial_custom_function.py

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+# -*- coding: utf-8 -*-
+"""
+PyTorch: Defining New autograd Functions
+----------------------------------------
+
+A third order polynomial, trained to predict :math:`y=\sin(x)` from :math:`-\pi`
+to :math:`pi` by minimizing squared Euclidean distance. Instead of writing the
+polynomial as :math:`y=a+bx+cx^2+dx^3`, we write the polynomial as
+:math:`y=a+b P_3(c+dx)` where :math:`P_3(x)=\frac{1}{2}\left(5x^3-3x\right)` is
+the `Legendre polynomial`_ of degree three.
+
+.. _Legendre polynomial:
+    https://en.wikipedia.org/wiki/Legendre_polynomials
+
+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
+:math:`P_3'(x)`. By mathematics, :math:`P_3'(x)=\frac{3}{2}\left(5x^2-1\right)`
+"""
+import torch
+import math
+
+
+class LegendrePolynomial3(torch.autograd.Function):
+    """
+    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
+    def backward(ctx, grad_output):
+        """
+        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.
+# Setting requires_grad=True indicates that we want to compute gradients with
+# 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
+    with torch.no_grad():
+        a -= learning_rate * a.grad
+        b -= learning_rate * b.grad
+        c -= learning_rate * c.grad
+        d -= learning_rate * d.grad
+
+        # Manually zero the gradients after updating weights
+        a.grad = None
+        b.grad = None
+        c.grad = None
+        d.grad = None
+
+print(f'Result: y = {a.item()} + {b.item()} * P3({c.item()} + {d.item()} x)')