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Compute Gradient
Use automatic differentiation to find the slope of a polynomial at a given point. The function is:
f(x) = x**3 + 2*x**2 + x
Your task
Implement compute_grad(x). The input is a Python number. Return a scalar
float32 tensor of shape (), containing the derivative at x. Return the
derivative, not the function value or a one-element vector.
For x=1, the function value is 4, but its derivative is 8. The derivative measures how quickly the function changes near that input.
Use the analytical formula 3*x**2 + 4*x + 1 to check your result, but compute
the answer with automatic differentiation for this exercise. For x=0 the
slope is 1; for x=-1 it is 0.
In PyTorch, create a float32 tensor with requires_grad=True, evaluate the
polynomial and call backward() or torch.autograd.grad. Read the derivative
from the input, not from the polynomial output.
In JAX, define a function returning the scalar polynomial value and call
jax.grad(f) on a float32 input. Keep the computation in tensor operations;
converting intermediate values to Python numbers disconnects differentiation.
Understand the ideas behind this exercise. The Problem tab contains the exact implementation requirements.
Compute derivatives by following the operations
Automatic differentiation computes derivatives by applying the chain rule to a program's operations. It does not estimate them by evaluating nearby inputs, as finite differences do, or rearrange the whole expression using symbolic algebra.
For f(g(x)), the chain rule combines the derivative of g with the derivative
of f at g(x). Repeating this process lets a framework differentiate a long
sequence of operations.
Forward mode follows an input change
For a function with n inputs and m outputs, the Jacobian J is an (m, n)
matrix of derivatives. Entry (i, j) describes how output i changes with input
j.
Forward-mode differentiation computes Jv: the output's rate of change when the
input moves in direction v. For example, if f(x) = sum(x_i^2), then at
x = [1, 2] the gradient is [2, 4]. In direction v = [3, 0], the rate of
change is 2 * 3 + 4 * 0 = 6.
This is a directional derivative. It gives one Jacobian column only when v
selects one input coordinate, such as [1, 0].
Reverse mode works backward from an output
Reverse mode computes J^T u, where u weights the outputs. For a scalar loss,
choosing u = 1 gives the gradient with respect to all inputs. This makes reverse
mode useful for training: a model can have many parameters but just one loss.
Forward mode can be useful when you need sensitivity to only a few input directions. Neither mode needs to construct the full Jacobian for these products. That can save substantial memory. Higher-order differentiation can similarly compute a Hessian-vector product without storing the full matrix of second derivatives.
Changing gradient flow changes what is differentiated
A stop-gradient operation keeps a value in the forward calculation but treats it as constant during differentiation. A custom derivative supplies a derivative rule explicitly. It should match the intended function unless the exercise asks for a surrogate, such as an approximation used during training.
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