Jacobian with jacrev

Compute a Jacobian with reverse-mode differentiation

A Jacobian contains the derivatives of every output with respect to every input. jax.jacrev constructs it using reverse-mode differentiation. For a vector input of length d and vector output of length k, the result has shape (k, d): outputs in rows, inputs in columns.

import jax
import jax.numpy as jnp

def f(x):
    return jnp.array([x[0], x[0] + x[1]])

J = jax.jacrev(f)(jnp.array([1.0, 2.0]))
# [[1.0, 0.0],
#  [1.0, 1.0]]

The first output depends only on x[0], giving the row [1, 0]. The second adds both inputs, giving [1, 1]. Pass the function itself to jax.jacrev, then evaluate the returned function at x.

Choose a mode without changing the derivative

Reverse-mode builds Jacobian rows from output weights. Forward-mode jax.jacfwd builds columns from input directions. For a full Jacobian, reverse-mode often suits functions with many inputs and few outputs. Actual time and memory use also depend on the function and compiler; the number of inputs or outputs alone is not a complete cost model.

For the smooth functions here, both methods compute the same mathematical Jacobian, allowing for floating-point differences. Composing them, as in jax.jacrev(jax.jacfwd(f)), differentiates the Jacobian again. For this exercise's three-output function, that would produce second derivatives with shape (3, d, d), not the requested first derivatives with shape (3, d).

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.