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(jax-301-cookbook)=
# The autodiff cookbook: JVPs and VJPs

<!--* freshness: { reviewed: '2026-07-16' } *-->

JAX has a pretty general automatic differentiation system. In this document
we'll go through a whole bunch of neat autodiff ideas that you can cherry-pick
for your own work. The basics of `jax.grad` (`argnums`, differentiating with
respect to containers, `value_and_grad`, checking derivatives against finite
differences) are covered in the 101 docs ({ref}`jax-101-transformations`), so
here we start one level up: Hessian-vector products, full Jacobians, the
`jvp`/`vjp` machinery underneath it all, and differentiation with complex
numbers.

```{code-cell}
import jax.numpy as jnp
from jax import grad, jit, vmap
from jax import random

key = random.key(0)
```

## Setup: a running example

As a running example, we'll use the same logistic regression model as the
101 docs:

```{code-cell}
def sigmoid(x):
    return 0.5 * (jnp.tanh(x / 2) + 1)

# Outputs probability of a label being true.
def predict(W, b, inputs):
    return sigmoid(jnp.dot(inputs, W) + b)

# Build a toy dataset.
inputs = jnp.array([[0.52, 1.12,  0.77],
                   [0.88, -1.08, 0.15],
                   [0.52, 0.06, -1.30],
                   [0.74, -2.49, 1.39]])
targets = jnp.array([True, True, False, True])

# Training loss is the negative log-likelihood of the training examples.
def loss(W, b):
    preds = predict(W, b, inputs)
    label_probs = preds * targets + (1 - preds) * (1 - targets)
    return -jnp.sum(jnp.log(label_probs))

# Initialize random model coefficients
key, W_key, b_key = random.split(key, 3)
W = random.normal(W_key, (3,))
b = random.normal(b_key, ())
```

## Jacobians and Hessians using `jacfwd` and `jacrev`

+++

You can compute full Jacobian matrices using the `jacfwd` and `jacrev` functions:

```{code-cell}
from jax import jacfwd, jacrev

# Isolate the function from the weight matrix to the predictions
f = lambda W: predict(W, b, inputs)

J = jacfwd(f)(W)
print("jacfwd result, with shape", J.shape)
print(J)

J = jacrev(f)(W)
print("jacrev result, with shape", J.shape)
print(J)
```

These two functions compute the same values (up to machine numerics), but differ in their implementation: `jacfwd` uses forward-mode automatic differentiation, which is more efficient for "tall" Jacobian matrices (more outputs than inputs), while `jacrev` uses reverse-mode, which is more efficient for "wide" Jacobian matrices (more inputs than outputs). For matrices that are near-square, `jacfwd` probably has an edge over `jacrev`.

+++

You can also use `jacfwd` and `jacrev` with container types:

```{code-cell}
def predict_dict(params, inputs):
    return predict(params['W'], params['b'], inputs)

J_dict = jacrev(predict_dict)({'W': W, 'b': b}, inputs)
for k, v in J_dict.items():
    print("Jacobian from {} to logits is".format(k))
    print(v)
```

For more details on forward- and reverse-mode, as well as how to implement `jacfwd` and `jacrev` as efficiently as possible, read on!

+++

Using a composition of two of these functions gives us a way to compute dense Hessian matrices:

```{code-cell}
def hessian(f):
    return jacfwd(jacrev(f))

H = hessian(f)(W)
print("hessian, with shape", H.shape)
print(H)
```

This shape makes sense: if we start with a function $f : \mathbb{R}^n \to \mathbb{R}^m$, then at a point $x \in \mathbb{R}^n$ we expect to get the shapes

* $f(x) \in \mathbb{R}^m$, the value of $f$ at $x$,
* $\partial f(x) \in \mathbb{R}^{m \times n}$, the Jacobian matrix at $x$,
* $\partial^2 f(x) \in \mathbb{R}^{m \times n \times n}$, the Hessian at $x$,

and so on.

To implement `hessian`, we could have used `jacfwd(jacrev(f))` or `jacrev(jacfwd(f))` or any other composition of the two. But forward-over-reverse is typically the most efficient. That's because in the inner Jacobian computation we're often differentiating a function with a wide Jacobian (maybe like a loss function $f : \mathbb{R}^n \to \mathbb{R}$), while in the outer Jacobian computation we're differentiating a function with a square Jacobian (since $\nabla f : \mathbb{R}^n \to \mathbb{R}^n$), which is where forward-mode wins out.

+++

(jax-301-stopping-gradients)=
## Stopping gradients

Sometimes you want autodiff to treat a value as a constant: use it as usual
in the forward pass, but propagate no derivative through it. That's what
`jax.lax.stop_gradient` does: it's the identity function, with a derivative
that's identically zero.

One classic use is optimizing against a target built from the same
parameters you're differentiating. In TD(0) reinforcement-learning updates,
for example, the value-function target depends on the parameters, but a
correct update requires treating it as a constant:

```{code-cell}
from jax import lax

def value_fn(theta, state):
  return jnp.dot(theta, state)

def td_loss(theta, s_prev, r, s):
  v_prev = value_fn(theta, s_prev)
  target = r + value_fn(theta, s)
  return -0.5 * (lax.stop_gradient(target) - v_prev) ** 2

theta = jnp.array([0.1, -0.1, 0.])
s_prev, r, s = jnp.array([1., 2., -1.]), 1., jnp.array([2., 1., 0.])
print(grad(td_loss)(theta, s_prev, r, s))
```

Without the `stop_gradient`, the gradient would include a second term
flowing through `target`, a different (and here, unwanted) algorithm.

Another classic is the *straight-through estimator*: apply a
non-differentiable (or zero-derivative) function in the forward pass, but
differentiate as if it were the identity. There's a well-known
`stop_gradient` one-liner for it:

```{code-cell}
def straight_through_round(x):
  return x + lax.stop_gradient(jnp.round(x) - x)

print(straight_through_round(1.7))        # rounds like round
print(grad(straight_through_round)(1.7))  # differentiates like the identity
```

In the forward pass the `x`s cancel, leaving `round(x)`; under
differentiation the `stop_gradient` term contributes zero, leaving the
derivative of the identity. (For more principled ways to attach this kind of
custom derivative behavior to functions or types, see
{doc}`custom-jvp-vjp` and the straight-through example in
{ref}`jax-301-hijax-types`.)

+++

## How it's made: two foundational autodiff functions

+++

(jax-301-jvp)=

### Jacobian-Vector products (JVPs, aka forward-mode autodiff)

JAX includes efficient and general implementations of both forward- and reverse-mode automatic differentiation. The familiar `grad` function is built on reverse-mode, but to explain the difference between the two modes, and when each can be useful, we need a bit of math background.

#### JVPs in math

Mathematically, given a function $f : \mathbb{R}^n \to \mathbb{R}^m$, the Jacobian of $f$ evaluated at an input point $x \in \mathbb{R}^n$, denoted $\partial f(x)$, is often thought of as a matrix:

$\qquad \partial f(x) \in \mathbb{R}^{m \times n}$.

But we can also think of $\partial f(x)$ as a linear map, which maps the tangent space of the domain of $f$ at the point $x$ (which is just another copy of $\mathbb{R}^n$) to the tangent space of the codomain of $f$ at the point $f(x)$ (a copy of $\mathbb{R}^m$):

$\qquad \partial f(x) : \mathbb{R}^n \to \mathbb{R}^m$.

This map is called the [pushforward map](https://en.wikipedia.org/wiki/Pushforward_(differential)) of $f$ at $x$. The Jacobian matrix is just the matrix for this linear map in a standard basis.

If we don't commit to one specific input point $x$, then we can think of the function $\partial f$ as first taking an input point and returning the Jacobian linear map at that input point:

$\qquad \partial f : \mathbb{R}^n \to \mathbb{R}^n \to \mathbb{R}^m$.

In particular, we can uncurry things so that given input point $x \in \mathbb{R}^n$ and a tangent vector $v \in \mathbb{R}^n$, we get back an output tangent vector in $\mathbb{R}^m$. We call that mapping, from $(x, v)$ pairs to output tangent vectors, the *Jacobian-vector product*, and write it as

$\qquad (x, v) \mapsto \partial f(x) v$

#### JVPs in JAX code

Back in Python code, JAX's `jvp` function models this transformation. Given a Python function that evaluates $f$, JAX's `jvp` is a way to get a Python function for evaluating $(x, v) \mapsto (f(x), \partial f(x) v)$.

```{code-cell}
from jax import jvp

# Isolate the function from the weight matrix to the predictions
f = lambda W: predict(W, b, inputs)

key, subkey = random.split(key)
v = random.normal(subkey, W.shape)

# Push forward the vector `v` along `f` evaluated at `W`
y, u = jvp(f, (W,), (v,))
```

In terms of [Haskell-like type signatures](https://wiki.haskell.org/Type_signature),
we could write

```haskell
jvp :: (a -> b) -> a -> T a -> (b, T b)
```

where we use `T a` to denote the type of the tangent space for `a`. In words, `jvp` takes as arguments a function of type `a -> b`, a value of type `a`, and a tangent vector value of type `T a`. It gives back a pair consisting of a value of type `b` and an output tangent vector of type `T b`.

+++

The `jvp`-transformed function is evaluated much like the original function, but paired up with each primal value of type `a` it pushes along tangent values of type `T a`. For each primitive numerical operation that the original function would have applied, the `jvp`-transformed function executes a "JVP rule" for that primitive that both evaluates the primitive on the primals and applies the primitive's JVP at those primal values.

That evaluation strategy has some immediate implications about computational complexity: since we evaluate JVPs as we go, we don't need to store anything for later, and so the memory cost is independent of the depth of the computation. In addition, the FLOP cost of the `jvp`-transformed function is about 3x the cost of just evaluating the function (one unit of work for evaluating the original function, for example `sin(x)`; one unit for linearizing, like `cos(x)`; and one unit for applying the linearized function to a vector, like `cos_x * v`). Put another way, for a fixed primal point $x$, we can evaluate $v \mapsto \partial f(x) \cdot v$ for about the same marginal cost as evaluating $f$.

That memory complexity sounds pretty compelling! So why don't we see forward-mode very often in machine learning?

To answer that, first think about how you could use a JVP to build a full Jacobian matrix. If we apply a JVP to a one-hot tangent vector, it reveals one column of the Jacobian matrix, corresponding to the nonzero entry we fed in. So we can build a full Jacobian one column at a time, and to get each column costs about the same as one function evaluation. That will be efficient for functions with "tall" Jacobians, but inefficient for "wide" Jacobians.

If you're doing gradient-based optimization in machine learning, you probably want to minimize a loss function from parameters in $\mathbb{R}^n$ to a scalar loss value in $\mathbb{R}$. That means the Jacobian of this function is a very wide matrix: $\partial f(x) \in \mathbb{R}^{1 \times n}$, which we often identify with the gradient vector $\nabla f(x) \in \mathbb{R}^n$. Building that matrix one column at a time, with each call taking a similar number of FLOPs to evaluate the original function, sure seems inefficient! In particular, for training neural networks, where $f$ is a training loss function and $n$ can be in the millions or billions, this approach just won't scale.

To do better for functions like this, we just need to use reverse-mode.

+++

(jax-301-vjp)=

### Vector-Jacobian products (VJPs, aka reverse-mode autodiff)

Where forward-mode gives us back a function for evaluating Jacobian-vector products, which we can then use to build Jacobian matrices one column at a time, reverse-mode is a way to get back a function for evaluating vector-Jacobian products (equivalently Jacobian-transpose-vector products), which we can use to build Jacobian matrices one row at a time.

#### VJPs in math

Let's again consider a function $f : \mathbb{R}^n \to \mathbb{R}^m$.
Starting from our notation for JVPs, the notation for VJPs is pretty simple:

$\qquad (x, v) \mapsto v \partial f(x)$,

where $v$ is an element of the cotangent space of the codomain of $f$ at $f(x)$ (isomorphic to another copy of $\mathbb{R}^m$). When being rigorous, we should think of $v$ as a linear map $v : \mathbb{R}^m \to \mathbb{R}$, and when we write $v \partial f(x)$ we mean function composition $v \circ \partial f(x)$, where the types work out because $\partial f(x) : \mathbb{R}^n \to \mathbb{R}^m$. But in the common case we can identify $v$ with a vector in $\mathbb{R}^m$ and use the two almost interchangeably, just like we might sometimes flip between "column vectors" and "row vectors" without much comment.

With that identification, we can alternatively think of the linear part of a VJP as the transpose (or adjoint) of the linear part of a JVP:

$\qquad (x, v) \mapsto \partial f(x)^\mathsf{T} v$.

For a given point $x$, we can write the signature as

$\qquad \partial f(x)^\mathsf{T} : \mathbb{R}^m \to \mathbb{R}^n$.

The corresponding map on cotangent spaces is often called the [pullback](https://en.wikipedia.org/wiki/Pullback_(differential_geometry))
of $f$ at $x$. The key for our purposes is that it goes from something that looks like the output of $f$ to something that looks like the input of $f$, just like we might expect from a transposed linear function.

#### VJPs in JAX code

Switching from math back to Python, the JAX function `vjp` can take a Python function for evaluating $f$ and give us back a Python function for evaluating the VJP $(x, v) \mapsto (f(x), v^\mathsf{T} \partial f(x))$.

```{code-cell}
from jax import vjp

# Isolate the function from the weight matrix to the predictions
f = lambda W: predict(W, b, inputs)

y, vjp_fun = vjp(f, W)

key, subkey = random.split(key)
u = random.normal(subkey, y.shape)

# Pull back the covector `u` along `f` evaluated at `W`
v = vjp_fun(u)
```

In terms of [Haskell-like type signatures](https://wiki.haskell.org/Type_signature),
we could write

```haskell
vjp :: (a -> b) -> a -> (b, CT b -> CT a)
```

where we use `CT a` to denote the type for the cotangent space for `a`. In words, `vjp` takes as arguments a function of type `a -> b` and a point of type `a`, and gives back a pair consisting of a value of type `b` and a linear map of type `CT b -> CT a`.

This is great because it lets us build Jacobian matrices one row at a time, and the FLOP cost for evaluating $(x, v) \mapsto (f(x), v^\mathsf{T} \partial f(x))$ is only about three times the cost of evaluating $f$. In particular, if we want the gradient of a function $f : \mathbb{R}^n \to \mathbb{R}$, we can do it in just one call. That's how `grad` is efficient for gradient-based optimization, even for objectives like neural network training loss functions on millions or billions of parameters.

There's a cost, though: while the FLOPs are friendly, memory scales with the depth of the computation. Also, the implementation is traditionally more complex than that of forward-mode, though JAX has a trick up its sleeve: as we'll see next, it builds reverse-mode out of forward-mode.

For more on how reverse-mode works, see [this tutorial video from the Deep Learning Summer School in 2017](http://videolectures.net/deeplearning2017_johnson_automatic_differentiation/).

+++

### `jax.linearize`: one forward pass, many JVPs

Every call to `jvp(f, (x,), (v,))` redoes the forward pass: it evaluates $f(x)$ alongside $\partial f(x) v$. If you want Jacobian-vector products against many different vectors $v$ at the *same* point $x$, that's wasteful: the work that depends only on $x$ could be done once. That's what `jax.linearize` provides:

```{code-cell}
from jax import linearize

f = lambda W: predict(W, b, inputs)

y, f_jvp = linearize(f, W)
print(y)
```

In terms of Haskell-like type signatures,

```haskell
linearize :: (a -> b) -> a -> (b, T a -> T b)
```

Like `vjp`, `linearize` returns the primal output paired with a function; but here the function is the *pushforward* $v \mapsto \partial f(x) v$, with the forward pass already done and the intermediates it needs saved. Each call to `f_jvp` costs roughly the linear part of the work alone:

```{code-cell}
key, k1, k2 = random.split(key, 3)
v1, v2 = random.normal(k1, W.shape), random.normal(k2, W.shape)
print(f_jvp(v1))
print(f_jvp(v2))
```

Besides the efficiency, `linearize` gives us a way to think about reverse-mode. The function `f_jvp` is guaranteed to be *linear*, and a linear function can be transposed (JAX exposes that operation as `jax.linear_transpose`). Reverse-mode is the composition of the two: linearize the function at $x$, then transpose the resulting linear map:

```{code-cell}
from jax import linear_transpose, vjp

f_vjp = linear_transpose(f_jvp, W)      # transpose of the linearization...

y2, f_vjp_ref = vjp(f, W)               # ...is exactly what vjp computes

u = jnp.ones_like(y)
print(jnp.allclose(f_vjp(u)[0], f_vjp_ref(u)[0]))
```

This "linearize, then transpose" decomposition is how JAX implements `vjp` and `grad` internally, and the rest of this documentation section uses its vocabulary: when we say one operation "transposes to" another (as in {doc}`sharding-ad` and {doc}`custom-derivatives`), we mean this transposition of the linearized computation.

+++

### Vector-valued gradients with VJPs

If you're interested in taking vector-valued gradients (like `tf.gradients`):

```{code-cell}
from jax import vjp

def vgrad(f, x):
  y, vjp_fn = vjp(f, x)
  return vjp_fn(jnp.ones(y.shape))[0]

print(vgrad(lambda x: 3*x**2, jnp.ones((2, 2))))
```

### Hessian-vector products using both forward- and reverse-mode

+++

A Hessian-vector product function can be useful in a [truncated Newton Conjugate-Gradient algorithm](https://en.wikipedia.org/wiki/Truncated_Newton_method) for minimizing smooth convex functions, or for studying the curvature of neural network training objectives (e.g. [1](https://arxiv.org/abs/1406.2572), [2](https://arxiv.org/abs/1811.07062), [3](https://arxiv.org/abs/1706.04454), [4](https://arxiv.org/abs/1802.03451)). The trick is to evaluate products $v \mapsto \partial^2 f(x) \cdot v$ *without instantiating the Hessian matrix*: for a scalar loss on millions or billions of parameters, the full $n \times n$ Hessian is impossible to store, but a Hessian-vector product costs only a small constant multiple of evaluating $f$.

The most efficient recipe uses forward-over-reverse composition.

Mathematically, given a function $f : \mathbb{R}^n \to \mathbb{R}$ to differentiate, a point $x \in \mathbb{R}^n$ at which to linearize the function, and a vector $v \in \mathbb{R}^n$, the Hessian-vector product function we want is

$(x, v) \mapsto \partial^2 f(x) v$

Consider the helper function $g : \mathbb{R}^n \to \mathbb{R}^n$ defined to be the derivative (or gradient) of $f$, namely $g(x) = \partial f(x)$. All we need is its JVP, since that will give us

$(x, v) \mapsto \partial g(x) v = \partial^2 f(x) v$.

We can translate that almost directly into code:

```{code-cell}
from jax import jvp, grad

# forward-over-reverse
def hvp(f, primals, tangents):
  return jvp(grad(f), primals, tangents)[1]
```

As a bonus, since we didn't have to call `jnp.dot` directly, this `hvp` function works with arrays of any shape and with arbitrary container types (like vectors stored as nested lists/dicts/tuples), and doesn't even have a dependence on `jax.numpy`.

Here's an example of how to use it:

```{code-cell}
def f(X):
  return jnp.sum(jnp.tanh(X)**2)

key, subkey1, subkey2 = random.split(key, 3)
X = random.normal(subkey1, (30, 40))
V = random.normal(subkey2, (30, 40))

ans1 = hvp(f, (X,), (V,))
ans2 = jnp.tensordot(hessian(f)(X), V, 2)

print(jnp.allclose(ans1, ans2, 1e-4, 1e-4))
```

Another way you might consider writing this is using reverse-over-forward:

```{code-cell}
# reverse-over-forward
def hvp_revfwd(f, primals, tangents):
  g = lambda primals: jvp(f, primals, tangents)[1]
  return grad(g)(primals)
```

That's not quite as good, though, because forward-mode has less overhead than reverse-mode, and since the outer differentiation operator here has to differentiate a larger computation than the inner one, keeping forward-mode on the outside works best. For comparison, here's reverse-over-reverse too, with all three timed against materializing the full Hessian:

```{code-cell}
# reverse-over-reverse, only works for single arguments
def hvp_revrev(f, primals, tangents):
  x, = primals
  v, = tangents
  return grad(lambda x: jnp.vdot(grad(f)(x), v))(x)


print("Forward over reverse")
%timeit -n10 -r3 hvp(f, (X,), (V,))
print("Reverse over forward")
%timeit -n10 -r3 hvp_revfwd(f, (X,), (V,))
print("Reverse over reverse")
%timeit -n10 -r3 hvp_revrev(f, (X,), (V,))

print("Naive full Hessian materialization")
%timeit -n10 -r3 jnp.tensordot(hessian(f)(X), V, 2)
```

## Composing VJPs, JVPs, and `vmap`

+++

### Jacobian-Matrix and Matrix-Jacobian products

Now that we have `jvp` and `vjp` transformations that give us functions to push forward or pull back one vector at a time, we can use JAX's `vmap` transformation ({ref}`jax-101-transformations`) to push and pull entire bases at once. In particular, we can use that to write fast matrix-Jacobian and Jacobian-matrix products.

```{code-cell}
# Isolate the function from the weight matrix to the predictions
f = lambda W: predict(W, b, inputs)

# Pull back the covectors `m_i` along `f`, evaluated at `W`, for all `i`.
# First, use a list comprehension to loop over rows in the matrix M.
def loop_mjp(f, x, M):
    y, vjp_fun = vjp(f, x)
    return jnp.vstack([jnp.asarray(vjp_fun(mi)) for mi in M])

# Now, use vmap to build a computation that does a single fast matrix-matrix
# multiply, rather than an outer loop over vector-matrix multiplies.
def vmap_mjp(f, x, M):
    y, vjp_fun = vjp(f, x)
    outs, = vmap(vjp_fun)(M)
    return outs

key = random.key(0)
num_covecs = 128
U = random.normal(key, (num_covecs,) + y.shape)

loop_vs = loop_mjp(f, W, M=U)
print('Non-vmapped Matrix-Jacobian product')
%timeit -n10 -r3 loop_mjp(f, W, M=U)

print('\nVmapped Matrix-Jacobian product')
vmap_vs = vmap_mjp(f, W, M=U)
%timeit -n10 -r3 vmap_mjp(f, W, M=U)

assert jnp.allclose(loop_vs, vmap_vs), 'Vmap and non-vmapped Matrix-Jacobian Products should be identical'
```

```{code-cell}
def loop_jmp(f, W, M):
    # jvp immediately returns the primal and tangent values as a tuple,
    # so we'll compute and select the tangents in a list comprehension
    return jnp.vstack([jvp(f, (W,), (mi,))[1] for mi in M])

def vmap_jmp(f, W, M):
    _jvp = lambda s: jvp(f, (W,), (s,))[1]
    return vmap(_jvp)(M)

num_vecs = 128
S = random.normal(key, (num_vecs,) + W.shape)

loop_vs = loop_jmp(f, W, M=S)
print('Non-vmapped Jacobian-Matrix product')
%timeit -n10 -r3 loop_jmp(f, W, M=S)
vmap_vs = vmap_jmp(f, W, M=S)
print('\nVmapped Jacobian-Matrix product')
%timeit -n10 -r3 vmap_jmp(f, W, M=S)

assert jnp.allclose(loop_vs, vmap_vs), 'Vmap and non-vmapped Jacobian-Matrix products should be identical'
```

### The implementation of `jacfwd` and `jacrev`

+++

Now that we've seen fast Jacobian-matrix and matrix-Jacobian products, it's not hard to guess how to write `jacfwd` and `jacrev`. We just use the same technique to push forward or pull back an entire standard basis (isomorphic to an identity matrix) at once.

```{code-cell}
from jax import jacrev as builtin_jacrev

def our_jacrev(f):
    def jacfun(x):
        y, vjp_fun = vjp(f, x)
        # Use vmap to do a matrix-Jacobian product.
        # Here, the matrix is the Euclidean basis, so we get all
        # entries in the Jacobian at once.
        J, = vmap(vjp_fun, in_axes=0)(jnp.eye(len(y)))
        return J
    return jacfun

assert jnp.allclose(builtin_jacrev(f)(W), our_jacrev(f)(W)), 'Incorrect reverse-mode Jacobian results!'
```

```{code-cell}
from jax import jacfwd as builtin_jacfwd

def our_jacfwd(f):
    def jacfun(x):
        _jvp = lambda s: jvp(f, (x,), (s,))[1]
        Jt = vmap(_jvp, in_axes=1)(jnp.eye(len(x)))
        return jnp.transpose(Jt)
    return jacfun

assert jnp.allclose(builtin_jacfwd(f)(W), our_jacfwd(f)(W)), 'Incorrect forward-mode Jacobian results!'
```

Interestingly, [Autograd](https://github.com/hips/autograd) couldn't do this. Our [implementation](https://github.com/HIPS/autograd/blob/96a03f44da43cd7044c61ac945c483955deba957/autograd/differential_operators.py#L60) of reverse-mode `jacobian` in Autograd had to pull back one vector at a time with an outer-loop `map`. Pushing one vector at a time through the computation is much less efficient than batching it all together with `vmap`.

+++

Another thing that Autograd couldn't do is `jit`. No matter how much Python dynamism you use in the function being differentiated, we can always use `jit` on the linear part of the computation. For example:

```{code-cell}
def f(x):
    try:
        if x < 3:
            return 2 * x ** 3
        else:
            raise ValueError
    except ValueError:
        return jnp.pi * x

y, f_vjp = vjp(f, 4.)
print(jit(f_vjp)(1.))
```

In fact, the callable returned by `jax.vjp` is a first-class value in its
own right, a pytree whose leaves are the saved residuals. You can use it to
split the forward and backward passes into separately compiled functions,
schedule them yourself, and control what gets saved. That's the subject of
{doc}`vjp-objects`.

(jax-301-complex)=
## Complex numbers and differentiation

In JAX, differentiation of complex-valued functions is defined in terms of
the underlying real derivatives. Jacobian-vector products (JVPs) and
vector-Jacobian products (VJPs) operate on real-linear maps without requiring
holomorphy. The only convention choice is how covectors are represented as
complex numbers: JAX identifies them with vectors via a bilinear pairing
rather than a sesquilinear one, which shows up in `grad` as a complex
conjugate relative to the gradient vector.

### The unambiguous part: JVPs and VJPs

Under the identification $\mathbb{C} \cong \mathbb{R}^2$, any function $f :
\mathbb{C} \to \mathbb{C}$ corresponds to a real function $F : \mathbb{R}^2
\to \mathbb{R}^2$ defined by

$\qquad f(x + y i) = u(x, y) + v(x, y) i
\quad\leftrightarrow\quad F(x, y) = (u(x, y), v(x, y)).$

The derivative of $F$ at a point is the real $2 \times 2$ Jacobian matrix

$\qquad J = \begin{bmatrix} \partial_0 u & \partial_1 u \\ \partial_0 v & \partial_1 v \end{bmatrix}.$

The **JVP** of $f$ is the pushforward defined by this real linear map applied
to tangent vectors, with complex numbers serving as representations for pairs
of reals: for a tangent $t = t_1 + t_2 i$, the output tangent is the complex
representation of $J (t_1, t_2)$. This definition applies to all
differentiable functions regardless of holomorphy:

```{code-cell}
def u(x, y): return x**2 + jnp.sin(y)
def v(x, y): return x * y

def fun(z):  # not holomorphic!
  x, y = jnp.real(z), jnp.imag(z)
  return u(x, y) + v(x, y) * 1j

z = 1.5 + 0.5j
x, y = jnp.real(z), jnp.imag(z)
J = jnp.array([[grad(u, 0)(x, y), grad(u, 1)(x, y)],
               [grad(v, 0)(x, y), grad(v, 1)(x, y)]])

t = 0.7 - 0.3j
_, t_out = jvp(fun, (z,), (t,))
t_pair = J @ jnp.array([jnp.real(t), jnp.imag(t)])
print(jnp.allclose(t_out, t_pair[0] + t_pair[1] * 1j))
```

When $f$ is holomorphic, the Cauchy–Riemann equations imply that $J$ is a
scaled rotation corresponding to complex multiplication by $f'(z)$, and the
JVP reduces to $t \mapsto f'(z)\, t$.

The **VJP** is the pullback (dual map) of the derivative. The derivative's
dual map sends an output linear functional $\varphi$ to an input linear
functional by composition, $\varphi \mapsto \varphi \circ \partial F(x)$.
Because $f$ is in general only $\mathbb{R}$-differentiable, its tangent and
cotangent spaces are vector spaces over $\mathbb{R}$, and cotangents are
$\mathbb{R}$-linear functionals into $\mathbb{R}$, not complex numbers.

JAX's `vjp` nonetheless returns cotangents with the same type as the primal
values, so covectors must be *represented* as complex numbers, and that
requires identifying each functional with a number. A nondegenerate
real-valued pairing $\langle \cdot, \cdot \rangle$ is exactly such an
identification: it matches the number $w$ with the functional
$\langle w, \cdot \rangle$. Relative to chosen
pairings on the domain and codomain, the transpose $A^\mathsf{T}$ of an
$\mathbb{R}$-linear map $A$ is characterized by

$\qquad \langle w, A t \rangle = \langle A^\mathsf{T} w, t \rangle
\quad \text{for all } t, w.$

On $\mathbb{C} \cong \mathbb{R}^2$, there are two standard choices of
real-valued pairing:

* the **bilinear** pairing $\langle w, t \rangle = \operatorname{Re}(w t) =
  w_1 t_1 - w_2 t_2$;
* the **sesquilinear** pairing $\langle w, t \rangle =
  \operatorname{Re}(\bar{w} t) = w_1 t_1 + w_2 t_2$, which is the standard
  Euclidean inner product on $\mathbb{R}^2$.

These pairings differ by a conjugation in the first argument, so the
transposes they induce differ by an elementwise conjugation. **JAX adopts the
bilinear pairing**.

To unpack the functional from the number: a cotangent $w = w_1 + w_2 i$
returned by `vjp` encodes the $\mathbb{R}$-linear functional
$t \mapsto \operatorname{Re}(w t)$, which acts on the real pair
$(t_1, t_2)$ as $w_1 t_1 - w_2 t_2$. Its components against the standard
dual basis are therefore $(w_1, -w_2)$, the components of $\bar{w}$. This
is the conjugation that resurfaces in `grad` below.

Under this convention, `vjp` is characterized by

$\qquad \operatorname{Re}(w \cdot \texttt{jvp}(t)) \; = \;
\operatorname{Re}(\texttt{vjp}(w) \cdot t)
\qquad \text{for all } t, w,$

using standard complex products without explicit conjugation:

```{code-cell}
w = -0.2 + 1.1j

_, fun_vjp = vjp(fun, z)
w_out, = fun_vjp(w)

print(jnp.allclose(jnp.real(w * t_out), jnp.real(w_out * t)))       # True
print(jnp.allclose(jnp.real(jnp.conj(w) * t_out),
                   jnp.real(jnp.conj(w_out) * t)))                  # False!
```

Furthermore, if $f$ is holomorphic, both the JVP and the VJP simplify to
complex multiplication:

$\qquad \texttt{jvp}(t) = f'(z)\,t, \qquad \texttt{vjp}(w) = f'(z)\,w.$

```{code-cell}
_, t_out = jvp(jnp.sin, (z,), (t,))
_, sin_vjp = vjp(jnp.sin, z)
w_out, = sin_vjp(w)
print(jnp.allclose(t_out, jnp.cos(z) * t))
print(jnp.allclose(w_out, jnp.cos(z) * w))
```

Under the sesquilinear pairing, the VJP of a holomorphic function would
evaluate to $w \mapsto \overline{f'(z)}\, w$.

A general $\mathbb{R}$-differentiable map $\mathbb{C} \to \mathbb{C}$ has
four real degrees of freedom in its derivative. A single JVP or VJP computes
a two-dimensional projection; two evaluations on linearly independent
tangents (such as $1$ and $i$) recover the full Jacobian. For functions with
a real domain or codomain, a single evaluation suffices: one `jvp` determines
the derivative of an $\mathbb{R} \to \mathbb{C}$ function, and one `vjp` (or
`grad`) determines the derivative of a $\mathbb{C} \to \mathbb{R}$ function.
When in doubt about what a complex derivative means, use `jvp` and `vjp`
directly: they are always well-defined, for any function.

### `grad` at complex inputs

For a scalar function $f : \mathbb{C} \to \mathbb{R}$ with $f(x + yi) = u(x,
y)$, JAX defines `grad(f)(z)` as `vjp(f, z)[1](1.0)[0]`. Applying the bilinear
transpose formula gives

$\qquad \texttt{grad}(f)(z) = \partial_0 u(x, y) - \partial_1 u(x, y)\, i.$

This is the complex conjugate of the gradient vector $(\partial_0 u,
\partial_1 u)$ in $\mathbb{R}^2$. Under the bilinear convention, directional
derivatives are given directly by the real part of the product:

$\qquad \lim_{\epsilon \to 0} \tfrac{1}{\epsilon}(f(z + \epsilon t) - f(z))
= \operatorname{Re}(\texttt{grad}(f)(z) \cdot t).$

Consequently, the direction of steepest ascent in the complex plane is
$\overline{\texttt{grad}(f)(z)}$, and gradient descent updates take the form
$z \leftarrow z - \eta\, \overline{\texttt{grad}(f)(z)}$. Updating along the
unconjugated gradient negates the imaginary component of the descent step:

```{code-cell}
def f(z):
  x, y = jnp.real(z), jnp.imag(z)
  return x**2 + y**2      # |z|^2, minimized at z = 0

print(grad(f)(3. + 4j))   # 6 - 8j: conjugate of the steepest-ascent 6 + 8j
```

```{code-cell}
z = 3. + 4j
for _ in range(100):
  z = z - 0.05 * jnp.conj(grad(f)(z))   # with the conjugate: descends
print(f(z))

z = 3. + 4j
for _ in range(100):
  z = z - 0.05 * grad(f)(z)             # without: the imaginary part grows!
print(f(z))
```

Under JAX's convention:

1. Optimization of a real-valued loss over complex parameters requires
  stepping along the conjugate of `grad`, $\overline{\texttt{grad}(f)(z)}$.
  Optimizer libraries written with real parameters in mind do not apply this
  conjugation, so audit any optimizer you use on complex parameters.
2. First-order Taylor approximations and directional derivatives use the
  unconjugated product: $f(z + t) \approx f(z) +
  \operatorname{Re}(\texttt{grad}(f)(z) \cdot t)$.

### Why the bilinear convention?

The two pairings differ by an involution, so they carry the same
information; choosing between them decides only where conjugations appear.
The choice is a tradeoff.

What the bilinear pairing provides:

1. **Holomorphic functions require no conjugation.** Both the JVP and the VJP are
   plain multiplication by $f'(z)$, and `grad(f, holomorphic=True)` returns
   $f'(z)$ itself rather than its conjugate.
2. **Plain-product identities.** The characterization identity above and
   the first-order Taylor expansion
   $f(z + t) \approx f(z) + \operatorname{Re}(\texttt{grad}(f)(z) \cdot t)$
   use ordinary complex products, with no conjugations.
3. **Derivative rules work unchanged from their real implementations.**
   Since $\operatorname{Re}((cw)t) = \operatorname{Re}(w(ct))$,
   multiplication by $c$ transposes to multiplication by $c$. So for a
   holomorphic primitive, the VJP is the regular complex derivative
   multiplied by the incoming cotangent $w$, the same expression as in the
   real case, and most simple math primitives don't need their derivative
   rules changed from their real implementations. (For example,
   the VJP rule for `sin` is $w \mapsto w \cos(z)$ in the real and complex
   cases alike.) This holds whether VJP rules are written directly, as in
   Autograd, or derived by transposing linearized computations, as in JAX;
   under the sesquilinear convention, every such rule would need a
   conjugation added.

What the sesquilinear pairing would provide instead:

1. **`grad` is steepest ascent directly.** The sesquilinear pairing is the
   Euclidean inner product on $\mathbb{R}^2$, so its identification of
   covectors with vectors is the Riesz representation: `grad` would return
   the gradient vector itself, and optimizers written for real parameters
   would step correctly with no conjugation. This is the convention PyTorch
   and TensorFlow adopt (see the Wirtinger discussion below).
2. **Familiar metric duality.** Transposes relative to it are the conjugate
   transposes of ordinary linear algebra, and
   $\langle w, w \rangle = |w|^2 \geq 0$.

In summary: the bilinear convention puts the conjugation into optimizer
steps, while the sesquilinear convention puts it into holomorphic
derivatives and into every transpose rule. The former is algebraic duality
(identifying $\mathbb{C}$ with its complex-linear dual), the latter metric
duality (Riesz representation).

### Relation to Wirtinger derivatives

Wirtinger derivatives provide an alternative coordinate representation of the
real Jacobian:

$\qquad \frac{\partial}{\partial z} = \tfrac{1}{2}\left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \qquad
\frac{\partial}{\partial \bar z} = \tfrac{1}{2}\left(\frac{\partial}{\partial
x} + i \frac{\partial}{\partial y}\right).$

The four real partial derivatives of a function $f : \mathbb{C} \to
\mathbb{C}$ are parameterized by the two complex quantities $\partial
f/\partial z$ and $\partial f/\partial \bar z$, and the JVP is expressed as

$\qquad \texttt{jvp}(t) = \frac{\partial f}{\partial z}\, t + \frac{\partial f}{\partial \bar z}\, \bar{t}.$

```{code-cell}
z = 1.5 + 0.5j                # the point where we computed J above

fx = J[0, 0] + J[1, 0] * 1j   # df/dx = du/dx + i dv/dx
fy = J[0, 1] + J[1, 1] * 1j   # df/dy = du/dy + i dv/dy
dfdz    = 0.5 * (fx - 1j * fy)
dfdzbar = 0.5 * (fx + 1j * fy)

_, t_out = jvp(fun, (z,), (t,))
print(jnp.allclose(t_out, dfdz * t + dfdzbar * jnp.conj(t)))
```

A function is holomorphic if and only if $\partial f/\partial \bar z = 0$
(the Cauchy–Riemann equations), in which case $\partial f/\partial z =
f'(z)$. For a real-valued function $f$, JAX's `grad` computes
$\texttt{grad}(f)(z) = 2\, \partial f/\partial z$, whereas the
steepest-ascent vector is $2\, \partial f/\partial \bar z =
\overline{\texttt{grad}(f)(z)}$. Other frameworks, including PyTorch and
TensorFlow, define the gradient of a real-valued loss $L$ as the
steepest-ascent vector $2\, \partial L/\partial \bar z$, incorporating the
conjugation into the returned derivative. Both approaches represent the same
underlying real derivative under different identification conventions.

### Holomorphic functions and `grad(f, holomorphic=True)`

For a function $f : \mathbb{C} \to \mathbb{C}$ with a complex output, `grad`
raises an error because the four real derivative components cannot be
uniquely represented by a single complex scalar. In general, `jax.vjp` or
`jax.jvp` should be used directly.

When $f$ is holomorphic, the derivative is completely characterized by the
single complex number $f'(z)$, as the Cauchy–Riemann equations restrict the
Jacobian to a scaled rotation. Setting `holomorphic=True` indicates that this
condition holds, causing `grad` to evaluate the VJP at cotangent $1.0$ and
return $f'(z)$:

```{code-cell}
print(grad(jnp.sin, holomorphic=True)(3. + 4j))
print(jnp.cos(3. + 4j))
```

The `holomorphic=True` parameter disables the error check for complex outputs
without verifying holomorphy. If applied to a non-holomorphic function, it
returns the derivative of the real component of the output, discarding the
imaginary component:

```{code-cell}
def f(z):
  return jnp.conjugate(z)   # not holomorphic!

grad(f, holomorphic=True)(3. + 4j)
```

Complex numbers are supported across JAX transformations and linear algebra
operations, including matrix factorizations. Here's a real-valued loss
computed through the Cholesky factorization of a complex Hermitian matrix.
Cholesky isn't holomorphic (it involves conjugation), so this is a job for
plain `grad`, and we can check the result against a finite difference using
the directional-derivative identity from above, summed over matrix entries:

```{code-cell}
A = jnp.array([[5.,    2.+3j,    5j],
              [2.-3j,   7.,  1.+7j],
              [-5j,  1.-7j,    12.]])

def f(X):
  L = jnp.linalg.cholesky(X)
  return jnp.sum(jnp.abs(L - jnp.sin(L))**2)   # real-valued

G = grad(f)(A)

T = jnp.array([[1., 1j, 0.], [-1j, 0., 2.], [0., 2., -1.]])  # a Hermitian direction
eps = 1e-3
print((f(A + eps * T) - f(A - eps * T)) / (2 * eps))  # finite difference
print(jnp.real(jnp.sum(G * T)))                        # Re(grad(f)(A) · T)
```

## More advanced autodiff

We worked through some easy, and then progressively more complicated,
applications of automatic differentiation in JAX. We hope you now feel that
taking derivatives in JAX is easy and powerful. The rest of this
documentation section goes deeper:

- {doc}`vjp-objects` — the VJP object as a pytree: splitting the forward and
  backward passes into separately compiled functions, run on your own
  schedule.
- {doc}`sharding-ad` — how autodiff interacts with sharding: the same
  cotangent-type reasoning as this page, extended to distributed arrays.
- {doc}`custom-derivatives` — defining your own derivative rules with hijax
  primitives (including efficient derivatives at fixed points), and
  {doc}`custom-jvp-vjp` for the classic decorator APIs.
- {doc}`refs` — how autodiff interacts with mutable arrays: plumbing values
  out of backward passes and accumulating gradients in place.
- {doc}`remat` — controlling what autodiff saves versus recomputes, to trade
  memory for FLOPs.
- {ref}`jax-301-hijax-types` — differentiating with respect to entirely new
  types.