# Whirlwind Tour¶

## What is functorch?¶

functorch is a library for JAX-like composable function transforms in PyTorch.

A “function transform” is a higher-order function that accepts a numerical function and returns a new function that computes a different quantity.

functorch has auto-differentiation transforms (

`grad(f)`

returns a function that computes the gradient of`f`

), a vectorization/batching transform (`vmap(f)`

returns a function that computes`f`

over batches of inputs), and others.These function transforms can compose with each other arbitrarily. For example, composing

`vmap(grad(f))`

computes a quantity called per-sample-gradients that stock PyTorch cannot efficiently compute today.

Furthermore, we also provide an experimental compilation transform in the `functorch.compile`

namespace. Our compilation transform, named AOT (ahead-of-time) Autograd, returns to you an FX graph (that optionally contains a backward pass), of which compilation via various backends is one path you can take.

## Why composable function transforms?¶

There are a number of use cases that are tricky to do in PyTorch today:

computing per-sample-gradients (or other per-sample quantities)

running ensembles of models on a single machine

efficiently batching together tasks in the inner-loop of MAML

efficiently computing Jacobians and Hessians

efficiently computing batched Jacobians and Hessians

Composing `vmap`

, `grad`

, `vjp`

, and `jvp`

transforms allows us to express the above without designing a separate subsystem for each.

## What are the transforms?¶

### grad (gradient computation)¶

`grad(func)`

is our gradient computation transform. It returns a new function that computes the gradients of `func`

. It assumes `func`

returns a single-element Tensor and by default it computes the gradients of the output of `func`

w.r.t. to the first input.

```
import torch
from functorch import grad
x = torch.randn([])
cos_x = grad(lambda x: torch.sin(x))(x)
assert torch.allclose(cos_x, x.cos())
# Second-order gradients
neg_sin_x = grad(grad(lambda x: torch.sin(x)))(x)
assert torch.allclose(neg_sin_x, -x.sin())
```

### vmap (auto-vectorization)¶

Note: vmap imposes restrictions on the code that it can be used on. For more details, please read its docstring.

`vmap(func)(*inputs)`

is a transform that adds a dimension to all Tensor operations in `func`

. `vmap(func)`

returns a new function that maps `func`

over some dimension (default: 0) of each Tensor in inputs.

vmap is useful for hiding batch dimensions: one can write a function func that runs on examples and then lift it to a function that can take batches of examples with `vmap(func)`

, leading to a simpler modeling experience:

```
import torch
from functorch import vmap
batch_size, feature_size = 3, 5
weights = torch.randn(feature_size, requires_grad=True)
def model(feature_vec):
# Very simple linear model with activation
assert feature_vec.dim() == 1
return feature_vec.dot(weights).relu()
examples = torch.randn(batch_size, feature_size)
result = vmap(model)(examples)
```

When composed with `grad`

, `vmap`

can be used to compute per-sample-gradients:

```
from functorch import vmap
batch_size, feature_size = 3, 5
def model(weights,feature_vec):
# Very simple linear model with activation
assert feature_vec.dim() == 1
return feature_vec.dot(weights).relu()
def compute_loss(weights, example, target):
y = model(weights, example)
return ((y - target) ** 2).mean() # MSELoss
weights = torch.randn(feature_size, requires_grad=True)
examples = torch.randn(batch_size, feature_size)
targets = torch.randn(batch_size)
inputs = (weights,examples, targets)
grad_weight_per_example = vmap(grad(compute_loss), in_dims=(None, 0, 0))(*inputs)
```

### vjp (vector-Jacobian product)¶

The `vjp`

transform applies `func`

to `inputs`

and returns a new function that computes the vector-Jacobian product (vjp) given some `cotangents`

Tensors.

```
from functorch import vjp
inputs = torch.randn(3)
func = torch.sin
cotangents = (torch.randn(3),)
outputs, vjp_fn = vjp(func, inputs); vjps = vjp_fn(*cotangents)
```

### jvp (Jacobian-vector product)¶

The `jvp`

transforms computes Jacobian-vector-products and is also known as “forward-mode AD”. It is not a higher-order function unlike most other transforms, but it returns the outputs of `func(inputs)`

as well as the jvps.

```
from functorch import jvp
x = torch.randn(5)
y = torch.randn(5)
f = lambda x, y: (x * y)
_, output = jvp(f, (x, y), (torch.ones(5), torch.ones(5)))
assert torch.allclose(output, x + y)
```

### jacrev, jacfwd, and hessian¶

The `jacrev`

transform returns a new function that takes in `x`

and returns the Jacobian of the function
with respect to `x`

using reverse-mode AD.

```
from functorch import jacrev
x = torch.randn(5)
jacobian = jacrev(torch.sin)(x)
expected = torch.diag(torch.cos(x))
assert torch.allclose(jacobian, expected)
```

Use `jacrev`

to compute the jacobian. This can be composed with `vmap`

to produce batched jacobians:

```
x = torch.randn(64, 5)
jacobian = vmap(jacrev(torch.sin))(x)
assert jacobian.shape == (64, 5, 5)
```

`jacfwd`

is a drop-in replacement for `jacrev`

that computes Jacobians using forward-mode AD:

```
from functorch import jacfwd
x = torch.randn(5)
jacobian = jacfwd(torch.sin)(x)
expected = torch.diag(torch.cos(x))
assert torch.allclose(jacobian, expected)
```

Composing `jacrev`

with itself or `jacfwd`

can produce hessians:

```
def f(x):
return x.sin().sum()
x = torch.randn(5)
hessian0 = jacrev(jacrev(f))(x)
hessian1 = jacfwd(jacrev(f))(x)
```

The `hessian`

is a convenience function that combines `jacfwd`

and `jacrev`

:

```
from functorch import hessian
def f(x):
return x.sin().sum()
x = torch.randn(5)
hess = hessian(f)(x)
```

## Conclusion¶

Check out our other tutorials (in the left bar) for more detailed explanations of how to apply functorch transforms for various use cases. `functorch`

is very much a work in progress and we’d love to hear how you’re using it – we encourage you to start a conversation at our issues tracker to discuss your use case.