# 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(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.

from functorch import grad
x = torch.randn([])
assert torch.allclose(cos_x, x.cos())

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

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

examples = torch.randn(batch_size, feature_size)
targets = torch.randn(batch_size)
inputs = (weights,examples, targets)


### 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.