Note

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# Forward-mode Automatic Differentiation (Beta)¶

This tutorial demonstrates how to use forward-mode AD to compute directional derivatives (or equivalently, Jacobian-vector products).

The tutorial below uses some APIs only available in versions >= 1.11 (or nightly builds).

Also note that forward-mode AD is currently in beta. The API is subject to change and operator coverage is still incomplete.

## Basic Usage¶

Unlike reverse-mode AD, forward-mode AD computes gradients eagerly
alongside the forward pass. We can use forward-mode AD to compute a
directional derivative by performing the forward pass as before,
except we first associate our input with another tensor representing
the direction of the directional derivative (or equivalently, the `v`

in a Jacobian-vector product). When an input, which we call “primal”, is
associated with a “direction” tensor, which we call “tangent”, the
resultant new tensor object is called a “dual tensor” for its connection
to dual numbers[0].

As the forward pass is performed, if any input tensors are dual tensors, extra computation is performed to propogate this “sensitivity” of the function.

```
import torch
import torch.autograd.forward_ad as fwAD
primal = torch.randn(10, 10)
tangent = torch.randn(10, 10)
def fn(x, y):
return x ** 2 + y ** 2
# All forward AD computation must be performed in the context of
# a ``dual_level`` context. All dual tensors created in such a context
# will have their tangents destroyed upon exit. This is to ensure that
# if the output or intermediate results of this computation are reused
# in a future forward AD computation, their tangents (which are associated
# with this computation) won't be confused with tangents from the later
# computation.
with fwAD.dual_level():
# To create a dual tensor we associate a tensor, which we call the
# primal with another tensor of the same size, which we call the tangent.
# If the layout of the tangent is different from that of the primal,
# The values of the tangent are copied into a new tensor with the same
# metadata as the primal. Otherwise, the tangent itself is used as-is.
#
# It is also important to note that the dual tensor created by
# ``make_dual`` is a view of the primal.
dual_input = fwAD.make_dual(primal, tangent)
assert fwAD.unpack_dual(dual_input).tangent is tangent
# To demonstrate the case where the copy of the tangent happens,
# we pass in a tangent with a layout different from that of the primal
dual_input_alt = fwAD.make_dual(primal, tangent.T)
assert fwAD.unpack_dual(dual_input_alt).tangent is not tangent
# Tensors that do not not have an associated tangent are automatically
# considered to have a zero-filled tangent of the same shape.
plain_tensor = torch.randn(10, 10)
dual_output = fn(dual_input, plain_tensor)
# Unpacking the dual returns a namedtuple with ``primal`` and ``tangent``
# as attributes
jvp = fwAD.unpack_dual(dual_output).tangent
assert fwAD.unpack_dual(dual_output).tangent is None
```

## Usage with Modules¶

To use `nn.Module`

with forward AD, replace the parameters of your
model with dual tensors before performing the forward pass. At the
time of writing, it is not possible to create dual tensor
`nn.Parameter`s. As a workaround, one must register the dual tensor
as a non-parameter attribute of the module.

```
import torch.nn as nn
model = nn.Linear(5, 5)
input = torch.randn(16, 5)
params = {name: p for name, p in model.named_parameters()}
tangents = {name: torch.rand_like(p) for name, p in params.items()}
with fwAD.dual_level():
for name, p in params.items():
delattr(model, name)
setattr(model, name, fwAD.make_dual(p, tangents[name]))
out = model(input)
jvp = fwAD.unpack_dual(out).tangent
```

## Using Modules stateless API (experimental)¶

Another way to use `nn.Module`

with forward AD is to utilize
the stateless API. NB: At the time of writing the stateless API is still
experimental and may be subject to change.

```
from torch.nn.utils._stateless import functional_call
# We need a fresh module because the functional call requires the
# the model to have parameters registered.
model = nn.Linear(5, 5)
dual_params = {}
with fwAD.dual_level():
for name, p in params.items():
# Using the same ``tangents`` from the above section
dual_params[name] = fwAD.make_dual(p, tangents[name])
out = functional_call(model, dual_params, input)
jvp2 = fwAD.unpack_dual(out).tangent
# Check our results
assert torch.allclose(jvp, jvp2)
```

## Custom autograd Function¶

Custom Functions also support forward-mode AD. To create custom Function
supporting forward-mode AD, register the `jvp()`

static method. It is
possible, but not mandatory for custom Functions to support both forward
and backward AD. See the
documentation
for more information.

```
class Fn(torch.autograd.Function):
@staticmethod
def forward(ctx, foo):
result = torch.exp(foo)
# Tensors stored in ctx can be used in the subsequent forward grad
# computation.
ctx.result = result
return result
@staticmethod
def jvp(ctx, gI):
gO = gI * ctx.result
# If the tensor stored in ctx will not also be used in the backward pass,
# one can manually free it using ``del``
del ctx.result
return gO
fn = Fn.apply
primal = torch.randn(10, 10, dtype=torch.double, requires_grad=True)
tangent = torch.randn(10, 10)
with fwAD.dual_level():
dual_input = fwAD.make_dual(primal, tangent)
dual_output = fn(dual_input)
jvp = fwAD.unpack_dual(dual_output).tangent
# It is important to use ``autograd.gradcheck`` to verify that your
# custom autograd Function computes the gradients correctly. By default,
# gradcheck only checks the backward-mode (reverse-mode) AD gradients. Specify
# ``check_forward_ad=True`` to also check forward grads. If you did not
# implement the backward formula for your function, you can also tell gradcheck
# to skip the tests that require backward-mode AD by specifying
# ``check_backward_ad=False``, ``check_undefined_grad=False``, and
# ``check_batched_grad=False``.
torch.autograd.gradcheck(Fn.apply, (primal,), check_forward_ad=True,
check_backward_ad=False, check_undefined_grad=False,
check_batched_grad=False)
```

```
True
```

## Functional API (beta)¶

We also offer a higher-level functional API in functorch for computing Jacobian-vector products that you may find simpler to use depending on your use case.

The benefit of the functional API is that there isn’t a need to understand or use the lower-level dual tensor API and that you can compose it with other functorch transforms (like vmap); the downside is that it offers you less control.

Note that the remainder of this tutorial will require functorch (https://github.com/pytorch/functorch) to run. Please find installation instructions at the specified link.

```
import functorch as ft
primal0 = torch.randn(10, 10)
tangent0 = torch.randn(10, 10)
primal1 = torch.randn(10, 10)
tangent1 = torch.randn(10, 10)
def fn(x, y):
return x ** 2 + y ** 2
# Here is a basic example to compute the JVP of the above function.
# The jvp(func, primals, tangents) returns func(*primals) as well as the
# computed jvp. Each primal must be associated with a tangent of the same shape.
primal_out, tangent_out = ft.jvp(fn, (primal0, primal1), (tangent0, tangent1))
# functorch.jvp requires every primal to be associated with a tangent.
# If we only want to associate certain inputs to `fn` with tangents,
# then we'll need to create a new function that captures inputs without tangents:
primal = torch.randn(10, 10)
tangent = torch.randn(10, 10)
y = torch.randn(10, 10)
import functools
new_fn = functools.partial(fn, y=y)
primal_out, tangent_out = ft.jvp(new_fn, (primal,), (tangent,))
```

## Using the functional API with Modules¶

To use `nn.Module`

with functorch.jvp to compute Jacobian-vector products
with respect to the model parameters, we need to reformulate the
`nn.Module`

as a function that accepts both the model parameters and inputs
to the module.

```
model = nn.Linear(5, 5)
input = torch.randn(16, 5)
tangents = tuple([torch.rand_like(p) for p in model.parameters()])
# Given a torch.nn.Module, ft.make_functional_with_buffers extracts the state
# (params and buffers) and returns a functional version of the model that
# can be invoked like a function.
# That is, the returned ``func`` can be invoked like
# ``func(params, buffers, input)``.
# ft.make_functional_with_buffers is analogous to the nn.Modules stateless API
# that you saw previously and we're working on consolidating the two.
func, params, buffers = ft.make_functional_with_buffers(model)
# Because jvp requires every input to be associated with a tangent, we need to
# create a new function that, when given the parameters, produces the output
def func_params_only(params):
return func(params, buffers, input)
model_output, jvp_out = ft.jvp(func_params_only, (params,), (tangents,))
```

[0] https://en.wikipedia.org/wiki/Dual_number

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