自动微分的正向模式 (Beta)

Created On: Dec 07, 2021 | Last Updated: Apr 18, 2023 | Last Verified: Nov 05, 2024

本教程演示如何使用正向模式 AD 计算方向导数（或同义的雅可比矢量积）。

下面的教程使用的部分 API 仅适用于版本 >= 1.11（或 nightly 版本）。

请注意，正向模式 AD 目前仍处于测试阶段。API 可能会发生变化，并且操作覆盖范围尚不完整。

基本用法

与反向模式 AD 不同，正向模式 AD 在前向传播过程中即刻计算梯度。我们可以通过执行前向传播并将输入与另一个表示方向导数方向（或雅可比矢量积中的 v）的张量相关联，使用正向模式 AD 计算方向导数。当输入（称为“主”或“primal”）与表示方向的张量（称为“切线”或“tangent”）相关联时，生成的新张量对象称为“对偶张量”，它与对偶数有关[0]。

在执行前向传播时，如果任何输入张量是对偶张量，则会进行额外计算以传播函数的这种“敏感性”。

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 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

与模块一起使用

要将正向 AD 与 nn.Module 一起使用，需要在前向传播之前将模型的参数替换为对偶张量。目前无法创建对偶张量的 nn.Parameter。作为一种解决方法，可以将对偶张量注册为模块的非参数属性。

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

使用函数式模块 API（测试阶段）

使用正向 AD 的另一个方法是利用函数式模块 API（也称为无状态模块 API）。

from torch.func 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)

自定义自动微分函数

自定义函数也支持正向模式 AD。要创建支持正向模式 AD 的自定义函数，需要注册 jvp() 静态方法。自定义函数可以同时支持正向和反向 AD，但这不是强制要求。有关更多信息，请参阅 文档。

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)

函数式 API（测试阶段）

我们还在 functorch 中提供了更高级别的函数式 API，您可能会发现根据使用场景，这种方法更简单。

函数式 API 的优点是无需了解或使用底层的对偶张量 API，并且可以将其与其他 functorch 转换（例如 vmap） 组合使用；其缺点是提供的控制较少。

请注意，本文余下部分需要 functorch 才能运行（https://github.com/pytorch/functorch）。安装说明请参阅指定链接。

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 Jacobian-vector product (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,))

结合模块使用函数式 API

要通过 functorch.jvp 使用 nn.Module 计算关于模型参数的雅可比矢量积，我们需要将 nn.Module 表述为一个接受模型参数和模块输入的函数。

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