427 lines
12 KiB
Plaintext
427 lines
12 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"%matplotlib inline"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"Autograd: 自动求导机制\n",
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"===================================\n",
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"\n",
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"PyTorch 中所有神经网络的核心是 ``autograd`` 包。\n",
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"我们先简单介绍一下这个包，然后训练第一个简单的神经网络。\n",
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"\n",
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"``autograd``包为张量上的所有操作提供了自动求导。\n",
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"它是一个在运行时定义的框架，这意味着反向传播是根据你的代码来确定如何运行，并且每次迭代可以是不同的。\n",
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"\n",
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"\n",
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"示例\n",
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"\n",
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"张量（Tensor）\n",
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"--------\n",
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"\n",
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"``torch.Tensor``是这个包的核心类。如果设置\n",
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"``.requires_grad`` 为 ``True``，那么将会追踪所有对于该张量的操作。 \n",
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"当完成计算后通过调用 ``.backward()``，自动计算所有的梯度，\n",
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"这个张量的所有梯度将会自动积累到 ``.grad`` 属性。\n",
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"\n",
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"要阻止张量跟踪历史记录，可以调用``.detach()``方法将其与计算历史记录分离，并禁止跟踪它将来的计算记录。\n",
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"\n",
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"为了防止跟踪历史记录（和使用内存），可以将代码块包装在``with torch.no_grad()：``中。\n",
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"在评估模型时特别有用，因为模型可能具有`requires_grad = True`的可训练参数，但是我们不需要梯度计算。\n",
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"\n",
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"在自动梯度计算中还有另外一个重要的类``Function``.\n",
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"\n",
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"``Tensor`` and ``Function`` are interconnected and build up an acyclic\n",
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"graph, that encodes a complete history of computation. Each tensor has\n",
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"a ``.grad_fn`` attribute that references a ``Function`` that has created\n",
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"the ``Tensor`` (except for Tensors created by the user - their\n",
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"``grad_fn is None``).\n",
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"\n",
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"``Tensor`` 和 ``Function``互相连接并生成一个非循环图，它表示和存储了完整的计算历史。\n",
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"每个张量都有一个``.grad_fn``属性，这个属性引用了一个创建了``Tensor``的``Function``（除非这个张量是用户手动创建的，即，这个张量的\n",
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"``grad_fn`` 是 ``None``）。\n",
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"\n",
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"如果需要计算导数，你可以在``Tensor``上调用``.backward()``。 \n",
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"如果``Tensor``是一个标量（即它包含一个元素数据）则不需要为``backward()``指定任何参数，\n",
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"但是如果它有更多的元素，你需要指定一个``gradient`` 参数来匹配张量的形状。\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"***译者注：在其他的文章中你可能会看到说将Tensor包裹到Variable中提供自动梯度计算，Variable 这个在0.41版中已经被标注为过期了，现在可以直接使用Tensor，官方文档在这里：***\n",
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"(https://pytorch.org/docs/stable/autograd.html#variable-deprecated) \n",
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"\n",
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"具体的后面会有详细说明"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"import torch"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"创建一个张量并设置 requires_grad=True 用来追踪他的计算历史\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([[1., 1.],\n",
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" [1., 1.]], requires_grad=True)\n"
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]
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}
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],
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"source": [
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"x = torch.ones(2, 2, requires_grad=True)\n",
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"print(x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"对张量进行操作:\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([[3., 3.],\n",
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" [3., 3.]], grad_fn=<AddBackward0>)\n"
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]
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}
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],
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"source": [
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"y = x + 2\n",
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"print(y)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"结果``y``已经被计算出来了，所以，``grad_fn``已经被自动生成了。\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"<AddBackward0 object at 0x000002004F7CC248>\n"
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]
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}
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],
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"source": [
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"print(y.grad_fn)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"对y进行一个操作\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([[27., 27.],\n",
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" [27., 27.]], grad_fn=<MulBackward0>) tensor(27., grad_fn=<MeanBackward0>)\n"
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]
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}
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],
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"source": [
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"z = y * y * 3\n",
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"out = z.mean()\n",
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"\n",
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"print(z, out)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"``.requires_grad_( ... )`` 可以改变现有张量的 ``requires_grad``属性。\n",
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"如果没有指定的话，默认输入的flag是 ``False``。\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"False\n",
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"True\n",
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"<SumBackward0 object at 0x000002004F7D5608>\n"
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]
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}
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],
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"source": [
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"a = torch.randn(2, 2)\n",
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"a = ((a * 3) / (a - 1))\n",
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"print(a.requires_grad)\n",
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"a.requires_grad_(True)\n",
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"print(a.requires_grad)\n",
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"b = (a * a).sum()\n",
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"print(b.grad_fn)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"梯度\n",
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"---------\n",
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"反向传播\n",
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"因为 ``out``是一个纯量（scalar），``out.backward()`` 等于``out.backward(torch.tensor(1))``。\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"out.backward()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"print gradients d(out)/dx\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([[4.5000, 4.5000],\n",
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" [4.5000, 4.5000]])\n"
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]
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}
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],
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"source": [
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"print(x.grad)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"得到矩阵 ``4.5``.将 ``out``叫做\n",
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"*Tensor* “$o$”.\n",
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"\n",
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"得到 $o = \\frac{1}{4}\\sum_i z_i$,\n",
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"$z_i = 3(x_i+2)^2$ 和 $z_i\\bigr\\rvert_{x_i=1} = 27$.\n",
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"\n",
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"因此,\n",
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"$\\frac{\\partial o}{\\partial x_i} = \\frac{3}{2}(x_i+2)$, 则\n",
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"$\\frac{\\partial o}{\\partial x_i}\\bigr\\rvert_{x_i=1} = \\frac{9}{2} = 4.5$.\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"在数学上，如果我们有向量值函数 $\\vec{y} = f(\\vec{x})$ ，且 $\\vec{y}$ 关于 $\\vec{x}$ 的梯度是一个雅可比矩阵(Jacobian matrix)：\n",
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"\n",
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"$J = \\begin{pmatrix} \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{1}}{\\partial x_{n}} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\frac{\\partial y_{m}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}} \\end{pmatrix}$\n",
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"\n",
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"一般来说，`torch.autograd`就是用来计算vector-Jacobian product的工具。也就是说，给定任一向量 $v=(v_{1}\\;v_{2}\\;\\cdots\\;v_{m})^{T}$ ，计算 $v^{T}\\cdot J$ ，如果 $v$ 恰好是标量函数 $l=g(\\vec{y})$ 的梯度，也就是说 $v=(\\frac{\\partial l}{\\partial y_{1}}\\;\\cdots\\;\\frac{\\partial l}{\\partial y_{m}})^{T}$，那么根据链式法则，vector-Jacobian product 是 $l$ 关于 $\\vec{x}$ 的梯度：\n",
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"\n",
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"$J^{T}\\cdot v = \\begin{pmatrix} \\frac{\\partial y_{1}}{\\partial x_{1}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{1}} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\frac{\\partial y_{1}}{\\partial x_{n}} & \\cdots & \\frac{\\partial y_{m}}{\\partial x_{n}} \\end{pmatrix} \\begin{pmatrix} \\frac{\\partial l}{\\partial y_{1}}\\\\ \\vdots \\\\ \\frac{\\partial l}{\\partial y_{m}} \\end{pmatrix} = \\begin{pmatrix} \\frac{\\partial l}{\\partial x_{1}}\\\\ \\vdots \\\\ \\frac{\\partial l}{\\partial x_{n}} \\end{pmatrix}$\n",
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"\n",
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"（注意，$v^{T}\\cdot J$ 给出了一个行向量，可以通过 $J^{T}\\cdot v$ 将其视为列向量）\n",
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"\n",
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"vector-Jacobian product 这种特性使得将外部梯度返回到具有非标量输出的模型变得非常方便。\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"现在让我们来看一个vector-Jacobian product的例子\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([ 293.4463, 50.6356, 1031.2501], grad_fn=<MulBackward0>)\n"
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]
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}
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],
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"source": [
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"x = torch.randn(3, requires_grad=True)\n",
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"\n",
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"y = x * 2\n",
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"while y.data.norm() < 1000:\n",
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" y = y * 2\n",
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"\n",
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"print(y)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"在这个情形中，`y`不再是个标量。`torch.autograd`无法直接计算出完整的雅可比行列，但是如果我们只想要vector-Jacobian product，只需将向量作为参数传入`backward`："
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tensor([5.1200e+01, 5.1200e+02, 5.1200e-02])\n"
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]
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}
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],
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"source": [
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"gradients = torch.tensor([0.1, 1.0, 0.0001], dtype=torch.float)\n",
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"y.backward(gradients)\n",
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"\n",
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"print(x.grad)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"如果``.requires_grad=True``但是你又不希望进行autograd的计算，\n",
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"那么可以将变量包裹在 ``with torch.no_grad()``中:\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"True\n",
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"True\n",
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"False\n"
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]
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}
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],
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"source": [
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"print(x.requires_grad)\n",
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"print((x ** 2).requires_grad)\n",
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"\n",
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"with torch.no_grad():\n",
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" print((x ** 2).requires_grad)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**稍后阅读:**\n",
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"\n",
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" ``autograd`` 和 ``Function`` 的官方文档 https://pytorch.org/docs/autograd\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Pytorch for Deeplearning",
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"language": "python",
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"name": "pytorch"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 1
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}
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