1102 lines
40 KiB
Plaintext
1102 lines
40 KiB
Plaintext
{
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"source": [
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"# softmax回归的简洁实现\n",
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":label:`sec_softmax_concise`\n",
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"\n",
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"在 :numref:`sec_linear_concise`中，\n",
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"我们发现(**通过深度学习框架的高级API能够使实现**)\n",
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"(~~softmax~~)\n",
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"线性(**回归变得更加容易**)。\n",
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"同样，通过深度学习框架的高级API也能更方便地实现softmax回归模型。\n",
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"本节如在 :numref:`sec_softmax_scratch`中一样，\n",
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"继续使用Fashion-MNIST数据集，并保持批量大小为256。\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": 1,
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"id": "7f81001f",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T06:57:16.212083Z",
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch import nn\n",
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"from d2l import torch as d2l"
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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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"id": "92e395a8",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T06:57:18.231806Z",
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"origin_pos": 5,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"batch_size = 256\n",
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"train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fb2b9199",
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"metadata": {
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"origin_pos": 6
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},
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"source": [
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"## 初始化模型参数\n",
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"\n",
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"如我们在 :numref:`sec_softmax`所述，\n",
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"[**softmax回归的输出层是一个全连接层**]。\n",
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"因此，为了实现我们的模型，\n",
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"我们只需在`Sequential`中添加一个带有10个输出的全连接层。\n",
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"同样，在这里`Sequential`并不是必要的，\n",
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"但它是实现深度模型的基础。\n",
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"我们仍然以均值0和标准差0.01随机初始化权重。\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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"id": "ebf37311",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T06:57:18.342288Z",
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"# PyTorch不会隐式地调整输入的形状。因此，\n",
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"# 我们在线性层前定义了展平层（flatten），来调整网络输入的形状\n",
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"net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))\n",
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"\n",
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"def init_weights(m):\n",
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" if type(m) == nn.Linear:\n",
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" nn.init.normal_(m.weight, std=0.01)\n",
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"\n",
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"net.apply(init_weights);"
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]
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},
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{
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"cell_type": "markdown",
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"id": "30a6c3c5",
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"metadata": {
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"origin_pos": 11
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},
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"source": [
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"## 重新审视Softmax的实现\n",
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":label:`subsec_softmax-implementation-revisited`\n",
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"\n",
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"在前面 :numref:`sec_softmax_scratch`的例子中，\n",
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"我们计算了模型的输出，然后将此输出送入交叉熵损失。\n",
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"从数学上讲，这是一件完全合理的事情。\n",
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"然而，从计算角度来看，指数可能会造成数值稳定性问题。\n",
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"\n",
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"回想一下，softmax函数$\\hat y_j = \\frac{\\exp(o_j)}{\\sum_k \\exp(o_k)}$，\n",
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"其中$\\hat y_j$是预测的概率分布。\n",
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"$o_j$是未规范化的预测$\\mathbf{o}$的第$j$个元素。\n",
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"如果$o_k$中的一些数值非常大，\n",
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||
"那么$\\exp(o_k)$可能大于数据类型容许的最大数字，即*上溢*（overflow）。\n",
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"这将使分母或分子变为`inf`（无穷大），\n",
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"最后得到的是0、`inf`或`nan`（不是数字）的$\\hat y_j$。\n",
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"在这些情况下，我们无法得到一个明确定义的交叉熵值。\n",
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"\n",
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"解决这个问题的一个技巧是：\n",
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"在继续softmax计算之前，先从所有$o_k$中减去$\\max(o_k)$。\n",
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||
"这里可以看到每个$o_k$按常数进行的移动不会改变softmax的返回值：\n",
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"\n",
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"$$\n",
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||
"\\begin{aligned}\n",
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"\\hat y_j & = \\frac{\\exp(o_j - \\max(o_k))\\exp(\\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))\\exp(\\max(o_k))} \\\\\n",
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||
"& = \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}.\n",
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"\\end{aligned}\n",
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||
"$$\n",
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"\n",
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"\n",
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||
"在减法和规范化步骤之后，可能有些$o_j - \\max(o_k)$具有较大的负值。\n",
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||
"由于精度受限，$\\exp(o_j - \\max(o_k))$将有接近零的值，即*下溢*（underflow）。\n",
|
||
"这些值可能会四舍五入为零，使$\\hat y_j$为零，\n",
|
||
"并且使得$\\log(\\hat y_j)$的值为`-inf`。\n",
|
||
"反向传播几步后，我们可能会发现自己面对一屏幕可怕的`nan`结果。\n",
|
||
"\n",
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||
"尽管我们要计算指数函数，但我们最终在计算交叉熵损失时会取它们的对数。\n",
|
||
"通过将softmax和交叉熵结合在一起，可以避免反向传播过程中可能会困扰我们的数值稳定性问题。\n",
|
||
"如下面的等式所示，我们避免计算$\\exp(o_j - \\max(o_k))$，\n",
|
||
"而可以直接使用$o_j - \\max(o_k)$，因为$\\log(\\exp(\\cdot))$被抵消了。\n",
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||
"\n",
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||
"$$\n",
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||
"\\begin{aligned}\n",
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||
"\\log{(\\hat y_j)} & = \\log\\left( \\frac{\\exp(o_j - \\max(o_k))}{\\sum_k \\exp(o_k - \\max(o_k))}\\right) \\\\\n",
|
||
"& = \\log{(\\exp(o_j - \\max(o_k)))}-\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)} \\\\\n",
|
||
"& = o_j - \\max(o_k) -\\log{\\left( \\sum_k \\exp(o_k - \\max(o_k)) \\right)}.\n",
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||
"\\end{aligned}\n",
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||
"$$\n",
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||
"\n",
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||
"我们也希望保留传统的softmax函数，以备我们需要评估通过模型输出的概率。\n",
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||
"但是，我们没有将softmax概率传递到损失函数中，\n",
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||
"而是[**在交叉熵损失函数中传递未规范化的预测，并同时计算softmax及其对数**]，\n",
|
||
"这是一种类似[\"LogSumExp技巧\"](https://en.wikipedia.org/wiki/LogSumExp)的聪明方式。\n"
|
||
]
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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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"id": "91c3ac45",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T06:57:18.353684Z",
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"iopub.status.busy": "2023-08-18T06:57:18.353410Z",
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"iopub.status.idle": "2023-08-18T06:57:18.358187Z",
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"shell.execute_reply": "2023-08-18T06:57:18.357079Z"
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},
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"origin_pos": 13,
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"tab": [
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"pytorch"
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]
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},
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||
"outputs": [],
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||
"source": [
|
||
"loss = nn.CrossEntropyLoss(reduction='none')"
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||
]
|
||
},
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{
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"cell_type": "markdown",
|
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"id": "c347cec2",
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"metadata": {
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"origin_pos": 15
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},
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"source": [
|
||
"## 优化算法\n",
|
||
"\n",
|
||
"在这里，我们(**使用学习率为0.1的小批量随机梯度下降作为优化算法**)。\n",
|
||
"这与我们在线性回归例子中的相同，这说明了优化器的普适性。\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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"id": "c4849ef8",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T06:57:18.362274Z",
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"iopub.status.busy": "2023-08-18T06:57:18.361998Z",
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"iopub.status.idle": "2023-08-18T06:57:18.366991Z",
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"shell.execute_reply": "2023-08-18T06:57:18.365798Z"
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},
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"origin_pos": 17,
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"tab": [
|
||
"pytorch"
|
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]
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},
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"outputs": [],
|
||
"source": [
|
||
"trainer = torch.optim.SGD(net.parameters(), lr=0.1)"
|
||
]
|
||
},
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{
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||
"cell_type": "markdown",
|
||
"id": "c2cf8941",
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"metadata": {
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"origin_pos": 20
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},
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"source": [
|
||
"## 训练\n",
|
||
"\n",
|
||
"接下来我们[**调用**] :numref:`sec_softmax_scratch`中(~~之前~~)\n",
|
||
"(**定义的训练函数来训练模型**)。\n"
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]
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},
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{
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"origin_pos": 21,
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"tab": [
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"pytorch"
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]
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"source": [
|
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"和以前一样，这个算法使结果收敛到一个相当高的精度，而且这次的代码比之前更精简了。\n",
|
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"\n",
|
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"## 小结\n",
|
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"\n",
|
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"* 使用深度学习框架的高级API，我们可以更简洁地实现softmax回归。\n",
|
||
"* 从计算的角度来看，实现softmax回归比较复杂。在许多情况下，深度学习框架在这些著名的技巧之外采取了额外的预防措施，来确保数值的稳定性。这使我们避免了在实践中从零开始编写模型时可能遇到的陷阱。\n",
|
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"\n",
|
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"## 练习\n",
|
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"\n",
|
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"1. 尝试调整超参数，例如批量大小、迭代周期数和学习率，并查看结果。\n",
|
||
"1. 增加迭代周期的数量。为什么测试精度会在一段时间后降低？我们怎么解决这个问题？\n"
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