语言模型与数据集.ipynb

{"cells":[{"cell_type":"markdown","metadata":{"graffitiCellId":"id_nporece","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"1054EE94F8104E6A80A156562A92897A","mdEditEnable":false},"source":"# 语言模型\n\n一段自然语言文本可以看作是一个离散时间序列，给定一个长度为$T$的词的序列$w_1, w_2, \\ldots, w_T$，语言模型的目标就是评估该序列是否合理，即计算该序列的概率：\n\n\n$$\nP(w_1, w_2, \\ldots, w_T).\n$$\n\n\n本节我们介绍基于统计的语言模型，主要是$n$元语法（$n$-gram）。在后续内容中，我们将会介绍基于神经网络的语言模型。"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_j2098vn","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"A05A3A43703846CDBCD2576B7BA3E21B","mdEditEnable":false},"source":"## 语言模型\n\n\n假设序列$w_1, w_2, \\ldots, w_T$中的每个词是依次生成的，我们有\n\n\n$$\n\n\\begin{align*}\nP(w_1, w_2, \\ldots, w_T)\n&= \\prod_{t=1}^T P(w_t \\mid w_1, \\ldots, w_{t-1})\\\\\n&= P(w_1)P(w_2 \\mid w_1) \\cdots P(w_T \\mid w_1w_2\\cdots w_{T-1})\n\\end{align*}\n\n$$\n\n\n例如，一段含有4个词的文本序列的概率\n\n\n$$\nP(w_1, w_2, w_3, w_4) =  P(w_1) P(w_2 \\mid w_1) P(w_3 \\mid w_1, w_2) P(w_4 \\mid w_1, w_2, w_3).\n$$\n\n\n语言模型的参数就是词的概率以及给定前几个词情况下的条件概率。设训练数据集为一个大型文本语料库，如维基百科的所有条目，词的概率可以通过该词在训练数据集中的相对词频来计算，例如，$w_1$的概率可以计算为：\n\n\n$$\n\n\\hat P(w_1) = \\frac{n(w_1)}{n}\n\n$$\n\n\n其中$n(w_1)$为语料库中以$w_1$作为第一个词的文本的数量，$n$为语料库中文本的总数量。\n\n类似的，给定$w_1$情况下，$w_2$的条件概率可以计算为：\n\n\n$$\n\n\\hat P(w_2 \\mid w_1) = \\frac{n(w_1, w_2)}{n(w_1)}\n\n$$\n\n\n其中$n(w_1, w_2)$为语料库中以$w_1$作为第一个词，$w_2$作为第二个词的文本的数量。"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_6801mjh","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"251399EF3DD149548D760A5F24266338","mdEditEnable":false},"source":"## n元语法\n\n序列长度增加，计算和存储多个词共同出现的概率的复杂度会呈指数级增加。$n$元语法通过马尔可夫假设简化模型，马尔科夫假设是指一个词的出现只与前面$n$个词相关，即$n$阶马尔可夫链（Markov chain of order $n$），如果$n=1$，那么有$P(w_3 \\mid w_1, w_2) = P(w_3 \\mid w_2)$。基于$n-1$阶马尔可夫链，我们可以将语言模型改写为\n\n\n$$\nP(w_1, w_2, \\ldots, w_T) = \\prod_{t=1}^T P(w_t \\mid w_{t-(n-1)}, \\ldots, w_{t-1}) .\n$$\n\n\n以上也叫$n$元语法（$n$-grams），它是基于$n - 1$阶马尔可夫链的概率语言模型。例如，当$n=2$时，含有4个词的文本序列的概率就可以改写为：\n\n\n$$\n\n\\begin{align*}\nP(w_1, w_2, w_3, w_4)\n&= P(w_1) P(w_2 \\mid w_1) P(w_3 \\mid w_1, w_2) P(w_4 \\mid w_1, w_2, w_3)\\\\\n&= P(w_1) P(w_2 \\mid w_1) P(w_3 \\mid w_2) P(w_4 \\mid w_3)\n\\end{align*}\n\n$$\n\n\n当$n$分别为1、2和3时，我们将其分别称作一元语法（unigram）、二元语法（bigram）和三元语法（trigram）。例如，长度为4的序列$w_1, w_2, w_3, w_4$在一元语法、二元语法和三元语法中的概率分别为\n\n\n$$\n\n\\begin{aligned}\nP(w_1, w_2, w_3, w_4) &=  P(w_1) P(w_2) P(w_3) P(w_4) ,\\\\\nP(w_1, w_2, w_3, w_4) &=  P(w_1) P(w_2 \\mid w_1) P(w_3 \\mid w_2) P(w_4 \\mid w_3) ,\\\\\nP(w_1, w_2, w_3, w_4) &=  P(w_1) P(w_2 \\mid w_1) P(w_3 \\mid w_1, w_2) P(w_4 \\mid w_2, w_3) .\n\\end{aligned}\n\n$$\n\n\n当$n$较小时，$n$元语法往往并不准确。例如，在一元语法中，由三个词组成的句子“你走先”和“你先走”的概率是一样的。然而，当$n$较大时，$n$元语法需要计算并存储大量的词频和多词相邻频率。"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_3wfv5oo","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"CFDDB4CFC6CA4F558608DA01CD07F9A0","mdEditEnable":false},"source":"思考：$n$元语法可能有哪些缺陷？"},{"metadata":{"id":"E0358FB3352B4843883573902E9AEFDF","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"mdEditEnable":false},"cell_type":"markdown","source":"1. 参数空间过大\n2. 数据稀疏"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_cpdv382","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"855C64A9C75A4C6C88134AC099FCD6D6","mdEditEnable":false},"source":"# 语言模型数据集"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_0419ugr","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"131B4EA95293494D8D7E089964514961","mdEditEnable":false},"source":"## 读取数据集"},{"cell_type":"code","execution_count":1,"metadata":{"graffitiCellId":"id_210oytz","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"434D33D454FB42C38C6F9816E9E1FBF4","collapsed":false,"scrolled":false},"outputs":[{"output_type":"stream","text":"63282\n想要有直升机\n想要和你飞到宇宙去\n想要和你融化在一起\n融化在宇宙里\n我每天每天每\n","name":"stdout"}],"source":"with open('/home/kesci/input/jaychou_lyrics4703/jaychou_lyrics.txt') as f:\n    corpus_chars = f.read()\nprint(len(corpus_chars))\nprint(corpus_chars[: 40])\ncorpus_chars = corpus_chars.replace('\\n', ' ').replace('\\r', ' ')\ncorpus_chars = corpus_chars[: 10000]"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_ls5ngie","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"FAC600069D7A4E8E84F4E9D8C1232B27","mdEditEnable":false},"source":"## 建立字符索引"},{"cell_type":"code","execution_count":2,"metadata":{"graffitiCellId":"id_rpxalgy","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"B76CFE1C0F69419AA0019EC506E6A7EB","collapsed":false,"scrolled":false},"outputs":[{"output_type":"stream","text":"1027\nchars: 想要有直升机 想要和你飞到宇宙去 想要和\nindices: [1022, 648, 1025, 366, 208, 792, 199, 1022, 648, 641, 607, 625, 26, 155, 130, 5, 199, 1022, 648, 641]\n","name":"stdout"}],"source":"idx_to_char = list(set(corpus_chars)) # 去重，得到索引到字符的映射\nchar_to_idx = {char: i for i, char in enumerate(idx_to_char)} # 字符到索引的映射\nvocab_size = len(char_to_idx)\nprint(vocab_size)\n\ncorpus_indices = [char_to_idx[char] for char in corpus_chars]  # 将每个字符转化为索引，得到一个索引的序列\nsample = corpus_indices[: 20]\nprint('chars:', ''.join([idx_to_char[idx] for idx in sample]))\nprint('indices:', sample)"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_4k8xe1z","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"AEC7A5F1A73E4E5B8B5C87B28A82E553","mdEditEnable":false},"source":"定义函数`load_data_jay_lyrics`，在后续章节中直接调用。"},{"cell_type":"code","execution_count":3,"metadata":{"graffitiCellId":"id_g98p1e0","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"82B3F014ECB14F099961B196CF98D42F","collapsed":false,"scrolled":false},"outputs":[],"source":"def load_data_jay_lyrics():\n    with open('/home/kesci/input/jaychou_lyrics4703/jaychou_lyrics.txt') as f:\n        corpus_chars = f.read()\n    corpus_chars = corpus_chars.replace('\\n', ' ').replace('\\r', ' ')\n    corpus_chars = corpus_chars[0:10000]\n    idx_to_char = list(set(corpus_chars))\n    char_to_idx = dict([(char, i) for i, char in enumerate(idx_to_char)])\n    vocab_size = len(char_to_idx)\n    corpus_indices = [char_to_idx[char] for char in corpus_chars]\n    return corpus_indices, char_to_idx, idx_to_char, vocab_size"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_ugfju1n","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"20B12ED94FF145F089127459E501DC68","mdEditEnable":false},"source":"## 时序数据的采样\n\n在训练中我们需要每次随机读取小批量样本和标签。与之前章节的实验数据不同的是，时序数据的一个样本通常包含连续的字符。假设时间步数为5，样本序列为5个字符，即“想”“要”“有”“直”“升”。该样本的标签序列为这些字符分别在训练集中的下一个字符，即“要”“有”“直”“升”“机”，即$X$=“想要有直升”，$Y$=“要有直升机”。\n\n现在我们考虑序列“想要有直升机，想要和你飞到宇宙去”，如果时间步数为5，有以下可能的样本和标签：\n* $X$：“想要有直升”，$Y$：“要有直升机”\n* $X$：“要有直升机”，$Y$：“有直升机，”\n* $X$：“有直升机，”，$Y$：“直升机，想”\n* ...\n* $X$：“要和你飞到”，$Y$：“和你飞到宇”\n* $X$：“和你飞到宇”，$Y$：“你飞到宇宙”\n* $X$：“你飞到宇宙”，$Y$：“飞到宇宙去”\n\n可以看到，如果序列的长度为$T$，时间步数为$n$，那么一共有$T-n$个合法的样本，但是这些样本有大量的重合，我们通常采用更加高效的采样方式。我们有两种方式对时序数据进行采样，分别是随机采样和相邻采样。\n\n### 随机采样\n\n下面的代码每次从数据里随机采样一个小批量。其中批量大小`batch_size`是每个小批量的样本数，`num_steps`是每个样本所包含的时间步数。\n在随机采样中，每个样本是原始序列上任意截取的一段序列，相邻的两个随机小批量在原始序列上的位置不一定相毗邻。"},{"cell_type":"code","execution_count":4,"metadata":{"graffitiCellId":"id_07sma6k","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"31D18572438E4FE5856F1F24BB0C430C","collapsed":false,"scrolled":false},"outputs":[],"source":"import torch\nimport random\ndef data_iter_random(corpus_indices, batch_size, num_steps, device=None):\n    # 减1是因为对于长度为n的序列，X最多只有包含其中的前n - 1个字符\n    num_examples = (len(corpus_indices) - 1) // num_steps  # 下取整，得到不重叠情况下的样本个数\n    example_indices = [i * num_steps for i in range(num_examples)]  # 每个样本的第一个字符在corpus_indices中的下标\n    random.shuffle(example_indices)\n\n    def _data(i):\n        # 返回从i开始的长为num_steps的序列\n        return corpus_indices[i: i + num_steps]\n    if device is None:\n        device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n    \n    for i in range(0, num_examples, batch_size):\n        # 每次选出batch_size个随机样本\n        batch_indices = example_indices[i: i + batch_size]  # 当前batch的各个样本的首字符的下标\n        X = [_data(j) for j in batch_indices]\n        Y = [_data(j + 1) for j in batch_indices]\n        yield torch.tensor(X, device=device), torch.tensor(Y, device=device)"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_edvdo82","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"5901DE6ECDA04E6F87E3FF2B0F697847","mdEditEnable":false},"source":"测试一下这个函数，我们输入从0到29的连续整数作为一个人工序列，设批量大小和时间步数分别为2和6，打印随机采样每次读取的小批量样本的输入`X`和标签`Y`。"},{"cell_type":"code","execution_count":5,"metadata":{"graffitiCellId":"id_30cx4t4","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"5E8FDD7CDC6E43D884D4976DAB8A9191","collapsed":false,"scrolled":false},"outputs":[{"output_type":"stream","text":"X:  tensor([[ 6,  7,  8,  9, 10, 11],\n        [12, 13, 14, 15, 16, 17]]) \nY: tensor([[ 7,  8,  9, 10, 11, 12],\n        [13, 14, 15, 16, 17, 18]]) \n\nX:  tensor([[ 0,  1,  2,  3,  4,  5],\n        [18, 19, 20, 21, 22, 23]]) \nY: tensor([[ 1,  2,  3,  4,  5,  6],\n        [19, 20, 21, 22, 23, 24]]) \n\n","name":"stdout"}],"source":"my_seq = list(range(30))\nfor X, Y in data_iter_random(my_seq, batch_size=2, num_steps=6):\n    print('X: ', X, '\\nY:', Y, '\\n')"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_z179h28","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"B233BB0A25CA41A6998C3ECEE5B4A3FB","mdEditEnable":false},"source":"### 相邻采样\n\n在相邻采样中，相邻的两个随机小批量在原始序列上的位置相毗邻。"},{"cell_type":"code","execution_count":6,"metadata":{"graffitiCellId":"id_6omgvln","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"8A18E8E0E87641EE84B8FE61A1D9ADD6","collapsed":false,"scrolled":false},"outputs":[],"source":"def data_iter_consecutive(corpus_indices, batch_size, num_steps, device=None):\n    if device is None:\n        device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n    corpus_len = len(corpus_indices) // batch_size * batch_size  # 保留下来的序列的长度\n    corpus_indices = corpus_indices[: corpus_len]  # 仅保留前corpus_len个字符\n    indices = torch.tensor(corpus_indices, device=device)\n    indices = indices.view(batch_size, -1)  # resize成(batch_size, )\n    batch_num = (indices.shape[1] - 1) // num_steps\n    for i in range(batch_num):\n        i = i * num_steps\n        X = indices[:, i: i + num_steps]\n        Y = indices[:, i + 1: i + num_steps + 1]\n        yield X, Y"},{"cell_type":"markdown","metadata":{"graffitiCellId":"id_5isfefi","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"B30A869DE4E3466891AAAEF45FCAC7F9","mdEditEnable":false},"source":"同样的设置下，打印相邻采样每次读取的小批量样本的输入`X`和标签`Y`。相邻的两个随机小批量在原始序列上的位置相毗邻。"},{"cell_type":"code","execution_count":7,"metadata":{"graffitiCellId":"id_7t0gy91","jupyter":{},"tags":[],"slideshow":{"slide_type":"slide"},"id":"6C18F0586CE64AC290556069BD63B15A","collapsed":false,"scrolled":false},"outputs":[{"output_type":"stream","text":"X:  tensor([[ 0,  1,  2,  3,  4,  5],\n        [15, 16, 17, 18, 19, 20]]) \nY: tensor([[ 1,  2,  3,  4,  5,  6],\n        [16, 17, 18, 19, 20, 21]]) \n\nX:  tensor([[ 6,  7,  8,  9, 10, 11],\n        [21, 22, 23, 24, 25, 26]]) \nY: tensor([[ 7,  8,  9, 10, 11, 12],\n        [22, 23, 24, 25, 26, 27]]) \n\n","name":"stdout"}],"source":"for X, Y in data_iter_consecutive(my_seq, batch_size=2, num_steps=6):\n    print('X: ', X, '\\nY:', Y, '\\n')"}],"metadata":{"kernelspec":{"name":"python3","display_name":"Python 3","language":"python"},"language_info":{"name":"python","version":"3.7.3","mimetype":"text/x-python","codemirror_mode":{"name":"ipython","version":3},"pygments_lexer":"ipython3","nbconvert_exporter":"python","file_extension":".py"}},"nbformat":4,"nbformat_minor":1}