From 168f7b8846ded9b0f505d25649b0be8c577e21c1 Mon Sep 17 00:00:00 2001 From: wdndev Date: Wed, 7 Aug 2024 22:27:11 +0800 Subject: [PATCH] ngram --- code/01_ngram/README.md | 232 ++++++++++++++++++++++- code/01_ngram/image/image_1rJ6pKjJoI.png | Bin 0 -> 49498 bytes code/01_ngram/images/output.png | Bin 50888 -> 0 bytes 3 files changed, 228 insertions(+), 4 deletions(-) create mode 100644 code/01_ngram/image/image_1rJ6pKjJoI.png delete mode 100644 code/01_ngram/images/output.png diff --git a/code/01_ngram/README.md b/code/01_ngram/README.md index 6730946..7d1f7c9 100644 --- a/code/01_ngram/README.md +++ b/code/01_ngram/README.md @@ -1,8 +1,221 @@ +# 1.ngram +## 1.统计语言模型 -运行日志: +### 1.1 介绍 -```shell +通俗的说,语言模型就是用来计算一个句子的概率的**概率模型**,它通常被描述为字符串`s`的概率分布$P(s)$。这里的字符串`s`由`l`个“基元”组成,基元可以是字、词或者短语等。于是句子`s`的概率可以表示为所有基元的联合概率分布: + +$$ +p(s)=p\left(w_{1}, w_{2}, w_{3}, \ldots, w_{l}\right) +$$ + +利用bayes公式将其转变为: + +$$ +\begin{aligned} p(s) & =p\left(w_{1}\right) p\left(w_{2} \mid w_{1}\right) p\left(w_{3} \mid w_{1}, w_{2}\right) \ldots p\left(w_{l} \mid w 1, w_{2}, \ldots, w_{l-1}\right) \\ & =\prod_{i=1}^{l} p\left(w_{i} \mid w_{1} \ldots w_{i-1}\right)\end{aligned} +$$ + +而上式中的条件概率$p(w_1),p(w_2|w_1),p(w_3,|w_1,w_2)...$就是语言模型的参数。当这些模型都计算出的时候,对于任何一个给定的句子`s`都可以通过将对应的条件概率相乘的方法很快得到句子的概率了。不同的模型计算这些参数的值的方式不同,常用的方法有**n-gram,决策树,最大熵模型神经网络**等等。 + +### 1.2 模型缺点 + +上述模型看上去非常简单,但是通过简单估算可以发现其中存在的问题。对于一个大小为`N`的词典D来说,一个长度为`l`的句子`s`,这其中需要的参数就是$N^l$个。在这种计算方法中,第`i`个词是由前`i-1`个词(历史词)推出的,随着历史词个数的增长,参数个数按指数级增长。这就会导致两个问题: + +- **参数空间过大**,根本无法从训练语料中得到所有的参数,因为大部分的情形在训练语料中根本就没出现 +- **数据稀疏严重**,对于没有在语料中出现的组合,根据最大似然估计就会得到这个参数为0,这将导致整个句子的概率为0 + +## 2.N元语法(N-gram)模型 + +### 2.1 马尔可夫假设 + +为了解决之前讲的参数空间过大的问题,引入马尔科夫假设:**一个词出现的概率只和它前面出现的一个或有限的几个词有关**: + +$$ +p(s)=\prod_{i=1}^{l} p\left(w_{i} \mid w_{1} \ldots w_{i-1}=\prod_{i=1}^{l} p\left(w_{i} \mid w_{i-k}, w_{i-(k-1)}, \ldots, w_{i-1}\right)\right. +$$ + +可以看到通过这样的方式,模型的参数就得到了大大减少。 + +### 2.2 N元语法 + +满足上面的条件的模型就被成为n元语法,其中的n对应考虑的不同历史词个数。通常情况下,n不能取太大的数,否则的话,参数个数过多的问题还是会存在,常用`n=1,2,3`。 + +`n=1`时的模型被称为一元语言模型(Uni-gram),它表示的是句子中每个词之间是条件无关的,也就是说所需要考虑的历史词为0,即 + +$$ +p\left(w_{1}, w_{2}, \ldots, w_{l}\right)=p\left(w_{1}\right) p\left(w_{2}\right) \ldots p\left(w_{l}\right) +$$ + +对于`n≥2`的情况,每个词与其前`n-1`个词有关,即 + +$$ +p\left(w_{1}, w_{2} m . ., w_{l}\right)=\prod_{i=1}^{l+1} p\left(w_{i} \mid w_{i-n+1}^{i-1}\right) +$$ + +然后根据最大似然估计,这些条件概率的计算就是在语料中统计这些词组合所出现的概率,即: + +$$ +p\left(w_{k} \mid w_{k-n+1}^{k-1}\right)=\frac{\operatorname{count}\left(w_{k-n+1}^{k}\right)}{\operatorname{count}\left(w_{k-n+1}^{k-1}\right)} +$$ + +从中可以看出,n的增大会增加模型的复杂度。虽然理论上n越大,模型的效果越好。但是当n很大的时候,需要的训练语料也会很大,而且当n大到一定程度时,其对于模型效果的提升越来越少,**通常使用n=3就已经满足要求了**。 + +### 2.3 OOV与平滑/回退 + +当语料库有限,大概率会在实际预测的时候遇到没见过的词或短语,这就是**未登录词(OOV)**,这样就会造成概率计算的公式中,分子或分母为0,毕竟它们都只是频率。分子为0的话,整个句子的概率是连乘出来的结果是0;分母是0的话,数学上就根本没法计算了。 + +即使是使用n=2的情况,也不能保证语料空间能够涵盖真实的语言模型空间。如果单纯的考虑词频比例作为参数进行计算的话就会碰到两个问题: + +- 当$\operatorname{count}\left(w_{k-n+1}^{k}\right)=0$时,是不是就能说明概率$p\left(w_{k} \mid w_{1}^{k-1}\right)$就是0呢? +- 当$\operatorname{count}\left(w_{k-n+1}^{k}\right)=\operatorname{count}\left(w_{k-n+1}^{k-1}\right)$,能否认为$p\left(w_{k} \mid w_{1}^{k-1}\right)$为1呢? + +显然上面的两种情况都是不能,考虑0或1这两个极端情况都会导致整个句子的概率过大或过小,这都是不希望看到的。然而,无论语料库有多么大,都无法回避这个问题。**平滑化方法就是处理这种问题的**。 + +计算n-gram的一个常用工具是[SRILM](http://www.speech.sri.com/projects/srilm/download.html "SRILM"),它是基于C++的,而且里面也集成了很多平滑方法,使用起来非常方便。 + +#### (1)**平滑(smoothing)** + +为每个w对应的Count增加一个很小的值,目的是使所有的 N-gram 概率之和为 1、使所有的 N-gram 概率都不为 0。常见平滑方法: + +**Laplace Smoothing** + +**Add-one**:即强制让所有的n-gram至少出现一次,只需要在分子和分母上分别做加法即可。这个方法的弊端是,大部分n-gram都是没有出现过的,很容易为他们分配过多的概率空间。 + +$$ +p\left(w_{n} \mid w_{n-1}\right)=\frac{C\left(w_{n-1} w_{n}\right)+1}{C\left(w_{n-1}\right)+|V|} +$$ + +**Add-K**:在Add-one的基础上做了一点小改动,原本是加1,现在加上一个小于1的常数K。但是缺点是这个常数仍然需要人工确定,对于不同的语料库K可能不同。 + +$$ +p\left(w_{n} \mid w_{n-1}\right)=\frac{C\left(w_{n-1} w_{n}\right)+k}{C\left(w_{n-1}\right)+k|V|} +$$ + +其他平滑方法: + +- Good-Turing smoothing +- Jelinek-Mercer smoothing (interpolation) +- Catz smoothing +- Witten-Bell smoothing +- Absolute discounting +- Kneser-Ney smoothing + +#### (2)回退 **(Katz backoff)** + +从N-gram回退到(N-1)-gram,例如Count(the,dog)\~=Count(dog)。 + +### 2.4 N-gram小节 + +n-gram模型简单来说就是在给定的语料库中统计各个词串出现的次数,并进行适当的平滑处理。之后将这些值存储起来,在计算出现句子的概率时,找到对应的概率参数,相乘即得到总的概率值。 + +总结下基于统计的 n-gram 语言模型的优缺点: + +优点: + +- 采用极大似然估计,参数易训练; +- 完全包含了前 n-1 个词的全部信息; +- 可解释性强,直观易理解。 + +缺点: + +- 缺乏长期依赖,只能建模到前 n-1 个词; +- 随着 n 的增大,参数空间呈指数增长; +- 数据稀疏,难免会出现OOV的问题; +- 单纯的基于统计频次,泛化能力差。 + +## 3.代码解读 + +### 3.1 详细代码 + +整体来说,代码比较简单,具体实现了n-gram模型(`NgramModel`)和回退 n-gram 模型(`BackoffNgramModel`,未使用),再训练时也采用`smoothing`的方式处理OOV。详细代码见 `code/01_ngram/ngram.py`,有详细中文注释,部分代码如下: + +- n-gram 模型 + +```python +class NgramModel: + """ n-gram 模型类。 + """ + def __init__(self, vocab_size, seq_len, smoothing=0.0): + """初始化 n-gram 模型。 + + Args: + vocab_size (int): 词汇表大小。 + seq_len (int): 序列长度。 + smoothing (float): 平滑系数,默认为 0.0。 + """ + self.seq_len = seq_len + self.vocab_size = vocab_size + self.smoothing = smoothing + # 计数数组,形状为 (vocab_size,) * seq_len + self.counts = np.zeros((vocab_size,) * seq_len, dtype=np.uint32) + # 用于存储均匀分布的缓冲区,避免每次创建新的数组 + self.uniform = np.ones(self.vocab_size, dtype=np.float32) / self.vocab_size + + def train(self, tape): + """ 训练模型。 + + Args: + tape (list): 形状为 (seq_len,) 的训练数据序列。 + """ + assert isinstance(tape, list) + assert len(tape) == self.seq_len + self.counts[tuple(tape)] += 1 + + def get_counts(self, tape): + """ 获取给定上下文的计数。 + + Args: + tape (list): 形状为 (seq_len - 1,) 的上下文序列。 + + Returns: + np.ndarray: 给定上下文的计数数组。 + """ + assert isinstance(tape, list) + assert len(tape) == self.seq_len - 1 + return self.counts[tuple(tape)] + + def __call__(self, tape): + """ 计算下一个 token 的条件概率分布。 + + Args: + tape (list): 形状为 (seq_len - 1,) 的上下文序列。 + + Returns: + np.ndarray: 下一个标记的条件概率分布。 + """ + + assert isinstance(tape, list) + assert len(tape) == self.seq_len - 1 + # 获取计数并应用平滑处理 + counts = self.counts[tuple(tape)].astype(np.float32) + # 添加平滑处理("fake counts")到所有计数 + counts += self.smoothing + counts_sum = counts.sum() + # 如果计数总和大于 0,则计算概率;否则返回均匀分布 + probs = counts / counts_sum if counts_sum > 0 else self.uniform + return probs +``` + +### 3.2 代码运行 + +#### (1)数据集 + +数据集是2018年来自[ssa.gov](https://www.ssa.gov/oact/babynames/ "ssa.gov")的32,032个名字,这些名字在测试部分分为1000个名字,在val部分分为1000个名字,其余的在训练部分,所有这些名字都在`datasets/data`文件夹中。 因此,n-gram模型本质上将尝试学习这些名称中字符的统计信息,然后通过从模型中采样来生成新的名称。 + +#### (2)代码运行 + +注意:在 code 目录下运行 + +```bash +python 01_ngram/ngram.py +``` + +脚本首先“训练”一个小的字符级Tokenizer(所有26个小写英文字母和换行字符的词汇表大小为27),然后使用验证分割对n-gram模型进行一个小的网格搜索,该模型具有n-gram阶' n '和平滑因子的各种超参数设置。 + +默认设置的最优值是“n=4,平滑=0.1”。 然后取这个最佳模型,从中抽取200个字符,最后报告测试损失和困惑。 下面是完整的输出,它应该只需要几秒钟就能产生: + +```bash (llm) PS llm101n-zh\code> python .\01_ngram\ngram.py seq_len 3 | smoothing 0.03 | train_loss 2.1843 | val_loss 2.2443 seq_len 3 | smoothing 0.10 | train_loss 2.1870 | val_loss 2.2401 @@ -46,12 +259,17 @@ test_loss 2.106370, test_perplexity 8.218358 wrote datasets/dev\ngram_probs.npy to disk (for visualization) ``` -可视化 +`4-gram`模型抽样了一些相对合理的名称,如`felton`和`jasiel`,但也抽样了一些更奇怪的名称,如`nebjnvfobzadon`,但不能对一个`4-gram`字符级语言模型期望太高。 最后,测试困惑度报告为`~8.2`,因此模型对测试集中的每个字符都感到困惑,就好像它从8.2个等可能字符中随机选择一样。 + +#### (3)可视化 + +代码还将n-gram概率写入磁盘到`datasets/dev`文件夹中,然后可以使用dev/visualize\_probs.ipynb文件可视化。 ```python probs = np.load("ngram_probs.npy") probs.shape ``` + > (27, 27, 27, 27) ```python @@ -64,4 +282,10 @@ plt.axis('off') > (-0.5, 728.5, 728.5, -0.5) -![output](images/output.png) +![](image/image_1rJ6pKjJoI.png) + +## 4参考资料 + +1. [opensource.niutrans.com/mtbook/section2-3.html](https://opensource.niutrans.com/mtbook/section2-3.html "opensource.niutrans.com/mtbook/section2-3.html") +2. [Speech and Language Processing](https://web.stanford.edu/~jurafsky/slp3/3.pdf "Speech and Language Processing") +3. 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