Perplexity in language model

1.PPL

PPL是用在自然语言处理领域（NLP）中，衡量语言模型好坏的指标。它主要是根据每个词来估计一句话出现的概率，并用句子长度作normalize，公式为

由公式可知，perplexity越小，模型越好。从公式最后一部分，感觉更像是描述GPT这种生成模型。

还有一种说法是：

2.Language Model

  • autoregressive (AR) language model

GPT：

 

  • autoencoding (AE)language model

BERT(denoising auto-encoding)：

where mt = 1 indicates xt is masked.

3.Reference

[论文笔记][2020-WWW]Enhanced-RCNN: An Efficient Method for Learning Sentence Similarity

前面的Related可以用来做综述

一、Architecture

二、Detail

1.Input Encoding

(a)RNN Encoder

(b)CNN Encoder

2.Interactive Sentence Representation

(a)Soft-attention Alignment(类似于ESIM)

(b)Interaction Modeling

3.Similarity Modeling

(a)Fusion Layer

(b)Label Prediction

MLP+softmax

三、Reference

文本生成中的解码策略

一.Language model decoding

given a sequence of m tokens as context, the task is to generate the next n continuation tokens to obtain the completed sequence . We assume that models compute using the common left-to-right decomposition of the text probability,

which is used to generate the generation token-by-token using a particular decoding strategy

二.Decoding Strategies

1.Maximization-based decoding

解码器的输出层后面通常会跟一个softmax函数来将输出概率归一化。如果词表比较大的话，softmax的计算复杂度会比较高(分母需要计算整个词表大小的维度)

  • Greedy decoding

每次选最高概率的一项作为输出，直到遇到结束符号，将这些输出的概率相乘，得到该sequence的概率。

缺点：每次选择都是最大的，但是整体的概率不一定是最大的，有可能错过最优的结果

考虑到`Beam Size=2`，第一步选择概率最大的两个A和B，第二步选择AB和BB（橙色大箭头）。然后以选择的AB和BB继续向上传播，又出现了四种情况ABA/ABB/BBA/BBB，依然是选择综合概率最大的两个ABB/BBB。以此类推，直至句子结束。

缺点：

    • 由于beam search的概率是一個连续相乘的結果，越早遇到结束符号，所得到的概率会越大，所以更倾向于生成短文本。解决办法是对输出概率取log，然后相加求最大概率。
    • beam size过大会导致计算复杂度过高，时间复杂度为O(N*b*V)，N为序列长度，b为beam size，v为词表大小
    • 语言模型通常为格式良好的文本分配高分，但较长文本的最高分往往是通用的、重复的和尴尬的。beam search的输出，与人讲话的概率分布，很不一样。

2.Stochastic Decoding

  • Sampling with temperature

解码器的输出层计算softmax时，通过改变T可以控制概率的形貌。当t大的时候，概率分布趋向平均，随机性增大(uniform sampling/pure sampling)；当t小的时候，概率密度趋向于集中，即强者愈强，随机性降低(poisson distribution/greedy decoding)。
实验证明，降低t会改善生成的质量，但是会降低生成的多样性。
  • Top-k Sampling

从词表选择topk个词，使得概率之和最大，即

缺点：k的取值难以界定。k太小，会生成比较bland和generic的结果，即缺乏diversity；k太大，会退化成随机sampling(pure sampling)，会陷入语义表达错误，即sample from tail distribution

  • Top-p Sampling(nucleus sampling)

丛词表选择p个概率之和大于给定阈值的词，即

采样集的大小将根据每个时间步长的概率分布动态调整。对于高的阈值来说，这一个小部分的词汇，就占据了绝大多数的概率—-核。

三.Implementation

def top_k_top_p_filtering(logits, top_k=0, top_p=0.0, filter_value=-float('Inf')):
    """ Filter a distribution of logits using top-k and/or nucleus (top-p) filtering
        Args:
            logits: logits distribution shape (vocabulary size)
            top_k >0: keep only top k tokens with highest probability (top-k filtering).
            top_p >0.0: keep the top tokens with cumulative probability >= top_p (nucleus filtering).
                Nucleus filtering is described in Holtzman et al. (http://arxiv.org/abs/1904.09751)
    """
    assert logits.dim() == 1  # batch size 1 for now - could be updated for more but the code would be less clear
    top_k = min(top_k, logits.size(-1))  # Safety check
    if top_k > 0:
        # Remove all tokens with a probability less than the last token of the top-k
        indices_to_remove = logits < torch.topk(logits, top_k)[0][..., -1, None]
        logits[indices_to_remove] = filter_value

    if top_p > 0.0:
        sorted_logits, sorted_indices = torch.sort(logits, descending=True)
        cumulative_probs = torch.cumsum(F.softmax(sorted_logits, dim=-1), dim=-1)

        # Remove tokens with cumulative probability above the threshold
        sorted_indices_to_remove = cumulative_probs > top_p
        # Shift the indices to the right to keep also the first token above the threshold
        sorted_indices_to_remove[..., 1:] = sorted_indices_to_remove[..., :-1].clone()
        sorted_indices_to_remove[..., 0] = 0

        indices_to_remove = sorted_indices[sorted_indices_to_remove]
        logits[indices_to_remove] = filter_value
    return logits

# Here is how to use this function for top-p sampling
temperature = 1.0
top_k = 0
top_p = 0.9

# Get logits with a forward pass in our model (input is pre-defined)
logits = model(input)

# Keep only the last token predictions of the first batch item (batch size 1), apply a temperature coefficient and filter
logits = logits[0, -1, :] / temperature
filtered_logits = top_k_top_p_filtering(logits, top_k=top_k, top_p=top_p)

# Sample from the filtered distribution
probabilities = F.softmax(filtered_logits, dim=-1)
next_token = torch.multinomial(probabilities, 1)

四.Reference

大根堆/小根堆的实现

#include <iostream>
#include <vector>

using namespace std;

template<typename P>

class Heap {
private:
    vector<P> heap_elem{-1};//堆容器, 使得堆的编号从1开始
    int maxmin_heap;//选择大\小根堆

    void up_adjust(int now) {//上浮调整（默认大根堆）
        while (now > 1 && heap_elem[now] > heap_elem[now / 2]) {
            swap(heap_elem[now], heap_elem[now / 2]);
            now = now / 2;
        }
    };

    void down_adjust(P &top, int now) {//下沉调整
        heap_elem[now] = top;
        int next;
        while (now * 2 <= heap_elem.size() - 1) {
            if (now * 2 + 1 > heap_elem.size() - 1) {
                next = now * 2;
            } else {
                next = heap_elem[now * 2] > heap_elem[now * 2 + 1] ? now * 2 : now * 2 + 1;
            }
            if (heap_elem[next] > heap_elem[now]) {
                swap(heap_elem[next], heap_elem[now]);
                now = next;
            } else {
                break;
            }
        }
    };

public:
    Heap(int m) {//构造堆
        maxmin_heap = m;//default max_heap
    };

    P getTop() {//弹获取堆顶元素
        if (heap_elem.size() > 1)
            return heap_elem[1] * maxmin_heap;
        return heap_elem[0];//堆为空
    }

    bool empty() {//判断堆空
        return heap_elem.size() == 1;
    }

    void push(P x) {//送元素入堆
        heap_elem.push_back(x * maxmin_heap);
        int now = heap_elem.size() - 1;
        up_adjust(now);
    };

    P pop() {//堆顶元素弹出
        P res = getTop();
        P top = heap_elem.back();
        heap_elem.pop_back();
        int now = 1;
        down_adjust(top, now);
        cout << "pop " << res << " successfully." << endl;
        return res;
    };

    int size() {
        return heap_elem.size() - 1;
    }

    void print() {
        for (int i = 1; i < heap_elem.size(); i++) {
            cout << heap_elem[i] * maxmin_heap << ' ';
        }
        cout << endl;
    }

};

int main() {
    //Topk测试没问题
    int max_heap = 1, min_heap = -1;
    Heap<int> h(min_heap);//可更改为min_heap，来构建小根堆
    int x;
    for (int i = 0; i < 10; ++i) {
        x = rand() % 100;
        h.push(x);
        cout << "x: " << x << endl;
        h.print();
    }
    h.pop();
    h.print();
    return 0;
}

 

Beam search

1.简单实现

import torch

# Beam  search
samples = []
topk = 10
log_prob, v_idx = decoder_outputs.detach().topk(topk)
for k in range(topk):
    samples.append([[v_idx[0][k].item()], log_prob[0][k], decoder_state])
for i in range(max_len):
    new_samples = []
    for sample in samples:
        v_list, score, decoder_state = sample
        if v_list[-1] == de_vocab.item2index['_EOS_']:
            new_samples.append([v_list, score, decoder_state])
            continue

        decoder_inputs = torch.LongTensor([v_list[-1]])
        decoder_outputs, new_states = decoder(decoder_inputs, encoder_output, decoder_state)
        log_prob, v_idx = decoder_outputs.data.topk(topk)

        for k in range(topk):
            new_v_list = []
            new_v_list += v_list + [v_idx[0][k].item()]
            new_samples.append([new_v_list, score + log_prob[0][k], new_states])

    new_samples = sorted(new_samples, key=lambda sample: sample[1], reverse=True)
    samples = new_samples[:topk]
    
v_list, score, states = samples[0]
for v_idx in v_list:
    pred_sent.append(de_vocab.index2item[v_idx])

2.transformers实现