残差连接 - 从零开始大模型

我们用 4 章的篇幅完成了 GPT 模型最核心的部分：自注意力机制。

接下来，我们将聚焦 GPT 模型预训练的工程实现。

GPT 模型的一个显著特点就是“大力出奇迹”。这不仅表现在模型训练使用了海量的数据，还表现在巨大的参数规模，以及动则几十甚至上百层的深度网络结构。具体来说，GPT-3 模型里的 Transformer 模块已经达到 96 层，更深的模型层数更多。Transformer 模块堆得越多，训练越困难。梯度在反向传播时需要穿透几十、上百层，每经过一层就被“稀释”一点，到了最后几层几乎消失殆尽。这就是所谓的梯度消失（Vanishing Gradient）问题。

残差连接¶

2015 年，微软研究院的何恺明等人提出了残差连接（Residual Connection）的概念，极大缓解了深层神经网络的训练难题。这篇论文（ResNet）是深度学习历史上被引用最多的论文之一。

残差连接的核心思路极其简单：让每一层的输出 = 输入 + 变换。

\text{output} = x + f(x)

(1)

而不再是传统的： $\text{output} = f(x)$ 。

残差连接为什么有效¶

前向传播：即使 $f(x)$ 学得不好，输出至少还有 $x$ ，信息不会丢失；
反向传播：梯度可以通过 $x$ 回传，不需要完全依赖 $f(x)$ 的梯度，梯度消失问题大大缓解。

残差连接就像给梯度多修了一条高速公路：不管中间的 $f(x)$ 多么曲折，梯度总能畅通无阻地传回前面的层。

残差连接的位置¶

GPT 模型里，网络深度主要由 Transformer 模块构成。Transformer 模块包含两个子层：自注意力和前馈网络。每个子层都需要加上残差连接：

自注意力： $x = x + \text{attention}(x)$ ；
前馈网络： $x = x + \text{feedforward}(x)$ 。

这样，不管堆叠多少层 Transformer 模块，梯度都可以顺畅地传达到第一层，避免梯度消失。

加入残差连接的 GPT 基础模块示意图：注意力层和前馈网络层各自有一条绕过该层的残差连接

图例：加入残差连接的 GPT 基础模块

import numpy as np

from src.core import *

np.random.seed(42)

数据集¶

字符编码数据集¶

class CharDataset(Dataset):

    def __init__(self, filename, batch_size=1, context_size=64, stride=None, split=0.9):
        self.filename = filename
        self.context_size = context_size
        self.stride = stride if stride is not None else context_size // 2
        self.split = split
        super().__init__(batch_size)

    def load(self):
        with open(self.filename, encoding="utf-8") as f:
            text = f.read()

        self.vocab = sorted(set(text))
        self.vocab_size = len(self.vocab)
        self.stoi = {ch: i for i, ch in enumerate(self.vocab)}
        self.itos = {i: ch for i, ch in enumerate(self.vocab)}
        self.tokens = self.encode(text)

        split = int(len(self.tokens) * self.split)
        self.train_data = self._pack(self.tokens[:split])
        self.test_data = self._pack(self.tokens[split:])

    def _pack(self, tokens):
        x, y = [], []
        for i in range(0, len(tokens) - self.context_size - 1, self.stride):
            x.append(tokens[i: i + self.context_size])
            y.append(tokens[i + 1: i + self.context_size + 1])
        return x, y

    def encode(self, symbols):
        return [self.stoi[s] for s in symbols]

    def decode(self, tokens):
        return "".join(self.itos[t] for t in tokens)

下三角矩阵层¶

class Tril(Layer):

    def __init__(self, value=-1e9):
        super().__init__()
        self.value = value

    def forward(self, x: Tensor):
        keep = np.tril(np.ones(x.shape[-2:]))
        p = Tensor(np.where(keep, x.data, self.value))

        def gradient_fn():
            x.grad += p.grad * keep

        return p.attach(gradient_fn, {x})

激活函数¶

高斯误差线性单元激活函数（GELU）¶

class GELU(Layer):

    def __init__(self):
        super().__init__()
        self.c = np.sqrt(2.0 / np.pi)

    def forward(self, x: Tensor):
        tanh = np.tanh(self.c * (x.data + 0.044715 * x.data ** 3))
        a = Tensor(0.5 * x.data * (1.0 + tanh))

        def gradient_fn():
            grad = 0.5 * (1.0 + tanh) + 0.5 * x.data * (1.0 - tanh ** 2) * self.c * (1.0 + 3.0 * 0.044715 * x.data ** 2)
            x.grad += a.grad * grad

        return a.attach(gradient_fn, parents=[x])

损失函数¶

GPT 交叉熵损失函数¶

class GPTLoss(Loss):

    def loss(self, p: Tensor, y: Tensor):
        exp = np.exp(p.data - np.max(p.data, axis=-1, keepdims=True))
        softmax = exp / np.sum(exp, axis=-1, keepdims=True)

        flat_softmax = softmax.reshape(-1, softmax.shape[-1])
        flat_y = y.data.reshape(-1).astype(np.int64)
        n = len(flat_y)
        rows = np.arange(n)

        log = np.log(np.clip(flat_softmax[rows, flat_y], 1e-10, 1))
        ce = Tensor(0 - np.sum(log) / n)

        def gradient_fn():
            flat_grad = flat_softmax.copy()
            flat_grad[rows, flat_y] -= 1
            p.grad += ce.grad * flat_grad.reshape(softmax.shape) / n

        return ce.attach(gradient_fn, parents=[p])

模型¶

GPT 嵌入层¶

class GPTEmbedding(Composite):

    def __init__(self, vocab_size, context_size, embedding_size):
        self.embedding = Embedding(vocab_size, embedding_size)
        self.positional_embedding = Embedding(context_size, embedding_size)

        super().__init__([self.embedding,
                          self.positional_embedding])

    def forward(self, x: Tensor):
        token = self.embedding(x)
        position = self.positional_embedding(Tensor(range(x.shape[1])))
        return token + position

GPT 注意力层¶

我们在 GPT 注意力层的最后加上残差连接。将输出从 self.output(...) 变成 x + self.output(...)。

class GPTAttention(Composite):

    def __init__(self, embedding_size, heads=1):
        self.embedding_size = embedding_size
        self.heads = heads

        self.query = Linear(embedding_size, embedding_size)
        self.key = Linear(embedding_size, embedding_size)
        self.value = Linear(embedding_size, embedding_size)
        self.mask = Tril()
        self.softmax = Softmax()
        self.output = Linear(embedding_size, embedding_size)

        super().__init__([self.query,
                          self.key,
                          self.value,
                          self.mask,
                          self.softmax,
                          self.output])

    def forward(self, x: Tensor):
        head_embedding_size = self.embedding_size // self.heads
        multi_head_shape = (-1, x.shape[1], self.heads, head_embedding_size)
        shape = (-1, x.shape[1], self.embedding_size)

        query = self.query(x).reshape(multi_head_shape).transpose((0, 2, 1, 3))
        key = self.key(x).reshape(multi_head_shape).transpose((0, 2, 3, 1))
        value = self.value(x).reshape(multi_head_shape).transpose((0, 2, 1, 3))
        scale = Tensor(np.array(1.0 / np.sqrt(head_embedding_size)))
        scores = query @ key * scale
        weights = self.softmax(self.mask(scores))
        # 残差连接
        return x + self.output((weights @ value).transpose((0, 2, 1, 3)).reshape(shape))

GPT 前馈网络层¶

同样，我们在 GPT 前馈网络层的最后也加上残差连接。将输出从 self.output(...) 变成 x + self.output(...)。

class GPTFeedForward(Composite):

    def __init__(self, embedding_size):
        self.input = Linear(embedding_size, embedding_size * 4)
        self.gelu = GELU()
        self.output = Linear(embedding_size * 4, embedding_size)

        super().__init__([self.input,
                          self.gelu,
                          self.output])

    def forward(self, x: Tensor):
        h = self.gelu(self.input(x))
        # 残差连接
        return x + self.output(h)

GPT Transformer 模块¶

class GPTTransformer(Composite):

    def __init__(self, embedding_size, heads):
        self.attention = GPTAttention(embedding_size, heads)
        self.feed_forward = GPTFeedForward(embedding_size)

        super().__init__([self.attention,
                          self.feed_forward])

    def forward(self, x: Tensor):
        x = self.attention(x)
        return self.feed_forward(x)

GPT 输出层¶

class GPTOutput(Composite):

    def __init__(self, embedding_size, vocab_size):
        self.output = Linear(embedding_size, vocab_size)

        super().__init__([self.output])

    def forward(self, x: Tensor):
        return self.output(x)

GPT 模型¶

class GPT(Composite):

    def __init__(self, vocab_size, context_size, embedding_size, heads, blocks):
        self.embedding = GPTEmbedding(vocab_size, context_size, embedding_size)
        self.transformers = [GPTTransformer(embedding_size, heads) for _ in range(blocks)]
        self.output = GPTOutput(embedding_size, vocab_size)

        super().__init__([self.embedding] + self.transformers + [self.output])

    def forward(self, x: Tensor):
        x = self.embedding(x)
        for layer in self.transformers:
            x = layer(x)
        return self.output(x)

GPT 训练器¶

class GPTTrainer(Trainer):

    def train(self, dataset, epochs, scheduler=None):
        dataset.train()
        self.model.train()

        steps = 0
        for epoch in range(epochs):
            for i in range(len(dataset)):
                if scheduler is not None:
                    self.optimizer.lr = scheduler.step(steps)

                feature, label = dataset[i]

                self.optimizer.zero_grad()
                prediction = self.model(feature)
                loss = self.loss_fn(prediction, label)
                loss.backward()
                self.optimizer.clip_grad_norm()
                self.optimizer.step()
                steps += 1

    def test(self, dataset):
        dataset.eval()
        self.model.eval()

        predictions = []
        total_loss = 0.0
        with Tensor.no_grad():
            for i in range(len(dataset)):
                feature, label = dataset[i]
                prediction = self.model(feature)
                loss = self.loss_fn(prediction, label)
                predictions.append(prediction)
                total_loss += float(loss.data)

        loss = total_loss / len(dataset)
        dataset.train()
        self.model.train()
        return predictions, loss

    def generate(self, dataset, prompt, steps=256):
        self.model.eval()
        tokens = dataset.encode(prompt)

        with Tensor.no_grad():
            for _ in range(steps):
                feature = Tensor([tokens[-dataset.context_size:]])
                logits = self.model(feature)

                last_logits = logits.data[0, -1]
                exp = np.exp(last_logits - np.max(last_logits))
                probs = exp / np.sum(exp)
                token = np.random.choice(len(probs), p=probs)
                tokens.append(token)

        return dataset.decode(tokens)

超参数¶

学习率¶

LEARNING_RATE = 0.001

批大小¶

BATCH_SIZE = 4

轮数¶

EPOCHS = 2

上下文尺寸¶

CONTEXT_SIZE = 32

嵌入尺寸¶

EMBEDDING_SIZE = 64

头数¶

HEADS = 2

Transformer 模块数¶

BLOCKS = 2

建模¶

DATA_FILE = "../../../../data/tinyshakespeare.txt"

dataset = CharDataset(DATA_FILE, BATCH_SIZE, CONTEXT_SIZE)
model = GPT(dataset.vocab_size, CONTEXT_SIZE, EMBEDDING_SIZE, HEADS, BLOCKS)
loss_fn = GPTLoss()
optimizer = AdamWOptimizer(model.parameters, lr=LEARNING_RATE)
trainer = GPTTrainer(model, loss_fn, optimizer)

训练¶

scheduler = WarmupCosineScheduler(LEARNING_RATE, EPOCHS * len(dataset), 100, LEARNING_RATE / 10)
trainer.train(dataset, EPOCHS, scheduler)

评估¶

prediction, loss = trainer.test(dataset)
print(f'prediction:\t{len(prediction)} steps, each {prediction[0].data.shape}\nloss:\t{loss}')

prediction:	1742 steps, each (4, 32, 65)
loss:	1.8085668785700706

推理¶

print(trainer.generate(dataset, prompt="ROMEO:"))

ROMEO:
Well, I'll poisen, swearn thee, haprite,
Will me is is fatherior offenretcher rangershed is in tet Catie, there name the king,
Lets Comiou, who no: well I delieve what bow,
And
Berancionr her; and love stime and in him esceasured,
That try love.

SIY:
HtU

在这一章，我们给 Transformer 模块的两个子层都加上了残差连接。梯度有了直达通道，深度神经网络的一个训练难题，梯度消失，迎刃而解。

但是残差连接也有一个副作用：每一层的输出都是“输入 + 变换”，随着层数增加，数值会不断累积，越来越大，导致训练不稳定。下一章，我们将着手解决这个问题。

课后练习¶

Transformer 模块的两个子层都加上了残差连接。试试去掉其中一个，看看训练结果如何？