From 80bb41a86287dd1372a9b16fdbf7d3da0115a6fd Mon Sep 17 00:00:00 2001 From: Yan Lin Date: Fri, 12 Jul 2024 16:26:09 +0800 Subject: [PATCH] post llama2.c source code reading notes --- .../posts/llama2_c_src_reading/attention.png | Bin 0 -> 492045 bytes .../llama2_c_src_reading/attention_vis.png | Bin 0 -> 25727 bytes .../posts/llama2_c_src_reading/head_query.png | Bin 0 -> 299043 bytes .../llama2_c_src_reading.md | 520 ++++++++++++++++++ .../posts/llama2_c_src_reading/matrix_dim.png | Bin 0 -> 18934 bytes 5 files changed, 520 insertions(+) create mode 100644 content/posts/llama2_c_src_reading/attention.png create mode 100644 content/posts/llama2_c_src_reading/attention_vis.png create mode 100644 content/posts/llama2_c_src_reading/head_query.png create mode 100644 content/posts/llama2_c_src_reading/llama2_c_src_reading.md create mode 100644 content/posts/llama2_c_src_reading/matrix_dim.png diff --git a/content/posts/llama2_c_src_reading/attention.png 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/dev/null +++ b/content/posts/llama2_c_src_reading/llama2_c_src_reading.md @@ -0,0 +1,520 @@ ++++ +title = 'llama2.c 源码阅读' +date = 2024-07-06T13:50:00+08:00 +author = "Skyan" +tags = ["LLM", "programming"] +ShowToc = true +ShowBreadCrumbs = true ++++ + +## 1. 概述 +前OpenAI著名工程师Andrej Kapathy开源了[llama2.c](https://github.com/karpathy/llama2.c)项目,该项目是llama2模型推理代码的C语言实现,用大概970行C代码实现了LLama2模型的推理算法。整个项目代码简洁高效,值得深度阅读。对掌握大模型推理算法的细节有极大的帮助。 + +## 2. 源码阅读 +### 2.1 基础算法 +RMS归一化公式是: +$$ +o_i = w_i \times x_i \times \frac {1}{\sqrt{\frac{1}{n}\sum_{j=0}^{n-1} x_j^2 + \epsilon}} +$$ +其中,\(\epsilon\) 为防止分母为0的数值。还有RMS因子是对x的归一化,w变量是gain变量,重新缩放标准化后的输入向量。 + +```c +// ---------------------------------------------------------------------------- +// neural net blocks; the dynamics of the Transformer +void rmsnorm(float* o, float* x, float* weight, int size) { + // calculate sum of squares + float ss = 0.0f; + for (int j = 0; j < size; j++) { + ss += x[j] * x[j]; + } + ss /= size; + ss += 1e-5f; + ss = 1.0f / sqrtf(ss); + // normalize and scale + for (int j = 0; j < size; j++) { + o[j] = weight[j] * (ss * x[j]); + } +} +``` + +softmax函数公式是: +$$ +o_i = \frac {e^{x_i-x_{max}}}{\sum_{j=0}^{n-1} e^{x_j-x_{max}}} +$$ + +代码如下,注释说的很清楚,减去最大值是为了防止数值溢出,数值更稳定。通过简单数学变换可以得知,最终结果不变。 +```c +void softmax(float* x, int size) { + // find max value (for numerical stability) + float max_val = x[0]; + for (int i = 1; i < size; i++) { + if (x[i] > max_val) { + max_val = x[i]; + } + } + // exp and sum + float sum = 0.0f; + for (int i = 0; i < size; i++) { + x[i] = expf(x[i] - max_val); + sum += x[i]; + } + // normalize + for (int i = 0; i < size; i++) { + x[i] /= sum; + } +} +``` + +W (d,n) @ x (n,) -> xout (d,)的矩阵乘法,采用naive的矩阵乘法,即外层循环是行,内层循环是列。代码如下: +```c +void matmul(float* xout, float* x, float* w, int n, int d) { + // W (d,n) @ x (n,) -> xout (d,) + // by far the most amount of time is spent inside this little function + int i; + #pragma omp parallel for private(i) + for (i = 0; i < d; i++) { + float val = 0.0f; + for (int j = 0; j < n; j++) { + val += w[i * n + j] * x[j]; + } + xout[i] = val; + } +} +``` +### 2.2. forward计算 + +模型中一个attention block的计算如下图所示: +![attention](attention.png) + +项目代码是按照每一个token来计算QKV的,其中参数dim是transformer的向量维度。l是layer序号。 + +第一步是rmsnorm,即归一化。输入是x (d,),rms权重向量是w->rms_att_weight + l*dim,计算结果输出到s->xb (d,)中。 +```c +// attention rmsnorm +rmsnorm(s->xb, x, w->rms_att_weight + l*dim, dim); +``` + +第二步是QKV的矩阵乘法,注意kv_dim和dim的区别,是为了同时兼容multi head attention和grouped query attention两种算法。如下图所示: +![head_query](head_query.png) + +kv_dim是key和value的总维度,dim是transformer的向量总维度。在multi head attention中,kv_dim = dim。在grouped query attention中,kv_dim = dim * n_kv_heads / n_heads。以图中为例,n_kv_heads = 4, n_heads = 8,则kv_dim = dim / 2。 + +对于各矩阵的维度,以及在MHA、GQA等算法中的关系,参考下图: +![matrix_dim](matrix_dim.png) + + +Q、K、V三个向量计算的详细代码如下,即Wq(d,d) @ xb(d,) -> q(d,),Wk(dkv,d) @ xb(d,) -> k(dkv,), Wv(dkv,d) @ xb(d,) -> v(dkv,) +```c +// key and value point to the kv cache +int loff = l * p->seq_len * kv_dim; // kv cache layer offset for convenience +s->k = s->key_cache + loff + pos * kv_dim; +s->v = s->value_cache + loff + pos * kv_dim; + +// qkv matmuls for this position +matmul(s->q, s->xb, w->wq + l*dim*dim, dim, dim); +matmul(s->k, s->xb, w->wk + l*dim*kv_dim, dim, kv_dim); +matmul(s->v, s->xb, w->wv + l*dim*kv_dim, dim, kv_dim); +``` + +接下来需要给Q和K向量添加RoPE位置编码,按照如下公式计算,其中m就是当前token的序号pos。需要注意的是,llama模型是给每一层的Q和K向量都添加这个编码。 +$$ +\begin{aligned} +\theta_i &= \frac{1}{10000^{2i/hs}}= 10000^{-2i/hs} \\ +Q(i) &=Q(i)\cos (m\theta_i) - Q(i+1)\sin(m\theta_i)\\ +Q(i+1) &=Q(i)\sin (m \theta_i) + Q(i+1)\cos(m\theta_i)\\ +K(i) &=K(i)\cos (m \theta_i) - K(i+1)\sin(m\theta_i)\\ +K(i+1) &=K(i)\sin (m \theta_i) + K(i+1)\cos(m\theta_i)\\ +\end{aligned} +$$ + +详细代码如下,注意在GQA中,K的向量长度小于Q的向量长度,所以在i < kv_dim时,计算Q和K的向量。在i >= kv_dim时,只计算Q的向量。 + +```c +// RoPE relative positional encoding: complex-valued rotate q and k in each head +for (int i = 0; i < dim; i+=2) { + int head_dim = i % head_size; + float freq = 1.0f / powf(10000.0f, head_dim / (float)head_size); + float val = pos * freq; + float fcr = cosf(val); + float fci = sinf(val); + int rotn = i < kv_dim ? 2 : 1; // how many vectors? 2 = q & k, 1 = q only + for (int v = 0; v < rotn; v++) { + float* vec = v == 0 ? s->q : s->k; // the vector to rotate (query or key) + float v0 = vec[i]; + float v1 = vec[i+1]; + vec[i] = v0 * fcr - v1 * fci; + vec[i+1] = v0 * fci + v1 * fcr; + } +} +``` + +接下来针对每个头,计算attention score。attention score的计算公式如下: +$$ +score(i) = softmax(\frac{ Q_i K^T}{\sqrt{d}})V , \quad Q_i \in \R^{1 \times d},K \in \R^{n\times d},V\in\R^{n\times d} +$$ +具体计算的时候,先遍历每个head,在每个head中,先计算Qi和K的点积,然后除以sqrt(d),得到att (1,n)向量,最后softmax得到attention score。 + +在GQA中,由于分组共享了Q和K的向量,在计算attention score的时候,需要把Q和K的向量“展开”还原为(n,d)的矩阵,具体做法是通过h / kv_mul,保证 +kv_mul个Q和K向量共享一个权重。 + +然后计算attention score (1,n)和V (n,d)的乘积,得到xb (1,d)。这个计算并不是完全按照普通矩阵乘来计算的,而是把每个位置的attention score和V的 +每一行相乘,然后累加到xb中。这样计算的好处是对cache更加友好,是一种常见的矩阵乘算法。 + +对于每个头,每个token的attention score计算过程的可视化如图所示: +![attention_vis](attention_vis.png) + +图中可以清楚看出,每个token都计算了一遍和其他token的相关度,再进行加权求和得到最终的attention score。 + +具体代码如下: +```c +for (h = 0; h < p->n_heads; h++) { + // get the query vector for this head + float* q = s->q + h * head_size; + // attention scores for this head + float* att = s->att + h * p->seq_len; + // iterate over all timesteps, including the current one + for (int t = 0; t <= pos; t++) { + // get the key vector for this head and at this timestep + float* k = s->key_cache + loff + t * kv_dim + (h / kv_mul) * head_size; + // calculate the attention score as the dot product of q and k + float score = 0.0f; + for (int i = 0; i < head_size; i++) { + score += q[i] * k[i]; + } + score /= sqrtf(head_size); + // save the score to the attention buffer + att[t] = score; + } + + // softmax the scores to get attention weights, from 0..pos inclusively + softmax(att, pos + 1); + + // weighted sum of the values, store back into xb + float* xb = s->xb + h * head_size; + memset(xb, 0, head_size * sizeof(float)); + for (int t = 0; t <= pos; t++) { + // get the value vector for this head and at this timestep + float* v = s->value_cache + loff + t * kv_dim + (h / kv_mul) * head_size; + // get the attention weight for this timestep + float a = att[t]; + // accumulate the weighted value into xb + for (int i = 0; i < head_size; i++) { + xb[i] += a * v[i]; + } + } +} +``` + +从代码中也能看出,为什么需要把K和V的矩阵进行cache。因为对于一个位置的token而言,Q矩阵每次参与计算的只有当前位置的一行,而K和V矩阵,则是每行都需要 +参与计算。最终得到的也是该位置的(1,d)向量作为attention score。因此,为了减少计算量,把K和V矩阵进行cache也是理所当然。 + +接下来的计算就非常简单,注释也非常直观。详细步骤如下: + +1. 计算Wo (d,d) @ xb^T (d,)得到xb2 (d,) +1. 通过残差连接,叠加x (d,)向量:x += xb2 +1. x再经过一个RMSNorm(x),得到xb (d,) +1. 计算hb和hb2:W1(hd, d) @ xb (d,) -> hb1(hd,) , W3(hd, d) @ xb (d,) -> hb2(hd, ) +1. hb经过silu非线性激活函数变换,计算方式为:$$silu(hb) = hb (1/ (1 + e^{-hb}))$$ +1. 然后计算逐位相乘 hb * hb2, 得到hb (hd,) +1. 计算W2(d, hd) @ hb (hd,) -> xb (d,) +1. 最终再通过残差连接,叠加xb向量:x += xb + +```c +// final matmul to get the output of the attention +matmul(s->xb2, s->xb, w->wo + l*dim*dim, dim, dim); + +// residual connection back into x +for (int i = 0; i < dim; i++) { + x[i] += s->xb2[i]; +} + +// ffn rmsnorm +rmsnorm(s->xb, x, w->rms_ffn_weight + l*dim, dim); + +// Now for FFN in PyTorch we have: self.w2(F.silu(self.w1(x)) * self.w3(x)) +// first calculate self.w1(x) and self.w3(x) +matmul(s->hb, s->xb, w->w1 + l*dim*hidden_dim, dim, hidden_dim); +matmul(s->hb2, s->xb, w->w3 + l*dim*hidden_dim, dim, hidden_dim); + +// SwiGLU non-linearity +for (int i = 0; i < hidden_dim; i++) { + float val = s->hb[i]; + // silu(x)=x*σ(x), where σ(x) is the logistic sigmoid + val *= (1.0f / (1.0f + expf(-val))); + // elementwise multiply with w3(x) + val *= s->hb2[i]; + s->hb[i] = val; +} + +// final matmul to get the output of the ffn +matmul(s->xb, s->hb, w->w2 + l*dim*hidden_dim, hidden_dim, dim); + +// residual connection +for (int i = 0; i < dim; i++) { + x[i] += s->xb[i]; +} +``` + +继续每一层的计算,每一层的输入都是x,输出也是x,循环计算。在每一层都算完以后,最后再计算: + +1. RMSNorm(x),把x向量进行归一化。 +1. 计算Wc(dvoc, d) @ x (d,) -> logits (dvoc,),其中dvoc为词典大小。 + +至此,最终得到的logits就是该位置的在token词典中的分类概率。 + +```c +// final rmsnorm +rmsnorm(x, x, w->rms_final_weight, dim); + +// classifier into logits +matmul(s->logits, x, w->wcls, p->dim, p->vocab_size); +return s->logits; +``` + +### 2.3 抽样方法 +拿到logits之后,需要通过抽样来最终确定输出哪个token,常见的抽样方法有greedy(argmax),随机抽样,以及top-p (nucleus) 抽样。 + +#### 2.3.1 Greedy Sampling +Greedy Sampling是直接选择概率最大的token作为输出。代码简单直观,如下: +```c +int sample_argmax(float* probabilities, int n) { + // return the index that has the highest probability + int max_i = 0; + float max_p = probabilities[0]; + for (int i = 1; i < n; i++) { + if (probabilities[i] > max_p) { + max_i = i; + max_p = probabilities[i]; + } + } + return max_i; +} +``` + +#### 2.3.2 Random Sampling +Random Sampling是随机选择一个token作为输出。代码也很简单,如下: +```c +int sample_mult(float* probabilities, int n, float coin) { + // sample index from probabilities (they must sum to 1!) + // coin is a random number in [0, 1), usually from random_f32() + float cdf = 0.0f; + for (int i = 0; i < n; i++) { + cdf += probabilities[i]; + if (coin < cdf) { + return i; + } + } + return n - 1; // in case of rounding errors +} +``` + +#### 2.3.3 Top-p (Nucleus) Sampling +Top-p (Nucleus) Sampling是随机选择概率大于某个阈值的token作为输出。代码也很简单,如下: +```c +int sample_topp(float* probabilities, int n, float topp, ProbIndex* probindex, float coin) { + // top-p sampling (or "nucleus sampling") samples from the smallest set of + // tokens that exceed probability topp. This way we never sample tokens that + // have very low probabilities and are less likely to go "off the rails". + // coin is a random number in [0, 1), usually from random_f32() + + int n0 = 0; + // quicksort indices in descending order of probabilities + // values smaller than (1 - topp) / (n - 1) cannot be part of the result + // so for efficiency we crop these out as candidates before sorting + const float cutoff = (1.0f - topp) / (n - 1); + for (int i = 0; i < n; i++) { + if (probabilities[i] >= cutoff) { + probindex[n0].index = i; + probindex[n0].prob = probabilities[i]; + n0++; + } + } + qsort(probindex, n0, sizeof(ProbIndex), compare); + + // truncate the list where cumulative probability exceeds topp + float cumulative_prob = 0.0f; + int last_idx = n0 - 1; // in case of rounding errors consider all elements + for (int i = 0; i < n0; i++) { + cumulative_prob += probindex[i].prob; + if (cumulative_prob > topp) { + last_idx = i; + break; // we've exceeded topp by including last_idx + } + } + + // sample from the truncated list + float r = coin * cumulative_prob; + float cdf = 0.0f; + for (int i = 0; i <= last_idx; i++) { + cdf += probindex[i].prob; + if (r < cdf) { + return probindex[i].index; + } + } + return probindex[last_idx].index; // in case of rounding errors +} +``` + +#### 2.3.4 选择抽样策略 +具体执行抽样前,需要做一些变换,比如: +* 除以temperature,用来调整概率分布,温度越高,概率分布越平滑 +* 计算softmax(logits),得到概率分布 +代码如下所示: + +```c +// apply the temperature to the logits +for (int q=0; qvocab_size; q++) { logits[q] /= sampler->temperature; } +// apply softmax to the logits to get the probabilities for next token +softmax(logits, sampler->vocab_size); +``` + +然后根据不同的采样策略,选择不同的采样函数。 + +### 2.4 encode和decode +#### 2.4.1 encode +encode函数将输入文本转化为token id序列。token id为int类型,长度为max_len。encode算法非常直观,先是在tokenize词典中查询每个UTF-8字符。如果找不到,则将文本编码为byte fallback。注意每个UTF-8字符长度是1到3个字节之间,需要针对UTF-8编码的规范进行判断。 + +代码如下: +```c +// process the raw (UTF-8) byte sequence of the input string +for (char *c = text; *c != '\0'; c++) { + + // reset buffer if the current byte is ASCII or a leading byte + // 0xC0 is 11000000, so (*c & 0xC0) keeps the first 2 bits and zeros the rest + // 0x80 is 10000000 + // in UTF-8, all continuation bytes start with "10" in first two bits + // so in English this is: "if this byte is not a continuation byte" + if ((*c & 0xC0) != 0x80) { + // this byte must be either a leading byte (11...) or an ASCII char (0x...) + // => reset our location, as we're starting a new UTF-8 codepoint + str_len = 0; + } + + // append the current byte to the buffer + str_buffer[str_len++] = *c; // ++ is post-increment, incremented after this line + str_buffer[str_len] = '\0'; + + // while the next character is a continuation byte, continue appending + // but if there are too many of them, just stop to avoid overruning str_buffer size. + if ((*(c+1) & 0xC0) == 0x80 && str_len < 4) { + continue; + } + + // ok c+1 is not a continuation byte, so we've read in a full codepoint + int id = str_lookup(str_buffer, t->sorted_vocab, t->vocab_size); + + if (id != -1) { + // we found this codepoint in vocab, add it as a token + tokens[(*n_tokens)++] = id; + } else { + // byte_fallback encoding: just encode each byte as a token + // +3 is here because the first 3 vocab elements are , , + // so the individual bytes only start at index 3 + for (int i=0; i < str_len; i++) { + tokens[(*n_tokens)++] = (unsigned char)str_buffer[i] + 3; + } + } + str_len = 0; // protect against a sequence of stray UTF8 continuation bytes +} +``` + +其次,尝试合并临近的字符,并查询tokenize词典,如果存在,则将临近的token缩对应的字符串合并为一个token。 +并反复迭代,直到找不到相邻的两个token可以合并为一个token为止。代码也很直观,如下: + +```c +// merge the best consecutive pair each iteration, according the scores in vocab_scores +while (1) { + float best_score = -1e10; + int best_id = -1; + int best_idx = -1; + + for (int i=0; i < (*n_tokens-1); i++) { + // check if we can merge the pair (tokens[i], tokens[i+1]) + sprintf(str_buffer, "%s%s", t->vocab[tokens[i]], t->vocab[tokens[i+1]]); + int id = str_lookup(str_buffer, t->sorted_vocab, t->vocab_size); + if (id != -1 && t->vocab_scores[id] > best_score) { + // this merge pair exists in vocab! record its score and position + best_score = t->vocab_scores[id]; + best_id = id; + best_idx = i; + } + } + + if (best_idx == -1) { + break; // we couldn't find any more pairs to merge, so we're done + } + + // merge the consecutive pair (best_idx, best_idx+1) into new token best_id + tokens[best_idx] = best_id; + // delete token at position best_idx+1, shift the entire sequence back 1 + for (int i = best_idx+1; i < (*n_tokens-1); i++) { + tokens[i] = tokens[i+1]; + } + (*n_tokens)--; // token length decreased +} +``` + +#### 2.4.2 decode +decode函数将token id序列转化为文本。代码也直观,有一些比较tricky之处,代码也注释清楚: +```c +char* decode(Tokenizer* t, int prev_token, int token) { + char *piece = t->vocab[token]; + // following BOS (1) token, sentencepiece decoder strips any leading whitespace (see PR #89) + if (prev_token == 1 && piece[0] == ' ') { piece++; } + // careful, some tokens designate raw bytes, and look like e.g. '<0x01>' + // parse this and convert and return the actual byte + unsigned char byte_val; + if (sscanf(piece, "<0x%02hhX>", &byte_val) == 1) { + piece = (char*)t->byte_pieces + byte_val * 2; + } + return piece; +} +``` + +### 2.5 文本生成 +文本生成是最基础的inference逻辑,对话也是基于文本生成而实现的。整个代码逻辑也非常简单: +1. 将每一个token id逐个进行forward计算 +2. 判断当前token位置是否还在prompt长度内,如果不在则执行sampling策略,通过logits向量选取下一个token +3. 否则直接从prompt中读取下一个token。 +4. 将下一个token进行decode,并打印出来。 + +代码详见: +```c +while (pos < steps) { + + // forward the transformer to get logits for the next token + float* logits = forward(transformer, token, pos); + + // advance the state machine + if (pos < num_prompt_tokens - 1) { + // if we are still processing the input prompt, force the next prompt token + next = prompt_tokens[pos + 1]; + } else { + // otherwise sample the next token from the logits + next = sample(sampler, logits); + } + pos++; + + // data-dependent terminating condition: the BOS (=1) token delimits sequences + if (next == 1) { break; } + + // print the token as string, decode it with the Tokenizer object + char* piece = decode(tokenizer, token, next); + safe_printf(piece); // same as printf("%s", piece), but skips "unsafe" bytes + fflush(stdout); + token = next; + + // init the timer here because the first iteration can be slower + if (start == 0) { start = time_in_ms(); } +} +``` + +### 2.6 其他 +其他部分的代码就是一些简单的数据结构定义,以及helper函数和main函数,这里就不再赘述了。 + +## 3. 总结 +总体来说,这个项目是一个toy项目,代码逻辑比较简单,但是也提供了非常多的细节参考。特别是兼容了MHA和GQA算法,对于理解这些算法的原理非常有帮助。 + +但也要看出,这个代码中并没有实现prefill阶段,而是采用逐个token输入的方式填充kv cache。效率的确比较低,但好在逻辑清晰,容易理解。 + +如果需要进一步优化这个代码,其实有很多可优化点,例如prefill的并行加载优化,减少重复decode等,但这些都超出了这个项目的范围,留给读者自己探索。 diff --git a/content/posts/llama2_c_src_reading/matrix_dim.png b/content/posts/llama2_c_src_reading/matrix_dim.png new file mode 100644 index 0000000000000000000000000000000000000000..2d8ef1f7b81d1ccbaec25a001ef834c1e68ea879 GIT binary patch literal 18934 zcmeJFWmHvN_&y55rV*qRDGA9$>jsN?lbB7n=+l2?+^TLH>yb5)won z2??126CLcy#B?D6{~)_-$Vnkpf2G(0|9N7ir(mt3g2W7d#zaDnvO_{e90LB3fj>w{ zDEY`pDBxdY#J+sU-`_&y^HKi(jLd*I(5GEs4GBpCN#Tj4mJjk?E>@1p+U=)q68mVV zFb|H^0P!_+rv84AB?p5NiN4}weadH@W_iUEsffD5M^fU#iA998m(D*vO?$W99{GAW z^GtQ}cnf35Zt0nfQMfkbv-` zmQv3ktw8_Y0|`rA^2hD}Tn}p+KR?V8C8Fei?uuvx%K!ZL|CNm}jfX9?d#_Da8J@iL zi9?l#MSy9@DvtZ)q3mUZ3Ee224q4^GKiVvmY84C1Q-6j?sGV={1mS? 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