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init
Browse files- scaling_laws.ipynb +0 -0
- transformer_sizing.ipynb +402 -0
scaling_laws.ipynb
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transformer_sizing.ipynb
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{
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"cells": [
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{
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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Transformer Theoretical Model\n",
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"\n",
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"This notebook stores a bunch of analysis about a Transformer, e.g. estimates the number of FLOPs, parameters, peak memory footprint, checkpoint size, etc."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"from collections import OrderedDict"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"# config_args = {\n",
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"# 'gpt2': dict(n_layer=12, n_head=12, n_embd=768), # 124M params\n",
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"# 'gpt2-medium': dict(n_layer=24, n_head=16, n_embd=1024), # 350M params\n",
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"# 'gpt2-large': dict(n_layer=36, n_head=20, n_embd=1280), # 774M params\n",
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32 |
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"# 'gpt2-xl': dict(n_layer=48, n_head=25, n_embd=1600), # 1558M params\n",
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"# }[model_type]\n",
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"\n",
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"block_size = 1024\n",
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"vocab_size = 50257\n",
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"n_layer = 12\n",
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"n_head = 12\n",
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"n_embd = 768\n",
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"bias = False\n",
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41 |
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"assert not bias, \"this notebook assumes bias=False just for simplicity\""
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]
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43 |
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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48 |
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"we see: 124337664, expected: 124337664, match: True\n",
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54 |
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"name params ratio (%) \n",
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55 |
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"emebedding/position 786432 0.6325\n",
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56 |
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"embedding/token 38597376 31.0424\n",
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57 |
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"embedding 39383808 31.6749\n",
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58 |
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"attention/ln 768 0.0006\n",
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59 |
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"attention/kqv 1769472 1.4231\n",
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60 |
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"attention/proj 589824 0.4744\n",
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61 |
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"attention 2360064 1.8981\n",
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62 |
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"mlp/ln 768 0.0006\n",
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63 |
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"mlp/ffw 2359296 1.8975\n",
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64 |
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"mlp/proj 2359296 1.8975\n",
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65 |
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"mlp 4719360 3.7956\n",
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66 |
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"block 7079424 5.6937\n",
|
67 |
+
"transformer 84953088 68.3245\n",
|
68 |
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"ln_f 768 0.0006\n",
|
69 |
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"dense 0 0.0000\n",
|
70 |
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"total 124337664 100.0000\n"
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71 |
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]
|
72 |
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}
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73 |
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],
|
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"source": [
|
75 |
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"def params():\n",
|
76 |
+
" \"\"\" estimates the number of parameters in the model\"\"\"\n",
|
77 |
+
" out = OrderedDict()\n",
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+
"\n",
|
79 |
+
" # token and position embeddings\n",
|
80 |
+
" out['emebedding/position'] = n_embd * block_size\n",
|
81 |
+
" out['embedding/token'] = n_embd * vocab_size\n",
|
82 |
+
" out['embedding'] = out['emebedding/position'] + out['embedding/token']\n",
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83 |
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"\n",
|
84 |
+
" # attention blocks\n",
|
85 |
+
" out['attention/ln'] = n_embd # note, bias=False in our LN\n",
|
86 |
+
" out['attention/kqv'] = n_embd * 3*n_embd\n",
|
87 |
+
" out['attention/proj'] = n_embd**2\n",
|
88 |
+
" out['attention'] = out['attention/ln'] + out['attention/kqv'] + out['attention/proj']\n",
|
89 |
+
"\n",
|
90 |
+
" # MLP blocks\n",
|
91 |
+
" ffw_size = 4*n_embd # feed forward size\n",
|
92 |
+
" out['mlp/ln'] = n_embd\n",
|
93 |
+
" out['mlp/ffw'] = n_embd * ffw_size\n",
|
94 |
+
" out['mlp/proj'] = ffw_size * n_embd\n",
|
95 |
+
" out['mlp'] = out['mlp/ln'] + out['mlp/ffw'] + out['mlp/proj']\n",
|
96 |
+
" \n",
|
97 |
+
" # the transformer and the rest of it\n",
|
98 |
+
" out['block'] = out['attention'] + out['mlp']\n",
|
99 |
+
" out['transformer'] = n_layer * out['block']\n",
|
100 |
+
" out['ln_f'] = n_embd # final layernorm\n",
|
101 |
+
" out['dense'] = 0 # 0 because of parameter sharing. This layer uses the weights from the embedding layer\n",
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102 |
+
"\n",
|
103 |
+
" # total\n",
|
104 |
+
" out['total'] = out['embedding'] + out['transformer'] + out['ln_f'] + out['dense']\n",
|
105 |
+
"\n",
|
106 |
+
" return out\n",
|
107 |
+
"\n",
|
108 |
+
"# compare our param count to that reported by PyTorch\n",
|
109 |
+
"p = params()\n",
|
110 |
+
"params_total = p['total']\n",
|
111 |
+
"print(f\"we see: {params_total}, expected: {124337664}, match: {params_total == 124337664}\")\n",
|
112 |
+
"# create a header\n",
|
113 |
+
"print(f\"{'name':20s} {'params':10s} {'ratio (%)':10s}\")\n",
|
114 |
+
"for k,v in p.items():\n",
|
115 |
+
" print(f\"{k:20s} {v:10d} {v/params_total*100:10.4f}\")\n",
|
116 |
+
" "
|
117 |
+
]
|
118 |
+
},
|
119 |
+
{
|
120 |
+
"cell_type": "code",
|
121 |
+
"execution_count": 4,
|
122 |
+
"metadata": {},
|
123 |
+
"outputs": [
|
124 |
+
{
|
125 |
+
"name": "stdout",
|
126 |
+
"output_type": "stream",
|
127 |
+
"text": [
|
128 |
+
"est checkpoint size: 1.49 GB\n",
|
129 |
+
"measured with wc -c ckpt.pt: 1542470366\n",
|
130 |
+
"fluff ratio: 103.38%\n"
|
131 |
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]
|
132 |
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}
|
133 |
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],
|
134 |
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"source": [
|
135 |
+
"# we can now calculate the size of each checkpoint\n",
|
136 |
+
"# params are stored in fp32, and the AdamW optimizer has 2 additional buffers per param for statistics\n",
|
137 |
+
"params_bytes = params_total*4\n",
|
138 |
+
"params_and_buffers_bytes = params_bytes + 2*params_bytes\n",
|
139 |
+
"print(f\"est checkpoint size: {params_and_buffers_bytes/1e9:.2f} GB\")\n",
|
140 |
+
"measured_bytes = 1542470366 # from wc -c ckpt.pt\n",
|
141 |
+
"print(f\"measured with wc -c ckpt.pt: {measured_bytes}\")\n",
|
142 |
+
"print(f\"fluff ratio: {measured_bytes/params_and_buffers_bytes*100:.2f}%\")"
|
143 |
+
]
|
144 |
+
},
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145 |
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{
|
146 |
+
"attachments": {},
|
147 |
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"cell_type": "markdown",
|
148 |
+
"metadata": {},
|
149 |
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"source": [
|
150 |
+
"We can also estimate the ratio of our GPU memory that will be taken up just by the weights and the buffers inside the AdamW optimizer"
|
151 |
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]
|
152 |
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},
|
153 |
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{
|
154 |
+
"cell_type": "code",
|
155 |
+
"execution_count": 5,
|
156 |
+
"metadata": {},
|
157 |
+
"outputs": [
|
158 |
+
{
|
159 |
+
"name": "stdout",
|
160 |
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"output_type": "stream",
|
161 |
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"text": [
|
162 |
+
"memory ratio taken up just for parameters: 3.73%\n"
|
163 |
+
]
|
164 |
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}
|
165 |
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],
|
166 |
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"source": [
|
167 |
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"gpu_memory = 40e9 # 40 GB A100 GPU, roughly\n",
|
168 |
+
"print(f\"memory ratio taken up just for parameters: {params_and_buffers_bytes / gpu_memory * 100:.2f}%\")"
|
169 |
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]
|
170 |
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},
|
171 |
+
{
|
172 |
+
"attachments": {},
|
173 |
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"cell_type": "markdown",
|
174 |
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"metadata": {},
|
175 |
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"source": [
|
176 |
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"i.e. not that much of the memory for this tiny model, most of the memory is activations (forward and backward). This of course changes dramatically for larger and larger models."
|
177 |
+
]
|
178 |
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},
|
179 |
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{
|
180 |
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"attachments": {},
|
181 |
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"cell_type": "markdown",
|
182 |
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"metadata": {},
|
183 |
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"source": [
|
184 |
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"Let's estimate FLOPs for a single forward pass."
|
185 |
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]
|
186 |
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},
|
187 |
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{
|
188 |
+
"cell_type": "code",
|
189 |
+
"execution_count": 6,
|
190 |
+
"metadata": {},
|
191 |
+
"outputs": [
|
192 |
+
{
|
193 |
+
"name": "stdout",
|
194 |
+
"output_type": "stream",
|
195 |
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"text": [
|
196 |
+
"name flops ratio (%) \n",
|
197 |
+
"attention/kqv 3623878656 1.2426\n",
|
198 |
+
"attention/scores 1610612736 0.5522\n",
|
199 |
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"attention/reduce 1610612736 0.5522\n",
|
200 |
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"attention/proj 1207959552 0.4142\n",
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201 |
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"attention 8053063680 2.7612\n",
|
202 |
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"mlp/ffw1 4831838208 1.6567\n",
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203 |
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"mlp/ffw2 4831838208 1.6567\n",
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204 |
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"mlp 9663676416 3.3135\n",
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205 |
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"block 17716740096 6.0747\n",
|
206 |
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"transformer 212600881152 72.8963\n",
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207 |
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"dense 79047426048 27.1037\n",
|
208 |
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"forward_total 291648307200 100.0000\n",
|
209 |
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"backward_total 583296614400 200.0000\n",
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210 |
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"total 874944921600 300.0000\n"
|
211 |
+
]
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212 |
+
}
|
213 |
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],
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214 |
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"source": [
|
215 |
+
"def flops():\n",
|
216 |
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" # we only count Weight FLOPs, all other layers (LayerNorm, Softmax, etc) are effectively irrelevant\n",
|
217 |
+
" # we count actual FLOPs, not MACs. Hence 2* all over the place\n",
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218 |
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" # basically for any matrix multiply A (BxC) @ B (CxD) -> (BxD) flops are 2*B*C*D\n",
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219 |
+
"\n",
|
220 |
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" out = OrderedDict()\n",
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221 |
+
" head_size = n_embd // n_head\n",
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222 |
+
"\n",
|
223 |
+
" # attention blocks\n",
|
224 |
+
" # 1) the projection to key, query, values\n",
|
225 |
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" out['attention/kqv'] = 2 * block_size * (n_embd * 3*n_embd)\n",
|
226 |
+
" # 2) calculating the attention scores\n",
|
227 |
+
" out['attention/scores'] = 2 * block_size * block_size * n_embd\n",
|
228 |
+
" # 3) the reduction of the values (B, nh, T, T) x (B, nh, T, hs) -> (B, nh, T, hs)\n",
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229 |
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" out['attention/reduce'] = 2 * n_head * (block_size * block_size * head_size)\n",
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230 |
+
" # 4) the final linear projection\n",
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231 |
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" out['attention/proj'] = 2 * block_size * (n_embd * n_embd)\n",
|
232 |
+
" out['attention'] = sum(out['attention/'+k] for k in ['kqv', 'scores', 'reduce', 'proj'])\n",
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233 |
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"\n",
|
234 |
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" # MLP blocks\n",
|
235 |
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" ffw_size = 4*n_embd # feed forward size\n",
|
236 |
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" out['mlp/ffw1'] = 2 * block_size * (n_embd * ffw_size)\n",
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237 |
+
" out['mlp/ffw2'] = 2 * block_size * (ffw_size * n_embd)\n",
|
238 |
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" out['mlp'] = out['mlp/ffw1'] + out['mlp/ffw2']\n",
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239 |
+
"\n",
|
240 |
+
" # the transformer and the rest of it\n",
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241 |
+
" out['block'] = out['attention'] + out['mlp']\n",
|
242 |
+
" out['transformer'] = n_layer * out['block']\n",
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243 |
+
" out['dense'] = 2 * block_size * (n_embd * vocab_size)\n",
|
244 |
+
"\n",
|
245 |
+
" # forward,backward,total\n",
|
246 |
+
" out['forward_total'] = out['transformer'] + out['dense']\n",
|
247 |
+
" out['backward_total'] = 2 * out['forward_total'] # use common estimate of bwd = 2*fwd\n",
|
248 |
+
" out['total'] = out['forward_total'] + out['backward_total']\n",
|
249 |
+
"\n",
|
250 |
+
" return out\n",
|
251 |
+
" \n",
|
252 |
+
"# compare our param count to that reported by PyTorch\n",
|
253 |
+
"f = flops()\n",
|
254 |
+
"flops_total = f['forward_total']\n",
|
255 |
+
"print(f\"{'name':20s} {'flops':14s} {'ratio (%)':10s}\")\n",
|
256 |
+
"for k,v in f.items():\n",
|
257 |
+
" print(f\"{k:20s} {v:14d} {v/flops_total*100:10.4f}\")\n",
|
258 |
+
" "
|
259 |
+
]
|
260 |
+
},
|
261 |
+
{
|
262 |
+
"cell_type": "code",
|
263 |
+
"execution_count": 7,
|
264 |
+
"metadata": {},
|
265 |
+
"outputs": [
|
266 |
+
{
|
267 |
+
"name": "stdout",
|
268 |
+
"output_type": "stream",
|
269 |
+
"text": [
|
270 |
+
"palm_flops: 875062886400, flops: 874944921600, ratio: 1.0001\n"
|
271 |
+
]
|
272 |
+
}
|
273 |
+
],
|
274 |
+
"source": [
|
275 |
+
"# now here is an estimate copy pasted from the PaLM paper\n",
|
276 |
+
"# this formula is often used to calculate MFU (model flops utilization)\n",
|
277 |
+
"def palm_flops():\n",
|
278 |
+
" \"\"\"estimate of the model flops following PaLM paper formula\"\"\"\n",
|
279 |
+
" # non-embedding model parameters. note that we do not subtract the\n",
|
280 |
+
" # embedding/token params because those are tied and get used in the last layer.\n",
|
281 |
+
" N = params()['total'] - params()['emebedding/position']\n",
|
282 |
+
" L, H, Q, T = n_layer, n_head, n_embd//n_head, block_size\n",
|
283 |
+
" mf_per_token = 6*N + 12*L*H*Q*T\n",
|
284 |
+
" mf = mf_per_token * block_size\n",
|
285 |
+
" return mf\n",
|
286 |
+
"\n",
|
287 |
+
"print(f\"palm_flops: {palm_flops():d}, flops: {flops()['total']:d}, ratio: {palm_flops()/flops()['total']:.4f}\")"
|
288 |
+
]
|
289 |
+
},
|
290 |
+
{
|
291 |
+
"attachments": {},
|
292 |
+
"cell_type": "markdown",
|
293 |
+
"metadata": {},
|
294 |
+
"source": [
|
295 |
+
"Ok they are quite similar, giving some confidence that my math in flops() function was ~ok. Now, A100 is cited at 312TFLOPS bfloat16 on tensor cores. So what is our model flops utilization (MFU)? I trained the model above with a batch_size of 20 and grad_accum of 5, which runs in about 755ms on a single A100 GPU. We get:"
|
296 |
+
]
|
297 |
+
},
|
298 |
+
{
|
299 |
+
"cell_type": "code",
|
300 |
+
"execution_count": 8,
|
301 |
+
"metadata": {},
|
302 |
+
"outputs": [
|
303 |
+
{
|
304 |
+
"name": "stdout",
|
305 |
+
"output_type": "stream",
|
306 |
+
"text": [
|
307 |
+
"fraction of A100 used: 37.14%\n"
|
308 |
+
]
|
309 |
+
}
|
310 |
+
],
|
311 |
+
"source": [
|
312 |
+
"# here is what we currently roughly measure\n",
|
313 |
+
"batch_size = 20 * 5 # 5 is grad_accum, so total batch size is 100\n",
|
314 |
+
"measured_time = 0.755 # in seconds per iteration\n",
|
315 |
+
"measured_throughput = batch_size / measured_time\n",
|
316 |
+
"flops_achieved = f['total'] * measured_throughput\n",
|
317 |
+
"\n",
|
318 |
+
"# A100 is cited to be 312 TFLOPS of bloat16 running on tensor cores\n",
|
319 |
+
"a100_flops_promised = 312e12\n",
|
320 |
+
"\n",
|
321 |
+
"# the fraction of the A100 that we are using:\n",
|
322 |
+
"print(f\"fraction of A100 used: {flops_achieved / a100_flops_promised * 100:.2f}%\")"
|
323 |
+
]
|
324 |
+
},
|
325 |
+
{
|
326 |
+
"attachments": {},
|
327 |
+
"cell_type": "markdown",
|
328 |
+
"metadata": {},
|
329 |
+
"source": [
|
330 |
+
"For reference, we'd prefer to be somewhere around 50%+, and not just for a single GPU but for an entire DDP run. So we still have some work to do, but at least we're within a factor of ~2X of what is achievable with this GPU."
|
331 |
+
]
|
332 |
+
},
|
333 |
+
{
|
334 |
+
"cell_type": "code",
|
335 |
+
"execution_count": 9,
|
336 |
+
"metadata": {},
|
337 |
+
"outputs": [
|
338 |
+
{
|
339 |
+
"name": "stdout",
|
340 |
+
"output_type": "stream",
|
341 |
+
"text": [
|
342 |
+
"time needed to train the model: 3.46 days\n"
|
343 |
+
]
|
344 |
+
}
|
345 |
+
],
|
346 |
+
"source": [
|
347 |
+
"# Finally let's check out the 6ND approximation as total cost of training in FLOPs\n",
|
348 |
+
"model_size = params()['total'] # this is number of parameters, N\n",
|
349 |
+
"tokens_num = 300e9 # 300B tokens, this is dataset size in tokens, D\n",
|
350 |
+
"a100_flops = 312e12 # 312 TFLOPS\n",
|
351 |
+
"assumed_mfu = 0.3 # assume this model flops utilization (take the current 37% from above and add some DDP overhead)\n",
|
352 |
+
"flops_throughput = a100_flops * 8 * assumed_mfu # assume an 8XA100 node at 30% utilization\n",
|
353 |
+
"flops_needed = 6 * model_size * tokens_num # 6ND\n",
|
354 |
+
"time_needed_s = flops_needed / flops_throughput # in seconds\n",
|
355 |
+
"print(f\"time needed to train the model: {time_needed_s/3600/24:.2f} days\")"
|
356 |
+
]
|
357 |
+
},
|
358 |
+
{
|
359 |
+
"attachments": {},
|
360 |
+
"cell_type": "markdown",
|
361 |
+
"metadata": {},
|
362 |
+
"source": [
|
363 |
+
"This is not a bad estimate at all. I trained this model and it converged in roughly 4 days. Btw as a good reference for where 6ND comes from and some intuition around it I recommend [Dzmitry's post](https://medium.com/@dzmitrybahdanau/the-flops-calculus-of-language-model-training-3b19c1f025e4)."
|
364 |
+
]
|
365 |
+
},
|
366 |
+
{
|
367 |
+
"attachments": {},
|
368 |
+
"cell_type": "markdown",
|
369 |
+
"metadata": {},
|
370 |
+
"source": [
|
371 |
+
"Now, FLOPs are just one constraint, the other that we have to keep a close track of is the memory bandwidth. TODO estimate LOAD/STORE costs of our model later."
|
372 |
+
]
|
373 |
+
}
|
374 |
+
],
|
375 |
+
"metadata": {
|
376 |
+
"kernelspec": {
|
377 |
+
"display_name": "pytorch2",
|
378 |
+
"language": "python",
|
379 |
+
"name": "python3"
|
380 |
+
},
|
381 |
+
"language_info": {
|
382 |
+
"codemirror_mode": {
|
383 |
+
"name": "ipython",
|
384 |
+
"version": 3
|
385 |
+
},
|
386 |
+
"file_extension": ".py",
|
387 |
+
"mimetype": "text/x-python",
|
388 |
+
"name": "python",
|
389 |
+
"nbconvert_exporter": "python",
|
390 |
+
"pygments_lexer": "ipython3",
|
391 |
+
"version": "3.10.8"
|
392 |
+
},
|
393 |
+
"orig_nbformat": 4,
|
394 |
+
"vscode": {
|
395 |
+
"interpreter": {
|
396 |
+
"hash": "7f5833218766b48e6e35e4452ee875aac0e2188d05bbe5298f2c62b79f08b222"
|
397 |
+
}
|
398 |
+
}
|
399 |
+
},
|
400 |
+
"nbformat": 4,
|
401 |
+
"nbformat_minor": 2
|
402 |
+
}
|