diff --git a/transformer_sizing.ipynb b/transformer_sizing.ipynb new file mode 100644 index 0000000000..5fb60f882d --- /dev/null +++ b/transformer_sizing.ipynb @@ -0,0 +1,400 @@ +{ + "cells": [ + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Transformer Theoretical Model\n", + "\n", + "This notebook stores a bunch of analysis about a Transformer, e.g. estimates the number of FLOPs, parameters, peak memory footprint, checkpoint size, etc." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from collections import OrderedDict" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "# config_args = {\n", + "# 'gpt2': dict(n_layer=12, n_head=12, n_embd=768), # 124M params\n", + "# 'gpt2-medium': dict(n_layer=24, n_head=16, n_embd=1024), # 350M params\n", + "# 'gpt2-large': dict(n_layer=36, n_head=20, n_embd=1280), # 774M params\n", + "# 'gpt2-xl': dict(n_layer=48, n_head=25, n_embd=1600), # 1558M params\n", + "# }[model_type]\n", + "\n", + "block_size = 1024\n", + "vocab_size = 50257\n", + "n_layer = 12\n", + "n_head = 12\n", + "n_embd = 768\n", + "bias = False\n", + "assert not bias, \"this notebook assumes bias=False just for simplicity\"" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "we see: 124337664, expected: 124337664, match: True\n", + "name params ratio (%) \n", + "emebedding/position 786432 0.6325\n", + "embedding/token 38597376 31.0424\n", + "embedding 39383808 31.6749\n", + "attention/ln 768 0.0006\n", + "attention/kqv 1769472 1.4231\n", + "attention/proj 589824 0.4744\n", + "attention 2360064 1.8981\n", + "mlp/ln 768 0.0006\n", + "mlp/ffw 2359296 1.8975\n", + "mlp/proj 2359296 1.8975\n", + "mlp 4719360 3.7956\n", + "block 7079424 5.6937\n", + "transformer 84953088 68.3245\n", + "ln_f 768 0.0006\n", + "dense 0 0.0000\n", + "total 124337664 100.0000\n" + ] + } + ], + "source": [ + "def params():\n", + " \"\"\" estimates the number of parameters in the model\"\"\"\n", + " out = OrderedDict()\n", + "\n", + " # token and position embeddings\n", + " out['emebedding/position'] = n_embd * block_size\n", + " out['embedding/token'] = n_embd * vocab_size\n", + " out['embedding'] = out['emebedding/position'] + out['embedding/token']\n", + "\n", + " # attention blocks\n", + " out['attention/ln'] = n_embd # note, bias=False in our LN\n", + " out['attention/kqv'] = n_embd * 3*n_embd\n", + " out['attention/proj'] = n_embd**2\n", + " out['attention'] = out['attention/ln'] + out['attention/kqv'] + out['attention/proj']\n", + "\n", + " # MLP blocks\n", + " ffw_size = 4*n_embd # feed forward size\n", + " out['mlp/ln'] = n_embd\n", + " out['mlp/ffw'] = n_embd * ffw_size\n", + " out['mlp/proj'] = ffw_size * n_embd\n", + " out['mlp'] = out['mlp/ln'] + out['mlp/ffw'] + out['mlp/proj']\n", + " \n", + " # the transformer and the rest of it\n", + " out['block'] = out['attention'] + out['mlp']\n", + " out['transformer'] = n_layer * out['block']\n", + " out['ln_f'] = n_embd # final layernorm\n", + " out['dense'] = 0 # 0 because of parameter sharing. This layer uses the weights from the embedding layer\n", + "\n", + " # total\n", + " out['total'] = out['embedding'] + out['transformer'] + out['ln_f'] + out['dense']\n", + "\n", + " return out\n", + "\n", + "# compare our param count to that reported by PyTorch\n", + "p = params()\n", + "params_total = p['total']\n", + "print(f\"we see: {params_total}, expected: {124337664}, match: {params_total == 124337664}\")\n", + "# create a header\n", + "print(f\"{'name':20s} {'params':10s} {'ratio (%)':10s}\")\n", + "for k,v in p.items():\n", + " print(f\"{k:20s} {v:10d} {v/params_total*100:10.4f}\")\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "est checkpoint size: 1.49 GB\n", + "measured with wc -c ckpt.pt: 1542470366\n", + "fluff ratio: 103.38%\n" + ] + } + ], + "source": [ + "# we can now calculate the size of each checkpoint\n", + "# params are stored in fp32, and the AdamW optimizer has 2 additional buffers per param for statistics\n", + "params_bytes = params_total*4\n", + "params_and_buffers_bytes = params_bytes + 2*params_bytes\n", + "print(f\"est checkpoint size: {params_and_buffers_bytes/1e9:.2f} GB\")\n", + "measured_bytes = 1542470366 # from wc -c ckpt.pt\n", + "print(f\"measured with wc -c ckpt.pt: {measured_bytes}\")\n", + "print(f\"fluff ratio: {measured_bytes/params_and_buffers_bytes*100:.2f}%\")" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "memory ratio taken up just for parameters: 3.73%\n" + ] + } + ], + "source": [ + "gpu_memory = 40e9 # 40 GB A100 GPU, roughly\n", + "print(f\"memory ratio taken up just for parameters: {params_and_buffers_bytes / gpu_memory * 100:.2f}%\")" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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." + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's estimate FLOPs for a single forward pass." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "name flops ratio (%) \n", + "attention/kqv 3623878656 1.2426\n", + "attention/scores 1610612736 0.5522\n", + "attention/reduce 1610612736 0.5522\n", + "attention/proj 1207959552 0.4142\n", + "attention 8053063680 2.7612\n", + "mlp/ffw1 4831838208 1.6567\n", + "mlp/ffw2 4831838208 1.6567\n", + "mlp 9663676416 3.3135\n", + "block 17716740096 6.0747\n", + "transformer 212600881152 72.8963\n", + "dense 79047426048 27.1037\n", + "forward_total 291648307200 100.0000\n", + "backward_total 583296614400 200.0000\n", + "total 874944921600 300.0000\n" + ] + } + ], + "source": [ + "def flops():\n", + " # we only count Weight FLOPs, all other layers (LayerNorm, Softmax, etc) are effectively irrelevant\n", + " # we count actual FLOPs, not MACs. Hence 2* all over the place\n", + " # basically for any matrix multiply A (BxC) @ B (CxD) -> (BxD) flops are 2*B*C*D\n", + "\n", + " out = OrderedDict()\n", + " head_size = n_embd // n_head\n", + "\n", + " # attention blocks\n", + " # 1) the projection to key, query, values\n", + " out['attention/kqv'] = 2 * block_size * (n_embd * 3*n_embd)\n", + " # 2) calculating the attention scores\n", + " out['attention/scores'] = 2 * block_size * block_size * n_embd\n", + " # 3) the reduction of the values (B, nh, T, T) x (B, nh, T, hs) -> (B, nh, T, hs)\n", + " out['attention/reduce'] = 2 * n_head * (block_size * block_size * head_size)\n", + " # 4) the final linear projection\n", + " out['attention/proj'] = 2 * block_size * (n_embd * n_embd)\n", + " out['attention'] = sum(out['attention/'+k] for k in ['kqv', 'scores', 'reduce', 'proj'])\n", + "\n", + " # MLP blocks\n", + " ffw_size = 4*n_embd # feed forward size\n", + " out['mlp/ffw1'] = 2 * block_size * (n_embd * ffw_size)\n", + " out['mlp/ffw2'] = 2 * block_size * (ffw_size * n_embd)\n", + " out['mlp'] = out['mlp/ffw1'] + out['mlp/ffw2']\n", + "\n", + " # the transformer and the rest of it\n", + " out['block'] = out['attention'] + out['mlp']\n", + " out['transformer'] = n_layer * out['block']\n", + " out['dense'] = 2 * block_size * (n_embd * vocab_size)\n", + "\n", + " # forward,backward,total\n", + " out['forward_total'] = out['transformer'] + out['dense']\n", + " out['backward_total'] = 2 * out['forward_total'] # use common estimate of bwd = 2*fwd\n", + " out['total'] = out['forward_total'] + out['backward_total']\n", + "\n", + " return out\n", + " \n", + "# compare our param count to that reported by PyTorch\n", + "f = flops()\n", + "flops_total = f['forward_total']\n", + "print(f\"{'name':20s} {'flops':14s} {'ratio (%)':10s}\")\n", + "for k,v in f.items():\n", + " print(f\"{k:20s} {v:14d} {v/flops_total*100:10.4f}\")\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "palm_flops: 879894724608, flops: 874944921600, ratio: 1.0057\n" + ] + } + ], + "source": [ + "# now here is an estimate copy pasted from the PaLM paper\n", + "# this formula is often used to calculate MFU (model flops utilization)\n", + "def palm_flops():\n", + " \"\"\"estimate of the model flops following PaLM paper formula\"\"\"\n", + " N = params()['total']\n", + " L, H, Q, T = n_layer, n_head, n_embd//n_head, block_size\n", + " mf_per_token = 6*N + 12*L*H*Q*T\n", + " mf = mf_per_token * block_size\n", + " return mf\n", + "\n", + "print(f\"palm_flops: {palm_flops():d}, flops: {flops()['total']:d}, ratio: {palm_flops()/flops()['total']:.4f}\")" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "fraction of A100 used: 37.14%\n" + ] + } + ], + "source": [ + "# here is what we currently roughly measure\n", + "batch_size = 20 * 5 # 5 is grad_accum, so total batch size is 100\n", + "measured_time = 0.755 # in seconds per iteration\n", + "measured_throughput = batch_size / measured_time\n", + "flops_achieved = f['total'] * measured_throughput\n", + "\n", + "# A100 is cited to be 312 TFLOPS of bloat16 running on tensor cores\n", + "a100_flops_promised = 312e12\n", + "\n", + "# the fraction of the A100 that we are using:\n", + "print(f\"fraction of A100 used: {flops_achieved / a100_flops_promised * 100:.2f}%\")" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "time needed to train the model: 3.46 days\n" + ] + } + ], + "source": [ + "# Finally let's check out the 6ND approximation as total cost of training in FLOPs\n", + "model_size = params()['total'] # this is number of parameters, N\n", + "tokens_num = 300e9 # 300B tokens, this is dataset size in tokens, D\n", + "a100_flops = 312e12 # 312 TFLOPS\n", + "assumed_mfu = 0.3 # assume this model flops utilization (take the current 37% from above and add some DDP overhead)\n", + "flops_throughput = a100_flops * 8 * assumed_mfu # assume an 8XA100 node at 50% utilization\n", + "flops_needed = 6 * model_size * tokens_num # 6ND\n", + "time_needed_s = flops_needed / flops_throughput # in seconds\n", + "print(f\"time needed to train the model: {time_needed_s/3600/24:.2f} days\")" + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is not a bad estimate at all. I trained this model and it converged in roughly 4 days." + ] + }, + { + "attachments": {}, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "pytorch2", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.8" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "7f5833218766b48e6e35e4452ee875aac0e2188d05bbe5298f2c62b79f08b222" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}