math/README.md

# Handy Math


## Estimate model training time

in days:
```
 (X billion tokens)*(8* M billion parameters)/(N_GPUs * Achieved_TFLOPs * 1e12*60*60*24)
```

`Achieved_TFLOPs` is measured by running experiments that tune up the setup for the best throughput performance.

For example, for a 13 billion parameter model, trained for 300 billion tokens, on 256 GPUs at 45 TFLOPs would take: `(300 billion)*(8*13 billion)/(256*45*1 trillion *60*60*24) = ~31 days`

```
$ python -c 'Btokens=300; Bmodel=13; n_gpus=256; Tflops=45; \
print(f"{Btokens*1e9*8*Bmodel*1e9/(n_gpus*Tflops*1e12*60*60*24):0.2f} days")'
31.35 days
```

Notes:

- the factor of 8 can be broken into `(2 x (1+2+1))` where the factor of 2 is for multiple+add, the two ones are for forward propagation and recomputation in the backward and the 2 is for the backward propagation.

contributed by Samyam Rajbhandari


## Calculate TFLOPs

The following is an estimation formula which slightly under-reports the real TFLOPs:

TFLOPs: `model_size_in_B * 4 * 2 * seqlen * global_batch_size / (time_in_sec_per_interation * total_gpus * 1e3)`

The factor of 4 is when used with activation check-pointing, otherwise it will be 3, but for 100B+ model, activation check-pointing will always be on.

So the `3*2` is often called "model FLOPs" and `4*2` - "hardware FLOPs".

```
perl -le '$ng=64; $ms=52; $gbs=1024; $sp=127; $seqlen=2048; print $ms*4*2*$seqlen*$gbs / ( $sp * $ng * 1e3)'
```
(ng = total gpus, ms = model size in B, gbs = global batch size, sp = throughput in seconds)

same with bash env vars and broken down GBS into mbs*dp*gas (gas=pp_chunks):
```
echo "($MSIZE*4*2*SEQLEN*$MICRO_BATCH_SIZE*$DP_SIZE*$GAS)/($THROUGHPUT*$NNODES*4*1000)" | bc -l
```

- Automatically process slurm/ megatron log files, average the throughput (prints 'fail' on when the training failed w/o producing a single iteration stat):
```
find . -type f -name "*out" -exec perl -lne 'm|elapsed time per iteration .ms.: ([\d\.]+)| &&  do {$x+=$1; $c++}; END { print "$ARGV " . ($c ? int($x/$c/1000) : "fail")}' {} \; | sort | grep -v fail
```

The exact formula is in Equation 3 of Section 5.1 of the [Efficient Large-Scale Language Model Training on GPU Clusters Using Megatron-LM](https://arxiv.org/abs/2104.04473) paper. You can see the code [here](https://github.com/bigscience-workshop/Megatron-DeepSpeed/pull/251).

For Inference only it'd be:

`24Bsh^2 + 4𝐵s^2h` floating point operations per layer


## Model sizing

### Params as a function of the network size hyperparams

```
NHIDDEN=4096; NLAYERS=36; SEQ_LEN=512; VOCAB_SIZE=50257; python -c "h=$NHIDDEN; l=$NLAYERS; s=$SEQ_LEN; v=$VOCAB_SIZE; print(f'Model size: {(l*(12*h**2 + 13*h) + v*h + s*h + 2*h) / 10**9 :.0f}B, ratio={int(h/l)}')"
```

For full details see [Calculate model size](../experiments/gpt2-utils.md).

The BLOOM architecture hasn't used the normal positional embedding, so the formula is slightly different and it no longer depends on SEQLEN, and we have added an additional layer norm after the word embedding so `s/s*h + 2*h/4*h` in the formula above:
```
NHIDDEN=14336; NLAYERS=70; NHEADS=112; VOCAB_SIZE=250000; python -c "h=$NHIDDEN; l=$NLAYERS; n=$NHEADS;  v=$VOCAB_SIZE; print(f'Model size: {(l*(12*h**2 + 13*h) + v*h + 4*h) / 10**9 :.0f}B, hidden/layers ratio: {int(h/l)}, hidden/heads ratio: {int(h/n)}')"
```

### Width-depth tradeoff

From [The Depth-to-Width Interplay in Self-Attention](https://arxiv.org/abs/2006.12467):

```
NLAYERS=70; python -c "import math; l=$NLAYERS; a = 5.039; b = 5.55e-2; print(f'Optimal n_params: {12 * l * math.exp(2*a) * math.exp(2*b*l) / 10**9 :.0f}B')"
```
This seems to be less important as the number of parameters scales up, but is useful to ground the discussion.


## Estimate total training time

Training Time Estimates. Given these throughputs, we can also estimate the total amount of time needed for end-to-end training on 𝑇 tokens. Training requires 𝐼 = 𝑇 /(𝐵 · 𝑠) iterations. Using the value of 𝐹 from equation (3) and empirical end-to-end throughputs from Table 1 (denoted by 𝑋), we can estimate total training time. We note that for the configurations in Table 1, we have 6ℎ ≫ 𝑠, 16𝑙ℎ ≫ (𝑉 + 𝑠), and 12𝑙ℎ ≫ 𝑉 . Combining these observations with equations (2) and (3), we arrive at:

End-to-end training time (seconds) ≈ 8𝑇𝑃/𝑛𝑋

Let us consider the GPT-3 model with 𝑃 =175 billion parameters as an example. This model was trained on 𝑇 = 300 billion tokens. On 𝑛 = 1024 A100 GPUs using batch size 1536, we achieve 𝑋 = 140 teraFLOP/s per GPU. As a result, the time required to train this model is 34 days. For the 1 trillion parameter model, we assume that 450 billion tokens are needed for end-to-end training. With 3072 A100 GPUs, we can achieve a per-GPU throughput of 163 teraFLOP/s, and end-to-end training time of 84 days. We believe these training times (using a reasonable number of GPUs) are practical.


This math and discussion is quoted from [Efficient Large-Scale Language Model Training on GPU Clusters Using Megatron-LM](https://arxiv.org/abs/2104.04473).

Let's explain the formula: `8𝑇𝑃/𝑛𝑋`

In the formula:

- T: number of tokens used for training in Billions
- P: number of parameters in normal numbers
- n: number of GPUs
- X: throughput per GPU in TFLOPs
- The result is in seconds, so divide by 3600*24 to get days

Example:

- T = 300B
- P = 200_000_000
- X = 150 TFLOPs (more or less the best one can get on an efficient setup on A100)
- n = 350

gives us:

```
$ python -c 'print(f"{8*300*200_000_000/(350*150)/(3600*24):0.2f}", "days")'
105.82 days
```

## Finding the checkpoint that has the amount of tokens you want

Trying to find the step at which you reached the number of tokens you want for every model size
n_samples = n_tokens / 2048
The average batch size during rampup is rampup_batch_size = 0.5 * (global_batch_size + start_batch_size) (edited)
The number of steps is rampup_samples / rampup_batch_size + (n_samples - rampup_samples) / global_batch_size = rampup_samples / 0.5 / (global_batch_size + start_batch_size) + (n_tokens / 2048 - rampup_samples) / global_batch_size. Those will all change for each model. For example for [tr11f](https://github.com/bigscience-workshop/bigscience/blob/master/train/tr11-176B-ml/smaller_models/tr11f-6B3-ml.slurm) at 150B tokens we have:

> - $GLOBAL_BATCH_SIZE = 512
> - --rampup-batch-size 192 32 9_765_625 which gives:
>    - start_batch_size = 192
>    - rampup_samples = 9,765,625
>
> so n_steps = 9,765,625 / 0.5 / (512 + 192) + (150,000,000,000 / 2048 - 9,765,625) / 512 = 151721