#109581
At this point, the vanilla implementation (the default) is good.
Docs: https://docs-preview.pytorch.org/pytorch/pytorch/129905/generated/torch.optim.Adafactor.html#torch.optim.Adafactor
Specifically, the impl in this PR, which attempts to replicate the paper,
```
optim = torch.optim.Adafactor([weight])
```
is close enough to https://pytorch-optimizers.readthedocs.io/en/latest/optimizer/#pytorch_optimizer.AdaFactor
```
optim_c = AdaFactor([weight], betas=(0, 0.999), scale_parameter=False)
```
is close enough to https://www.tensorflow.org/api_docs/python/tf/keras/optimizers/Adafactor
```
optim = keras.optimizers.Adafactor(learning_rate=0.01)
```
The three results respectively for the same randomly generated weights:
```
# ours
tensor([[ 0.3807594, -0.3912092],
[ 0.0762539, 0.5377805],
[ 0.2459473, 0.4662207]])
# pytorch-optimizer
tensor([[ 0.3807592, -0.3912172],
[ 0.0762507, 0.5377818],
[ 0.2459457, 0.4662213]])
# keras
array([[ 0.38076326, -0.39121315],
[ 0.0762547 , 0.5377859 ],
[ 0.24594972, 0.46622536]], dtype=float32)
```
This gives me confidence to move forward in speeding up the implementation now that a baseline has been established. If you're curious about differences:
* keras assigns step_size (rho_t in their code) to `min(lr, 1 / sqrt(step)` whereas the OG impl uses a hardcoded 0.01 instead of lr. We do the same thing as keras, but our lr default is 0.01.
* We differ from the pytorch-optimizers default in that our default will not track momentum (thus `beta1=0`) and we do not apply parameter scaling.
<details>
Keras collab: https://colab.research.google.com/drive/1i3xF8ChL7TWKJGV_5v_5nMhXKnYmQQ06?usp=sharing
My script repro:
```
import torch
from pytorch_optimizer import AdaFactor
torch.set_printoptions(precision=7)
weight = torch.tensor([[ 0.37697506, -0.39500135],
[ 0.07246649, 0.53399765],
[ 0.24216151, 0.46243715]], dtype=torch.float32)
# bias = torch.tensor([0, 0], dtype=torch.float32)
weight.grad = torch.tensor([[-0.5940447, -0.7743838],
[-0.5940447, -0.7743838],
[-0.5940447, -0.7743838]], dtype=torch.float32)
# bias.grad = torch.tensor([-2.5027974, 1.5422692], dtype=torch.float32)
weight_c = weight.clone()
weight_c.grad = weight.grad.clone()
optim = torch.optim.Adafactor([weight])
optim.step()
print(weight)
optim_c = AdaFactor([weight_c], betas=(0, 0.999), scale_parameter=False)
optim_c.step()
print(weight_c)
```
<details>
Pull Request resolved: https://github.com/pytorch/pytorch/pull/129905
Approved by: https://github.com/albanD
