Grokfast: Accelerated Grokking by
Amplifying Slow Gradients

Jaerin Lee* · Bong Gyun Kang* · Kihoon Kim · Kyoung Mu Lee

Seoul National University

*Denotes equal contribution.

tl;dr: We accelerate the grokking phenomenon by amplifying low-frequencies of the parameter gradients with an augmented optimizer.

Abstract: One puzzling artifact in machine learning dubbed grokking is where delayed generalization is achieved tenfolds of iterations after near perfect overfitting to the training data. Focusing on the long delay itself on behalf of machine learning practitioners, our goal is to accelerate generalization of a model under grokking phenomenon. By regarding a series of gradients of a parameter over training iterations as a random signal over time, we can spectrally decompose the parameter trajectories under gradient descent into two components: the fast-varying, overfitting-yielding component and the slow-varying, generalization-inducing component. This analysis allows us to accelerate the grokking phenomenon more than $\times$ 50 with only a few lines of code that amplifies the slow-varying components of gradients. The experiments show that our algorithm applies to diverse tasks involving images, languages, and graphs, enabling practical availability of this peculiar artifact of sudden generalization.

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Usage

Installation

Grokfast doesn't require additional packages except for PyTorch. The file requirements.txt is only for reproduction of the experiments in the article, as described in the Reproduction section below.

Instructions

Grokfast can be applied by inserting a single line before the optimizer call.

1. Download a single file grokfast.py from our repository.

wget https://raw.githubusercontent.com/ironjr/grokfast/main/grokfast.py

2. Import the helper function.

from grokfast import gradfilter_ma, gradfilter_ema

3. Insert the following line before the training loop.

grads = None

4. Between loss.backward() and optimizer.step(), insert one of the following line. Make sure model is of type nn.Module and grads are initialized properly before the training loop:

# ... in the optimization loop.
loss.backwards() # Calculate the gradients.

### Option 1: Grokfast (has argument alpha, lamb)
grads = gradfilter_ema(model, grads=grads, alpha=alpha, lamb=lamb)
### Option 2: Grokfast-MA (has argument window_size, lamb)
# grads = gradfilter_ma(model, grads=grads, window_size=window_size, lamb=lamb)

optimizer.step() # Call the optimizer.
# ... logging & other codes.

Done!

(2-1) ...or, copy and paste the method directly into your code!

### Imports
from collections import deque
from typing import Dict, Optional, Literal
import torch
import torch.nn as nn


### Grokfast
def gradfilter_ema(
    m: nn.Module,
    grads: Optional[Dict[str, torch.Tensor]] = None,
    alpha: float = 0.99,
    lamb: float = 5.0,
) -> Dict[str, torch.Tensor]:
    if grads is None:
        grads = {n: p.grad.data.detach() for n, p in m.named_parameters() if p.requires_grad}

    for n, p in m.named_parameters():
        if p.requires_grad:
            grads[n] = grads[n] * alpha + p.grad.data.detach() * (1 - alpha)
            p.grad.data = p.grad.data + grads[n] * lamb

    return grads


### Grokfast-MA
def gradfilter_ma(
    m: nn.Module,
    grads: Optional[Dict[str, deque]] = None,
    window_size: int = 128,
    lamb: float = 5.0,
    filter_type: Literal['mean', 'sum'] = 'mean',
    warmup: bool = True,
    trigger: bool = False,
) -> Dict[str, deque]:
    if grads is None:
        grads = {n: deque(maxlen=window_size) for n, p in m.named_parameters() if p.requires_grad}

    for n, p in m.named_parameters():
        if p.requires_grad:
            grads[n].append(p.grad.data.detach())

            if not warmup or len(grads[n]) == window_size and not trigger:
                if filter_type == "mean":
                    avg = sum(grads[n]) / len(grads[n])
                elif filter_type == "sum":
                    avg = sum(grads[n])
                else:
                    raise ValueError(f"Unrecognized filter_type {filter_type}")
                p.grad.data = p.grad.data + avg * lamb

    return grads

Arguments

1. Grokfast (gradfilter_ema)

   • m: nn.Module: Model that contains every trainable parameters.
   • grads: Optional[Dict[str, torch.Tensor]] = None: Running memory (EMA). Initialize by setting it to None. Feed the output of the method recursively after on.
   • alpha: float = 0.98: Momentum hyperparmeter of the EMA.
   • lamb: float = 2.0: Amplifying factor hyperparameter of the filter.

2. Grokfast-MA (gradfilter_ma)

   • m: nn.Module: Model that contains every trainable parameters.
   • grads: Optional[Dict[str, deque]] = None: Running memory (Queue for windowed moving average). Initialize by setting it to None. Feed the output of the method recursively after on.
   • window_size: int = 100: The width of the filter window. Additional memory requirements increases linearly with respect to the windows size.
   • lamb: float = 5.0: Amplifying factor hyperparameter of the filter.
   • filter_type: Literal['mean', 'sum'] = 'mean': Aggregation method for the running queue.
   • warmup: bool = True: If true, filter is not applied until the queue is filled.
   • trigger: bool = False: For ablation study only. If true, the filter is simply not applied.

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Reproduction

We also note the additional computational resources required for each run. Time & memory costs are measured with a single GTX 1080 Ti GPU.

Installation

This will install the additional packages to preprocess each data and to summarize the results.

conda create -n grok python=3.10 && conda activate grok
git clone https://github.com/ironjr/grokfast
pip install -r requirements.txt

Algorithmic Data (Transformer decoder, Grokfast-MA)

Run	Iterations to Reach 95% Val. Acc.	Wall Clock Time to Reach 95% Val. Acc. (s)	VRAM Requirements (MB)	Latency Per Iteration (s)
Baseline	39890	5984	290	0.15
Grokfast-MA	790 ($\times$ 50.49 $\downarrow$)	292 ($\times$ 20.49 $\downarrow$)	458	0.37

# python main.py --label test # Baseline.
python main.py --label test --filter ma --window_size 100 --lamb 5.0 --weight_decay 0.01

Algorithmic Data (Transformer decoder, Grokfast)

Run	Iterations to Reach 95% Val. Acc.	Wall Clock Time to Reach 95% Val. Acc. (s)	VRAM Requirements (MB)	Latency Per Iteration (s)
Baseline	39890	5984	$290	0.15
Grokfast	910 ($\times$ 43.84 $\downarrow$)	137 ($\times$ 43.79 $\downarrow$)	294	0.15

# python main.py --label test # Baseline.
python main.py --label test --filter ema --alpha 0.98 --lamb 2.0 --weight_decay 0.005

MNIST (MLP)

Run	Iterations to Reach 95% Val. Acc.	Wall Clock Time to Reach 95% Val. Acc. (s)	VRAM Requirements (MB)	Latency Per Iteration (ms)
Baseline	44022	1928	196	43.8
Grokfast	2001 ($\times$ 22.00 $\downarrow$)	87.8 ($\times$ 21.96 $\downarrow$)	198	43.9

# python main_mnist.py --label test # Baseline.
python main_mnist.py --label test --alpha 0.8 --lamb 0.1 --weight_decay 2.0

IMDb (LSTM)

Run	Best Validation Acc.	Minimum Validation Loss	VRAM Requirements (MB)	Latency Per Iteration (ms)
Baseline	0.84	0.517	754	20.4
Grokfast	0.90	0.412	762	21.2

• Before training, download the IMDb dataset from Google Drive or Baidu Webdisk (pw: vdp7).

# python main_imdb.py --label test # Baseline.
python main_imdb.py --label test --alpha 0.98 --lamb 2.0 --weight_decay 10.0

QM9 (G-CNN)

Run	Minimum Validation Loss	VRAM Requirements (MB)	Latency Per Iteration (ms)
Baseline	0.00659	216	40.2
Grokfast	0.00348	216	41.4

# python main_qm9.py --label test # Baseline.
python main_qm9.py --label test --alpha 0.9 --lamb 1.0 --weight_decay 0.01

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FAQ

Choosing the right hyperparameters

These recommendations are based on my experiences during the experiments shown in the main manuscript. This may not work perfectly to every other problems, and maybe more intelligent techniques can do better jobs than this procedure. So, please take these as one possible starting guidelines for designing your own filters.

1. Cutoff parameters: The work uses MA/EMA filters to implement the filtering techniques. The cutoff frequency is determined by the window size for the MA filter, and the momentum parameter for the EMA filter.
   1. Roughly figure out the amount of acceleration you want to achieve. For example, in the main manuscript, the cutoff parameters are determined based on the original grokking report, where experiments shows generalization happening X100 slower than overfitting. Therefore, we want N=100 times faster acceleration.
   2. Set the pivotal values for the cutoff parameter search. For MA, I started to set the window size of "w=N=100" and for EMA, I began with the momentum parameter alpha that satisfies "alpha^{N} = alpha^{100} = 0.1" (which is roughly alpha ~ 0.98).
   3. Perform hyperparameter search near the pivot values. I swept across hyperparameter values near the values set in (1.b).
2. Weight decay: The weight decay is set in the optimizer constructor as usual (e.g., optimizer = optim.Adam(m.parameters(), weight_decay=wd)).
   1. Start from the default weight decay of that task. For example, the value chosen by the most widely used Github repository of that task.
   2. Fix the weight decay and try to find the optimal setting for the Grokfast filter parameters (momentum, window size, and amplitude) first. Although weight decay do affect the values of the optimal filter parameters, its effect seems to be insignificant in my experiences.
   3. Start increasing the weight decay value. Start from X1 then try (X2, X5, X10). I couldn't get better results with X100 scale of the default value.

Acknowledgement

Our code is heavily based on the following projects:

• Ziming Liu et al., "Omnigrok: Grokking Beyond Algorithmic Data," ICLR 2023. [arXiv] [code]
• Alethea Power et al., "Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets," arXiv preprint arXiv:2201.02177. [arXiv] [code]
• @danielmamay's Re-implementation of Grokking. [code]

Thank you all for providing useful references!

Citation

Please cite us if you find our project useful!

@article{lee2024grokfast,
    title={{Grokfast}: Accelerated Grokking by Amplifying Slow Gradients},
    author={Lee, Jaerin and Kang, Bong Gyun and Kim, Kihoon and Lee, Kyoung Mu},
    journal={arXiv preprint arXiv:2405.20233},
    year={2024}
}

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Contact

If you have any questions, please email jarin.lee@gmail.com.