#

Noam Optimizer

This is the PyTorch implementation of optimizer introduced in the paper Attention Is All You Need.

14from typing import Dict
15
16from labml_nn.optimizers import WeightDecay
17from labml_nn.optimizers.amsgrad import AMSGrad

#

Noam Optimizer

This class extends from Adam optimizer defined in adam.py .

20class Noam(AMSGrad):

#

Initialize the optimizer

params is the list of parameters
lr is the learning rate $α$
betas is a tuple of ( $β_{1}$ , $β_{2}$ )
eps is $\overset{ϵ}{^}$ or $ϵ$ based on optimized_update
weight_decay is an instance of class WeightDecay defined in __init__.py
'optimized_update' is a flag whether to optimize the bias correction of the second moment by doing it after adding $ϵ$
amsgrad is a flag indicating whether to use AMSGrad or fallback to plain Adam
warmup number of warmup steps
d_model model size; i.e. number of dimensions in the transformer
defaults is a dictionary of default for group values. This is useful when you want to extend the class AdamWarmup .

27    def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-16,
28                 weight_decay: WeightDecay = WeightDecay(),
29                 optimized_update: bool = True,
30                 amsgrad=False,
31                 warmup=0, d_model=512, defaults=None):

#

49        defaults = {} if defaults is None else defaults
50        defaults.update(dict(warmup=warmup))
51        super().__init__(params, lr, betas, eps, weight_decay, optimized_update, amsgrad, defaults)
52        self.d_model = d_model

#

Get learning-rate

$α \frac{1}{d _{m o d e l}} m i n (\frac{1}{t}, \frac{t}{w ^{3/2}})$ where $w$ is the number of warmup steps.

54    def get_lr(self, state: Dict[str, any], group: Dict[str, any]):

#

$m i n (\frac{1}{t}, \frac{t}{w ^{3/2}})$

62        factor = min(state['step'] ** (-0.5), state['step'] * group['warmup'] ** (-1.5))

#

$α \frac{1}{d _{m o d e l}} m i n (\frac{1}{t}, \frac{t}{w ^{3/2}})$

64        return group['lr'] * self.d_model ** (-0.5) * factor

#

Plot learning rate for different warmups and model sizes

Plot of learning rate

67def _test_noam_lr():

#

73    import matplotlib.pyplot as plt
74    import numpy as np
75    from torch import nn
76
77    model = nn.Linear(10, 10)
78    opts = [Noam(model.parameters(), d_model=512, warmup=4000, lr=1),
79            Noam(model.parameters(), d_model=512, warmup=8000, lr=1),
80            Noam(model.parameters(), d_model=2048, warmup=2000, lr=1)]
81    plt.plot(np.arange(1, 20000), [[opt.get_lr({'step': i}, opt.defaults) for opt in opts] for i in range(1, 20000)])
82    plt.legend(["512:4000", "512:8000", "2048:2000"])
83    plt.title("Learning Rate")
84    plt.show()
85
86
87if __name__ == '__main__':
88    _test_noam_lr()