#

Adam Optimizer

This is a PyTorch implementation of popular optimizer Adam from paper Adam: A Method for Stochastic Optimization.

Adam update is,

m_{t} v_{t} \overset{m}{^}_{t} \overset{v}{^}_{t} θ_{t} \leftarrow β_{1} m_{t - 1} + (1 - β_{1}) \cdot g_{t} \leftarrow β_{2} v_{t - 1} + (1 - β_{2}) \cdot g_{t}^{2} \leftarrow \frac{m _{t}}{1 - β _{1} ^{t}} \leftarrow \frac{v _{t}}{1 - β _{2} ^{t}} \leftarrow θ_{t - 1} - α \cdot \frac{m ^ _{t}}{v ^ _{t} + ϵ}

where $α$ , $β_{1}$ , $β_{2}$ and $ϵ$ are scalar hyper parameters. $m_{t}$ and $v_{t}$ are first and second order moments. $\overset{m}{^}_{t}$ and $\overset{v}{^}_{t}$ are biased corrected moments. $ϵ$ is used as a fix for division by zero error, but also acts as a form of a hyper-parameter that acts against variance in gradients.

Effective step taken assuming $ϵ = 0$ is, $Δ t = α \cdot \frac{m ^ _{t}}{v ^ _{t}}$ This is bounded by, $∣Δ t ∣ \leq α \cdot \frac{1 - β _{1}}{1 - β _{2}}$ when $1 - β_{1} > 1 - β_{2}$ and $∣Δ t ∣ \leq α$ otherwise. And in most common scenarios, $∣Δ t ∣ \approx α$

40import math
41from typing import Dict, Any, Tuple, Optional
42
43import torch
44from labml import tracker
45from torch import nn
46
47from labml_nn.optimizers import GenericAdaptiveOptimizer, WeightDecay

#

Adam Optimizer

We extend the class GenericAdaptiveOptimizer defined in __init__.py to implement the Adam optimizer.

50class Adam(GenericAdaptiveOptimizer):

#

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 $ϵ$
defaults is a dictionary of default for group values. This is useful when you want to extend the class Adam .

58    def __init__(self, params,
59                 lr: float = 1e-3, betas: Tuple[float, float] = (0.9, 0.999), eps: float = 1e-16,
60                 weight_decay: WeightDecay = WeightDecay(),
61                 optimized_update: bool = True,
62                 defaults: Optional[Dict[str, Any]] = None):

#

76        defaults = {} if defaults is None else defaults
77        defaults.update(weight_decay.defaults())
78        super().__init__(params, defaults, lr, betas, eps)
79
80        self.weight_decay = weight_decay
81        self.optimized_update = optimized_update

#

Initialize a parameter state

state is the optimizer state of the parameter (tensor)
group stores optimizer attributes of the parameter group
param is the parameter tensor $θ_{t - 1}$

83    def init_state(self, state: Dict[str, any], group: Dict[str, any], param: nn.Parameter):

#

This is the number of optimizer steps taken on the parameter, $t$

93        state['step'] = 0

#

Exponential moving average of gradients, $m_{t}$

95        state['exp_avg'] = torch.zeros_like(param, memory_format=torch.preserve_format)

#

Exponential moving average of squared gradient values, $v_{t}$

97        state['exp_avg_sq'] = torch.zeros_like(param, memory_format=torch.preserve_format)

#

Calculate $m_{t}$ and and $v_{t}$

state is the optimizer state of the parameter (tensor)
group stores optimizer attributes of the parameter group
grad is the current gradient tensor $g_{t}$ for the parameter $θ_{t - 1}$

99    def get_mv(self, state: Dict[str, Any], group: Dict[str, Any], grad: torch.Tensor):

#

Get $β_{1}$ and $β_{2}$

109        beta1, beta2 = group['betas']

#

Get $m_{t - 1}$ and $v_{t - 1}$

112        m, v = state['exp_avg'], state['exp_avg_sq']

#

In-place calculation of $m_{t}$ $m_{t} \leftarrow β_{1} m_{t - 1} + (1 - β_{1}) \cdot g_{t}$

116        m.mul_(beta1).add_(grad, alpha=1 - beta1)

#

In-place calculation of $v_{t}$ $v_{t} \leftarrow β_{2} v_{t - 1} + (1 - β_{2}) \cdot g_{t}^{2}$

119        v.mul_(beta2).addcmul_(grad, grad, value=1 - beta2)
120
121        return m, v

#

Get learning-rate

This returns the modified learning rate based on the state. For Adam this is just the specified learning rate for the parameter group, $α$ .

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

#

131        return group['lr']

#

Do the Adam parameter update

state is the optimizer state of the parameter (tensor)
group stores optimizer attributes of the parameter group
param is the parameter tensor $θ_{t - 1}$
m and v are the uncorrected first and second moments $m_{t}$ and $v_{t}$ .

This computes the following

θ_{t} \leftarrow θ_{t - 1} - α \cdot \frac{m ^ _{t}}{v ^ _{t} + ϵ}

Since $α$ , $β_{1}$ , $β_{2}$ and $ϵ$ are scalars and others are tensors we modify this calculation to optimize the computation.

θ_{t} θ_{t} θ_{t} \leftarrow θ_{t - 1} - α \cdot \frac{m ^ _{t}}{v ^ _{t} + ϵ} \leftarrow θ_{t - 1} - α \cdot \frac{m _{t} / ( 1 - β _{1} ^{t} )}{v _{t} / ( 1 - β _{2} ^{t} ) + ϵ} \leftarrow θ_{t - 1} - α \frac{1 - β _{2} ^{t}}{1 - β _{1} ^{t}} \cdot \frac{m _{t}}{v _{t} + ϵ ^}

where $\overset{ϵ}{^} = (1 - β_{2}^{t}) ϵ$ is what we should specify as the hyper-parameter.

133    def adam_update(self, state: Dict[str, any], group: Dict[str, any], param: torch.nn.Parameter,
134                    m: torch.Tensor, v: torch.Tensor):

#

Get $β_{1}$ and $β_{2}$

166        beta1, beta2 = group['betas']

#

Bias correction term for $\overset{m}{^}_{t}$ , $1 - β_{1}^{t}$

168        bias_correction1 = 1 - beta1 ** state['step']

#

Bias correction term for $\overset{v}{^}_{t}$ , $1 - β_{2}^{t}$

170        bias_correction2 = 1 - beta2 ** state['step']

#

Get learning rate

173        lr = self.get_lr(state, group)

#

Whether to optimize the computation

176        if self.optimized_update:

#

$v_{t} + \overset{ϵ}{^}$

178            denominator = v.sqrt().add_(group['eps'])

#

$α \frac{1 - β _{2} ^{t}}{1 - β _{1} ^{t}}$

180            step_size = lr * math.sqrt(bias_correction2) / bias_correction1

#

$θ_{t} \leftarrow θ_{t - 1} - α \frac{1 - β _{2} ^{t}}{1 - β _{1} ^{t}} \cdot \frac{m _{t}}{v _{t} + ϵ ^}$

183            param.data.addcdiv_(m, denominator, value=-step_size)

#

Computation without optimization

185        else:

#

$\frac{v _{t}}{1 - β _{2} ^{t}} + ϵ$

187            denominator = (v.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])

#

$\frac{α}{1 - β _{1} ^{t}}$

189            step_size = lr / bias_correction1

#

$θ_{t} \leftarrow θ_{t - 1} - α \cdot \frac{m ^ _{t}}{v ^ _{t} + ϵ}$

192            param.data.addcdiv_(m, denominator, value=-step_size)

#

Take an update step for a given parameter tensor

state is the optimizer state of the parameter (tensor)
group stores optimizer attributes of the parameter group
grad is the current gradient tensor $g_{t}$ for the parameter $θ_{t - 1}$
param is the parameter tensor $θ_{t - 1}$

194    def step_param(self, state: Dict[str, any], group: Dict[str, any], grad: torch.Tensor, param: torch.nn.Parameter):

#

Calculate weight decay

205        grad = self.weight_decay(param, grad, group)

#

Get $m_{t}$ and $v_{t}$

208        m, v = self.get_mv(state, group, grad)

#

Increment $t$ the number of optimizer steps

211        state['step'] += 1

#

Perform Adam update

214        self.adam_update(state, group, param, m, v)

Adam Optimizer

Adam Optimizer

Initialize the optimizer

Initialize a parameter state

Calculate mt​ and and vt​

Get learning-rate

Do the Adam parameter update

Take an update step for a given parameter tensor

Calculate $m_{t}$ and and $v_{t}$