This implementation removes the extra epsilon in the bias correlation from AdaBelief to investigate the effects of belief only.

AdaBelief_NoEpsilon.py

import math
import torch
from torch.optim.optimizer import Optimizer

version_higher = ( torch.__version__ >= "1.5.0" )

class AdaBelief(Optimizer):
    r"""Implements AdaBelief algorithm. Modified from Adam in PyTorch

    Caution:
    This implementation removes the extra epsilon in the bias correlation
    to investigate the effects of belief only.

    Arguments:
        params (iterable): iterable of parameters to optimize or dicts defining
            parameter groups
        lr (float, optional): learning rate (default: 1e-3)
        betas (Tuple[float, float], optional): coefficients used for computing
            running averages of gradient and its square (default: (0.9, 0.999))
        eps (float, optional): term added to the denominator to improve
            numerical stability (default: 1e-8)
        weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
        amsgrad (boolean, optional): whether to use the AMSGrad variant of this
            algorithm from the paper `On the Convergence of Adam and Beyond`_
            (default: False)
        weight_decouple (boolean, optional): ( default: False) If set as True, then
            the optimizer uses decoupled weight decay as in AdamW
        fixed_decay (boolean, optional): (default: False) This is used when weight_decouple
            is set as True.
            When fixed_decay == True, the weight decay is performed as
            $W_{new} = W_{old} - W_{old} \times decay$.
            When fixed_decay == False, the weight decay is performed as
            $W_{new} = W_{old} - W_{old} \times decay \times lr$. Note that in this case, the
            weight decay ratio decreases with learning rate (lr).
        rectify (boolean, optional): (default: False) If set as True, then perform the rectified
            update similar to RAdam

    reference: AdaBelief Optimizer, adapting stepsizes by the belief in observed gradients
               NeurIPS 2020 Spotlight
    """

    def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8,
                 weight_decay=0, amsgrad=False, weight_decouple = False, fixed_decay=False, rectify = False ):
        if not 0.0 <= lr:
            raise ValueError("Invalid learning rate: {}".format(lr))
        if not 0.0 <= eps:
            raise ValueError("Invalid epsilon value: {}".format(eps))
        if not 0.0 <= betas[0] < 1.0:
            raise ValueError("Invalid beta parameter at index 0: {}".format(betas[0]))
        if not 0.0 <= betas[1] < 1.0:
            raise ValueError("Invalid beta parameter at index 1: {}".format(betas[1]))
        defaults = dict(lr=lr, betas=betas, eps=eps,
                        weight_decay=weight_decay, amsgrad=amsgrad)
        super(AdaBelief, self).__init__(params, defaults)

        self.weight_decouple = weight_decouple
        self.rectify = rectify
        self.fixed_decay = fixed_decay
        if self.weight_decouple:
            print('Weight decoupling enabled in AdaBelief')
            if self.fixed_decay:
                print('Weight decay fixed')
        if self.rectify:
            print('Rectification enabled in AdaBelief')
        if amsgrad:
            print('AMS enabled in AdaBelief')
    def __setstate__(self, state):
        super(AdaBelief, self).__setstate__(state)
        for group in self.param_groups:
            group.setdefault('amsgrad', False)

    def reset(self):
        for group in self.param_groups:
            for p in group['params']:
                state = self.state[p]
                amsgrad = group['amsgrad']

                # State initialization
                state['step'] = 0
                # Exponential moving average of gradient values
                state['exp_avg'] = torch.zeros_like(p.data,
                                   memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)

                # Exponential moving average of squared gradient values
                state['exp_avg_var'] = torch.zeros_like(p.data,
                                    memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)
                if amsgrad:
                    # Maintains max of all exp. moving avg. of sq. grad. values
                    state['max_exp_avg_var'] = torch.zeros_like(p.data,
                                    memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)

    def step(self, closure=None):
        """Performs a single optimization step.

        Arguments:
            closure (callable, optional): A closure that reevaluates the model
                and returns the loss.
        """
        loss = None
        if closure is not None:
            loss = closure()

        for group in self.param_groups:
            for p in group['params']:
                if p.grad is None:
                    continue
                grad = p.grad.data
                if grad.is_sparse:
                    raise RuntimeError('AdaBelief does not support sparse gradients, please consider SparseAdam instead')
                amsgrad = group['amsgrad']

                state = self.state[p]
               
                beta1, beta2 = group['betas']

                # State initialization
                if len(state) == 0:
                    state['rho_inf'] = 2.0 / (1.0 - beta2) - 1.0
                    state['step'] = 0
                    # Exponential moving average of gradient values
                    state['exp_avg'] = torch.zeros_like(p.data,
                                    memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)
                    # Exponential moving average of squared gradient values
                    state['exp_avg_var'] = torch.zeros_like(p.data,
                                    memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)
                    if amsgrad:
                        # Maintains max of all exp. moving avg. of sq. grad. values
                        state['max_exp_avg_var'] = torch.zeros_like(p.data,
                                    memory_format=torch.preserve_format) if version_higher else torch.zeros_like(p.data)

                # get current state variable
                exp_avg, exp_avg_var = state['exp_avg'], state['exp_avg_var']

                state['step'] += 1
                bias_correction1 = 1 - beta1 ** state['step']
                bias_correction2 = 1 - beta2 ** state['step']

                # perform weight decay, check if decoupled weight decay
                if self.weight_decouple:
                    if not self.fixed_decay:
                        p.data.mul_(1.0 - group['lr'] * group['weight_decay'])
                    else:
                        p.data.mul_(1.0 - group['weight_decay'])
                else:
                    if group['weight_decay'] != 0:
                        grad.add_(group['weight_decay'], p.data)

                # Update first and second moment running average
                exp_avg.mul_(beta1).add_(1 - beta1, grad)
                grad_residual = grad - exp_avg
                exp_avg_var.mul_(beta2).addcmul_(1 - beta2, grad_residual, grad_residual)

                if amsgrad:
                    max_exp_avg_var = state['max_exp_avg_var']
                    # Maintains the maximum of all 2nd moment running avg. till now
                    torch.max(max_exp_avg_var, exp_avg_var, out=max_exp_avg_var)

                    # Use the max. for normalizing running avg. of gradient
                    denom = (max_exp_avg_var.add_(group['eps']).sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
                else:
                    denom = (exp_avg_var.sqrt() / math.sqrt(bias_correction2)).add_(group['eps']) # remove extra epsilon to investigate the effects of belief only.
                    #denom = (exp_avg_var.add_(group['eps']).sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])

                if not self.rectify:
                    # Default update
                    step_size = group['lr'] / bias_correction1
                    p.data.addcdiv_(-step_size, exp_avg, denom)

                else:# Rectified update
                    # calculate rho_t
                    state['rho_t'] = state['rho_inf'] - 2 * state['step'] * beta2 ** state['step'] / (
                            1.0 - beta2 ** state['step'])

                    if state['rho_t'] > 4: # perform Adam style update if variance is small
                        rho_inf, rho_t = state['rho_inf'], state['rho_t']
                        rt = (rho_t - 4.0) * (rho_t - 2.0) * rho_inf / (rho_inf - 4.0) / (rho_inf - 2.0) / rho_t
                        rt = math.sqrt(rt)

                        step_size = rt * group['lr'] / bias_correction1

                        p.data.addcdiv_(-step_size, exp_avg, denom)

                    else: # perform SGD style update
                        p.data.add_( -group['lr'], exp_avg)

        return loss