"""Implements Nadam algorithm (a variant of Adam based on Nesterov momentum).

It has been proposed in `Incorporating Nesterov Momentum into Adam`__.

Arguments:

params (iterable): iterable of parameters to optimize or dicts defining

parameter groups

lr (float, optional): learning rate (default: 2e-3)

betas (Tuple[float, float], optional): coefficients used for computing

running averages of gradient and its square

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)

schedule_decay (float, optional): momentum schedule decay (default: 4e-3)

__ http://cs229.stanford.edu/proj2015/054_report.pdf

__ http://www.cs.toronto.edu/~fritz/absps/momentum.pdf

Originally taken from: https://github.com/pytorch/pytorch/pull/1408

NOTE: Has potential issues but does work well on some problems.

"""

def __init__(self, params, lr=2e-3, betas=(0.9, 0.999), eps=1e-8,

weight_decay=0, schedule_decay=4e-3):

defaults = dict(lr=lr, betas=betas, eps=eps,

weight_decay=weight_decay, schedule_decay=schedule_decay)

super(Nadam, self).__init__(params, defaults)

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

state = self.state[p]

# State initialization

if len(state) == 0:

state['step'] = 0

state['m_schedule'] = 1.

state['exp_avg'] = grad.new().resize_as_(grad).zero_()

state['exp_avg_sq'] = grad.new().resize_as_(grad).zero_()

# Warming momentum schedule

m_schedule = state['m_schedule']

schedule_decay = group['schedule_decay']

exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']

beta1, beta2 = group['betas']

eps = group['eps']

state['step'] += 1

t = state['step']

if group['weight_decay'] != 0:

grad = grad.add(group['weight_decay'], p.data)

momentum_cache_t = beta1 * \

(1. - 0.5 * (0.96 ** (t * schedule_decay)))

momentum_cache_t_1 = beta1 * \

(1. - 0.5 * (0.96 ** ((t + 1) * schedule_decay)))

m_schedule_new = m_schedule * momentum_cache_t

m_schedule_next = m_schedule * momentum_cache_t * momentum_cache_t_1

state['m_schedule'] = m_schedule_new

# Decay the first and second moment running average coefficient

exp_avg.mul_(beta1).add_(1. - beta1, grad)

exp_avg_sq.mul_(beta2).addcmul_(1. - beta2, grad, grad)

exp_avg_sq_prime = exp_avg_sq / (1. - beta2 ** t)

denom = exp_avg_sq_prime.sqrt_().add_(eps)

p.data.addcdiv_(-group['lr'] * (1. - momentum_cache_t) / (1. - m_schedule_new), grad, denom)

p.data.addcdiv_(-group['lr'] * momentum_cache_t_1 / (1. - m_schedule_next), exp_avg, denom)