有一些論文對warmup進行了討論，使用 SGD 訓練神經網絡時，在初始使用較大學習率而后期改為較小學習率在各種任務場景下都是一種廣為使用的做法，在實踐中效果好且最近也有若干文章嘗試對其進行了理論解釋。例如《On Layer Normalization in the Transformer Architecture》等，論文中作者發現Post-LN Transformer在訓練的初始階段，輸出層附近的期望梯度非常大，所以沒有warm-up的話模型優化過程就會非常不穩定。

雖然在實踐中效果好且最近也有若干文章嘗試對其進行了理論解釋，但到底為何有效，目前還沒有被充分證明。

Transformer中的warm-up可以看作學習率 lr 隨迭代數 t 的函數：

學習率 lr 會以某種方式遞減，學習率從0開始增長，經過 Twarmup 次迭代達到最大。論文中對Adam，SGD等有無warmup做了實驗，

可以看到，warmup增加了訓練時間，同時在最初階段使用較大的學習率會導致Loss偏大，對模型的訓練的影響是巨大的。warmup在這里對SGD是非常重要的。

Rectified Adam針對warmup前期數據樣本不足導致的biased variance的問題提出了解決方案，論文中實驗結果看到還是有一定效果的。RAdam 由隨機初始化帶來的 Variance 比較小。即使隔離掉 warmup 部分的影響后Variance 也是要比 Adam 小的。

class AdamWarmup(Optimizer):# DOTAdef __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8, weight_decay=0, warmup = 0):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, warmup = warmup)super(AdamW, self).__init__(params, defaults)def __setstate__(self, state):super(AdamW, self).__setstate__(state)def step(self, closure=None):loss = Noneif closure is not None:loss = closure()for group in self.param_groups:for p in group['params']:if p.grad is None:continuegrad = p.grad.data.float()if grad.is_sparse:raise RuntimeError('Adam does not support sparse gradients, please consider SparseAdam instead')p_data_fp32 = p.data.float()state = self.state[p]if len(state) == 0:state['step'] = 0state['exp_avg'] = torch.zeros_like(p_data_fp32)state['exp_avg_sq'] = torch.zeros_like(p_data_fp32)else:state['exp_avg'] = state['exp_avg'].type_as(p_data_fp32)state['exp_avg_sq'] = state['exp_avg_sq'].type_as(p_data_fp32)exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']beta1, beta2 = group['betas']state['step'] += 1exp_avg_sq.mul_(beta2).addcmul_(1 - beta2, grad, grad)exp_avg.mul_(beta1).add_(1 - beta1, grad)denom = exp_avg_sq.sqrt().add_(group['eps'])bias_correction1 = 1 - beta1 ** state['step']bias_correction2 = 1 - beta2 ** state['step']if group['warmup'] > state['step']:scheduled_lr = 1e-8 + state['step'] * group['lr'] / group['warmup']else:scheduled_lr = group['lr']step_size = scheduled_lr * math.sqrt(bias_correction2) / bias_correction1if group['weight_decay'] != 0:p_data_fp32.add_(-group['weight_decay'] * scheduled_lr, p_data_fp32)p_data_fp32.addcdiv_(-step_size, exp_avg, denom)p.data.copy_(p_data_fp32)return loss

class RAdam(Optimizer):def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8, weight_decay=0, degenerated_to_sgd=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]))self.degenerated_to_sgd = degenerated_to_sgdif isinstance(params, (list, tuple)) and len(params) > 0 and isinstance(params[0], dict):for param in params:if 'betas' in param and (param['betas'][0] != betas[0] or param['betas'][1] != betas[1]):param['buffer'] = [[None, None, None] for _ in range(10)]defaults = dict(lr=lr, betas=betas, eps=eps, weight_decay=weight_decay, buffer=[[None, None, None] for _ in range(10)])super(RAdam, self).__init__(params, defaults)def __setstate__(self, state):super(RAdam, self).__setstate__(state)def step(self, closure=None):loss = Noneif closure is not None:loss = closure()for group in self.param_groups:for p in group['params']:if p.grad is None:continuegrad = p.grad.data.float()if grad.is_sparse:raise RuntimeError('RAdam does not support sparse gradients')p_data_fp32 = p.data.float()state = self.state[p]if len(state) == 0:state['step'] = 0state['exp_avg'] = torch.zeros_like(p_data_fp32)state['exp_avg_sq'] = torch.zeros_like(p_data_fp32)else:state['exp_avg'] = state['exp_avg'].type_as(p_data_fp32)state['exp_avg_sq'] = state['exp_avg_sq'].type_as(p_data_fp32)exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']beta1, beta2 = group['betas']exp_avg_sq.mul_(beta2).addcmul_(1 - beta2, grad, grad)exp_avg.mul_(beta1).add_(1 - beta1, grad)state['step'] += 1buffered = group['buffer'][int(state['step'] % 10)]if state['step'] == buffered[0]:N_sma, step_size = buffered[1], buffered[2]else:buffered[0] = state['step']beta2_t = beta2 ** state['step']N_sma_max = 2 / (1 - beta2) - 1N_sma = N_sma_max - 2 * state['step'] * beta2_t / (1 - beta2_t)buffered[1] = N_sma# more conservative since it's an approximated valueif N_sma >= 5:step_size = math.sqrt((1 - beta2_t) * (N_sma - 4) / (N_sma_max - 4) * (N_sma - 2) / N_sma * N_sma_max / (N_sma_max - 2)) / (1 - beta1 ** state['step'])elif self.degenerated_to_sgd:step_size = 1.0 / (1 - beta1 ** state['step'])else:step_size = -1buffered[2] = step_size# more conservative since it's an approximated valueif N_sma >= 5:if group['weight_decay'] != 0:p_data_fp32.add_(-group['weight_decay'] * group['lr'], p_data_fp32)denom = exp_avg_sq.sqrt().add_(group['eps'])p_data_fp32.addcdiv_(-step_size * group['lr'], exp_avg, denom)p.data.copy_(p_data_fp32)elif step_size > 0:if group['weight_decay'] != 0:p_data_fp32.add_(-group['weight_decay'] * group['lr'], p_data_fp32)p_data_fp32.add_(-step_size * group['lr'], exp_avg)p.data.copy_(p_data_fp32)return loss

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