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本文來源：煉丹筆記

有一些論文對(duì)warmup進(jìn)行了討論，使用 SGD 訓(xùn)練神經(jīng)網(wǎng)絡(luò)時(shí)，在初始使用較大學(xué)習(xí)率而后期改為較小學(xué)習(xí)率在各種任務(wù)場(chǎng)景下都是一種廣為使用的做法，在實(shí)踐中效果好且最近也有若干文章嘗試對(duì)其進(jìn)行了理論解釋。例如《On Layer Normalization in the Transformer Architecture》等，論文中作者發(fā)現(xiàn)Post-LN Transformer在訓(xùn)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(yōu)化過程就會(huì)非常不穩(wěn)定。

雖然在實(shí)踐中效果好且最近也有若干文章嘗試對(duì)其進(jìn)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Warmup

Transformer中的warm-up可以看作學(xué)習(xí)率 lr 隨迭代數(shù) t 的函數(shù)：

學(xué)習(xí)率 lr 會(huì)以某種方式遞減，學(xué)習(xí)率從0開始增長(zhǎng)，經(jīng)過 Twarmup 次迭代達(dá)到最大。論文中對(duì)Adam，SGD等有無warmup做了實(shí)驗(yàn)，

可以看到，warmup增加了訓(xùn)練時(shí)間，同時(shí)在最初階段使用較大的學(xué)習(xí)率會(huì)導(dǎo)致Loss偏大，對(duì)模型的訓(xùn)練的影響是巨大的。warmup在這里對(duì)SGD是非常重要的。

Rectified Adam

Rectified Adam針對(duì)warmup前期數(shù)據(jù)樣本不足導(dǎo)致的biased variance的問題提出了解決方案，論文中實(shí)驗(yàn)結(jié)果看到還是有一定效果的。RAdam 由隨機(jī)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。

Warmup代碼

class AdamWarmup(Optimizer):
    def __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 = 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.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'] = 0                    state['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'] += 1
                exp_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_correction1                                if 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


04
RAdam代碼

import mathimport torchfrom torch.optim.optimizer import Optimizer, required
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_sgd        if 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 = 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.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'] = 0                    state['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'] += 1                buffered = 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) - 1                    N_sma = N_sma_max - 2 * state['step'] * beta2_t / (1 - beta2_t)                    buffered[1] = N_sma
                    # more conservative since it's an approximated value                    if 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 = -1                    buffered[2] = step_size
                # more conservative since it's an approximated value                if 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