大梳理！深度学习优化算法：从 SGD 到 AdamW 原理和代码解读

绝地灬酷狼 2022-09-05 12:44 251阅读 0赞

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作者丨知乎 科技猛兽  极市平台 编辑

https://zhuanlan.zhihu.com/p/391947979

本文思想来自下面这篇大佬的文章：

Juliuszh：一个框架看懂优化算法之异同 SGD/AdaGrad/Adam

https://zhuanlan.zhihu.com/p/32230623

主要是对深度学习各种优化器 (从SGD到AdamW) 使用统一的框架做一次整理，本文相比于链接从源代码的角度理解这些优化器的思路。

**代码来自 PyTorch1.7.0 官方教程：**

https://pytorch.org/docs/1.7.0/optim.html

首先我们来回顾一下各类优化算法。

深度学习优化算法经历了 SGD -> SGDM -> NAG ->AdaGrad -> AdaDelta -> Adam -> Nadam -> AdamW 这样的发展历程。Google一下就可以看到很多的教程文章，详细告诉你这些算法是如何一步一步演变而来的。在这里，我们换一个思路，用一个框架来梳理所有的优化算法，做一个更加高屋建瓴的对比。

*  **统一框架：**

首先定义：待优化参数：，目标函数：，初始学习率。

而后，开始进行迭代优化。在每个epoch：

1 计算目标函数关于当前参数的梯度：

2 根据历史梯度计算一阶动量和二阶动量：

![b7adf2d44416ff29051afca0c37e25b2.png][]

3 计算当前时刻的下降梯度：

4 根据下降梯度进行更新：

掌握了这个框架，你可以轻轻松松设计自己的优化算法。

我们拿着这个框架，来照一照各种玄乎其玄的优化算法的真身。步骤3, 4对于各个算法都是一致的，主要的差别就体现在1和2上，也就是计算一阶动量和二阶动量时采用不同的套路。当计算好二者之后，都是使用固定的学习率与二者作用得到当前时刻的下降梯度，进而最后更新参数。

在所有优化器的代码里面有一些函数的作用是相通的：

> **共性的方法有：**

*  `add_param_group`(param\_group)：把参数放进优化器中，这在 Fine-tune 预训练网络时很有用，因为可以使冻结层可训练并随着训练的进行添加到优化器中。
 *  `load_state_dict`(state\_dict)：把优化器的状态加载进去。
 *  `state_dict`()：返回优化器的状态，以dict的形式返回。
 *  `step`(closure=None)：优化一步参数。
 *  `zero_grad`(set\_to\_none=False)：把所有的梯度值设为0。

> **使用方法：**

for input, target in dataset:
        def closure():
            optimizer.zero_grad()
            output = model(input)
            loss = loss_fn(output, target)
            loss.backward()
            return loss
        optimizer.step(closure)

下面正式开始。

## SGD ##

先来看SGD。SGD没有动量的概念，也就是说：

代入步骤3，可以看到下降梯度就是最简单的

SGD最大的缺点是下降速度慢，而且可能会在沟壑的两边持续震荡，停留在一个局部最优点。

## SGD with Momentum ##

为了抑制SGD的震荡，SGDM认为梯度下降过程可以加入惯性。下坡的时候，如果发现是陡坡，那就利用惯性跑的快一些。SGDM全称是SGD with momentum，在SGD基础上引入了一阶动量：

![695da8188d7244cb3cd059f1941f0549.png][]

一阶动量是各个时刻梯度方向的指数移动平均值，约等于最近个时刻的梯度向量和的平均值。

也就是说，时刻的下降方向，不仅由当前点的梯度方向决定，而且由此前累积的下降方向决定。的经验值为0.9，这就意味着下降方向主要是此前累积的下降方向，并略微偏向当前时刻的下降方向。想象高速公路上汽车转弯，在高速向前的同时略微偏向，急转弯可是要出事的。

## SGD with Nesterov Acceleration ##

SGD 还有一个问题是困在局部最优的沟壑里面震荡。想象一下你走到一个盆地，四周都是略高的小山，你觉得没有下坡的方向，那就只能待在这里了。可是如果你爬上高地，就会发现外面的世界还很广阔。因此，我们不能停留在当前位置去观察未来的方向，而要向前一步、多看一步、看远一些。

NAG全称Nesterov Accelerated Gradient，是在SGD、SGD-M的基础上的进一步改进，改进点在于步骤1。我们知道在时刻的主要下降方向是由累积动量决定的，自己的梯度方向说了也不算，那与其看当前梯度方向，不如先看看如果跟着累积动量走了一步，那个时候再怎么走。因此，NAG在步骤1，不计算当前位置的梯度方向，而是计算如果按照累积动量走了一步，那个时候的下降方向：

![864c38c0d6405c66b72c63650397cf3a.png][]

然后用下一个点的梯度方向，与历史累积动量相结合，计算步骤2中当前时刻的累积动量。

> **定义优化器：**

CLASS torch.optim.SGD(params, lr=<required parameter>, momentum=0, dampening=0, weight_decay=0, nesterov=False)

> **参数：**

*  **params** (iterable) – 优化器作用的模型参数。
 *  **lr** (float) – learning rate，相当于是统一框架中的。
 *  **momentum** (float, optional) – 动量参数。(默认值：0)
 *  **weight\_decay** (float, optional) – 权重衰减系数 weight decay (L2 penalty) (默认值：0)
 *  **dampening** (float, optional) – dampening for momentum (默认值：0)
 *  **nesterov** (bool, optional) – 允许 Nesterov momentum (默认值：False)

FLOAT：https://docs.python.org/3/library/functions.html\#float

bool:https://docs.python.org/3/library/functions.html\#bool

> **源码解读：**

import torch
    from .optimizer import Optimizer, required
    
    
    [docs]class SGD(Optimizer):
        r"""Implements stochastic gradient descent (optionally with momentum).
    
        Nesterov momentum is based on the formula from
        `On the importance of initialization and momentum in deep learning`__.
    
        Args:
            params (iterable): iterable of parameters to optimize or dicts defining
                parameter groups
            lr (float): learning rate
            momentum (float, optional): momentum factor (default: 0)
            weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
            dampening (float, optional): dampening for momentum (default: 0)
            nesterov (bool, optional): enables Nesterov momentum (default: False)
    
        Example:
            >>> optimizer = torch.optim.SGD(model.parameters(), lr=0.1, momentum=0.9)
            >>> optimizer.zero_grad()
            >>> loss_fn(model(input), target).backward()
            >>> optimizer.step()
    
        __ http://www.cs.toronto.edu/%7Ehinton/absps/momentum.pdf
    
        .. note::
            The implementation of SGD with Momentum/Nesterov subtly differs from
            Sutskever et. al. and implementations in some other frameworks.
    
            Considering the specific case of Momentum, the update can be written as
    
            .. math::
                \begin{aligned}
                    v_{t+1} & = \mu * v_{t} + g_{t+1}, \\
                    p_{t+1} & = p_{t} - \text{lr} * v_{t+1},
                \end{aligned}
    
            where :math:`p`, :math:`g`, :math:`v` and :math:`\mu` denote the 
            parameters, gradient, velocity, and momentum respectively.
    
            This is in contrast to Sutskever et. al. and
            other frameworks which employ an update of the form
    
            .. math::
                \begin{aligned}
                    v_{t+1} & = \mu * v_{t} + \text{lr} * g_{t+1}, \\
                    p_{t+1} & = p_{t} - v_{t+1}.
                \end{aligned}
    
            The Nesterov version is analogously modified.
        """
    
        def __init__(self, params, lr=required, momentum=0, dampening=0,
                     weight_decay=0, nesterov=False):
            if lr is not required and lr < 0.0:
                raise ValueError("Invalid learning rate: {}".format(lr))
            if momentum < 0.0:
                raise ValueError("Invalid momentum value: {}".format(momentum))
            if weight_decay < 0.0:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
    
            defaults = dict(lr=lr, momentum=momentum, dampening=dampening,
                            weight_decay=weight_decay, nesterov=nesterov)
            if nesterov and (momentum <= 0 or dampening != 0):
                raise ValueError("Nesterov momentum requires a momentum and zero dampening")
            super(SGD, self).__init__(params, defaults)
    
        def __setstate__(self, state):
            super(SGD, self).__setstate__(state)
            for group in self.param_groups:
                group.setdefault('nesterov', False)
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                weight_decay = group['weight_decay']
                momentum = group['momentum']
                dampening = group['dampening']
                nesterov = group['nesterov']
    
                for p in group['params']:
                    if p.grad is None:
                        continue
                    d_p = p.grad
                    if weight_decay != 0:
                        d_p = d_p.add(p, alpha=weight_decay)
                    if momentum != 0:
                        param_state = self.state[p]
                        if 'momentum_buffer' not in param_state:
                            buf = param_state['momentum_buffer'] = torch.clone(d_p).detach()
                        else:
                            buf = param_state['momentum_buffer']
                            buf.mul_(momentum).add_(d_p, alpha=1 - dampening)
                        if nesterov:
                            d_p = d_p.add(buf, alpha=momentum)
                        else:
                            d_p = buf
    
                    p.add_(d_p, alpha=-group['lr'])
    
            return loss

> 这里通过 d\_p=p.grad 得到每个参数的梯度，也就是1式的。
> 
> 如果使用 weight\_decay 的话，那么相当于目标函数加上，所以相当于是梯度相当于要再加上，所以使用了 d\_p = d\_p.add(p, alpha=weight\_decay)。
> 
> 通过 buf.mul\_(momentum).add\_(d\_p, alpha=1 - dampening) 来计算动量，momentum参数一般取0.9，就相当于是之前的动量buf乘以，再加上此次的梯度d\_p乘以。
> 
> 如果不通过nesterov方式更新参数，那么3式中的就相当于是上一步计算出的动量了。如果通过nesterov方式更新参数，那么3式中的就相当于，和不用nesterov方式相比，相差了  。
> 
> 最后通过 p.add\_(d\_p, alpha=-group\['lr'\]) 更新梯度，相当于是上面的 3 式。

## AdaGrad ##

此前我们都没有用到二阶动量。二阶动量的出现，才意味着“自适应学习率”优化算法时代的到来。SGD及其变种以同样的学习率更新每个参数，但深度神经网络往往包含大量的参数，这些参数并不是总会用得到（想想大规模的embedding）。对于经常更新的参数，我们已经积累了大量关于它的知识，不希望被单个样本影响太大，希望学习速率慢一些；对于偶尔更新的参数，我们了解的信息太少，希望能从每个偶然出现的样本身上多学一些，即学习速率大一些。

怎么样去度量历史更新频率呢？那就是二阶动量——该维度上，迄今为止所有梯度值的平方和：

我们再回顾一下步骤3中的下降梯度：

可以看出，此时实质上的学习率由变成了。一般为了避免分母为0，会在分母上加一个小的平滑项。因此是恒大于0的，而且参数更新越频繁，二阶动量越大，学习率就越小。

这一方法在稀疏数据场景下表现非常好。但也存在一些问题：因为是单调递增的，会使得学习率单调递减至0，可能会使得训练过程提前结束，即便后续还有数据也无法学到必要的知识。

> **定义优化器：**

CLASS torch.optim.Adagrad(params,lr=0.01,lr_decay=0,weight_decay=0,initial_accumulator_value=0,eps=1e-10)

> **参数：**

*  **params** (iterable) – 优化器作用的模型参数。
 *  **lr** (float) – learning rate – 相当于是统一框架中的。
 *  **lr\_decay**(float,optional) – 学习率衰减 (默认值：0)
 *  **weight\_decay** (float, optional) – 权重衰减系数 weight decay (L2 penalty) (默认值：0)
 *  **eps**(float,optional)：防止分母为0的一个小数 (默认值：1e-10)

> **源码解读：**

[docs]class Adagrad(Optimizer):
        """Implements Adagrad algorithm.
    
        It has been proposed in `Adaptive Subgradient Methods for Online Learning
        and Stochastic Optimization`_.
    
        Arguments:
            params (iterable): iterable of parameters to optimize or dicts defining
                parameter groups
            lr (float, optional): learning rate (default: 1e-2)
            lr_decay (float, optional): learning rate decay (default: 0)
            weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
            eps (float, optional): term added to the denominator to improve
                numerical stability (default: 1e-10)
    
        .. _Adaptive Subgradient Methods for Online Learning and Stochastic
            Optimization: http://jmlr.org/papers/v12/duchi11a.html
        """
    
        def __init__(self, params, lr=1e-2, lr_decay=0, weight_decay=0, initial_accumulator_value=0, eps=1e-10):
            if not 0.0 <= lr:
                raise ValueError("Invalid learning rate: {}".format(lr))
            if not 0.0 <= lr_decay:
                raise ValueError("Invalid lr_decay value: {}".format(lr_decay))
            if not 0.0 <= weight_decay:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
            if not 0.0 <= initial_accumulator_value:
                raise ValueError("Invalid initial_accumulator_value value: {}".format(initial_accumulator_value))
            if not 0.0 <= eps:
                raise ValueError("Invalid epsilon value: {}".format(eps))
    
            defaults = dict(lr=lr, lr_decay=lr_decay, eps=eps, weight_decay=weight_decay,
                            initial_accumulator_value=initial_accumulator_value)
            super(Adagrad, self).__init__(params, defaults)
    
            for group in self.param_groups:
                for p in group['params']:
                    state = self.state[p]
                    state['step'] = 0
                    state['sum'] = torch.full_like(p, initial_accumulator_value, memory_format=torch.preserve_format)
    
        def share_memory(self):
            for group in self.param_groups:
                for p in group['params']:
                    state = self.state[p]
                    state['sum'].share_memory_()
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                params_with_grad = []
                grads = []
                state_sums = []
                state_steps = []
    
                for p in group['params']:
                    if p.grad is not None:
                        params_with_grad.append(p)
                        grads.append(p.grad)
                        state = self.state[p]
                        state_sums.append(state['sum'])
                        # update the steps for each param group update
                        state['step'] += 1
                        # record the step after step update
                        state_steps.append(state['step'])
    
                F.adagrad(params_with_grad,
                          grads,
                          state_sums,
                          state_steps,
                          group['lr'],
                          group['weight_decay'],
                          group['lr_decay'],
                          group['eps'])
    
            return loss

## AdaDelta / RMSProp ##

由于AdaGrad单调递减的学习率变化过于激进，我们考虑一个改变二阶动量计算方法的策略：不累积全部历史梯度，而只关注过去一段时间窗口的下降梯度。这也就是AdaDelta名称中Delta的来历。

修改的思路很简单。前面我们讲到，指数移动平均值大约就是过去一段时间的平均值，因此我们用这一方法来计算二阶累积动量：

![72bb40f59c0f098bd090919bb891d7b0.png][]

接下来还是步骤3：

这就避免了二阶动量持续累积、导致训练过程提前结束的问题了。

### RMSProp ###

> **定义优化器：**

CLASS torch.optim.RMSprop(params, lr=0.01, alpha=0.99, eps=1e-08, weight_decay=0, momentum=0, centered=False)

> **参数：**

*  **params** (iterable) – 优化器作用的模型参数。
 *  **lr** (float) – learning rate – 相当于是统一框架中的。
 *  **momentum** (float, optional) – 动量参数。(默认值：0)。
 *  **alpha**(*float,optional*) – 平滑常数 (默认值：0.99)。
 *  **centered**(bool,optional) – if`True`, compute the centered RMSProp, the gradient is normalized by an estimation of its variance，就是这一项是 True 的话就把方差使用梯度作归一化。
 *  **weight\_decay** (float, optional) – 权重衰减系数 weight decay (L2 penalty) (默认值：0)
 *  **eps**(float,optional)：防止分母为0的一个小数 (默认值：1e-10)

> **源码解读：**

import torch
    from .optimizer import Optimizer
    
    
    [docs]class RMSprop(Optimizer):
        r"""Implements RMSprop algorithm.
    
        Proposed by G. Hinton in his
        `course <https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf>`_.
    
        The centered version first appears in `Generating Sequences
        With Recurrent Neural Networks <https://arxiv.org/pdf/1308.0850v5.pdf>`_.
    
        The implementation here takes the square root of the gradient average before
        adding epsilon (note that TensorFlow interchanges these two operations). The effective
        learning rate is thus :math:`\alpha/(\sqrt{v} + \epsilon)` where :math:`\alpha`
        is the scheduled learning rate and :math:`v` is the weighted moving average
        of the squared gradient.
    
        Arguments:
            params (iterable): iterable of parameters to optimize or dicts defining
                parameter groups
            lr (float, optional): learning rate (default: 1e-2)
            momentum (float, optional): momentum factor (default: 0)
            alpha (float, optional): smoothing constant (default: 0.99)
            eps (float, optional): term added to the denominator to improve
                numerical stability (default: 1e-8)
            centered (bool, optional) : if ``True``, compute the centered RMSProp,
                the gradient is normalized by an estimation of its variance
            weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
    
        """
    
        def __init__(self, params, lr=1e-2, alpha=0.99, eps=1e-8, weight_decay=0, momentum=0, centered=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 <= momentum:
                raise ValueError("Invalid momentum value: {}".format(momentum))
            if not 0.0 <= weight_decay:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
            if not 0.0 <= alpha:
                raise ValueError("Invalid alpha value: {}".format(alpha))
    
            defaults = dict(lr=lr, momentum=momentum, alpha=alpha, eps=eps, centered=centered, weight_decay=weight_decay)
            super(RMSprop, self).__init__(params, defaults)
    
        def __setstate__(self, state):
            super(RMSprop, self).__setstate__(state)
            for group in self.param_groups:
                group.setdefault('momentum', 0)
                group.setdefault('centered', False)
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                for p in group['params']:
                    if p.grad is None:
                        continue
                    grad = p.grad
                    if grad.is_sparse:
                        raise RuntimeError('RMSprop does not support sparse gradients')
                    state = self.state[p]
    
                    # State initialization
                    if len(state) == 0:
                        state['step'] = 0
                        state['square_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        if group['momentum'] > 0:
                            state['momentum_buffer'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        if group['centered']:
                            state['grad_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
    
                    square_avg = state['square_avg']
                    alpha = group['alpha']
    
                    state['step'] += 1
    
                    if group['weight_decay'] != 0:
                        grad = grad.add(p, alpha=group['weight_decay'])
    
                    square_avg.mul_(alpha).addcmul_(grad, grad, value=1 - alpha)
    
                    if group['centered']:
                        grad_avg = state['grad_avg']
                        grad_avg.mul_(alpha).add_(grad, alpha=1 - alpha)
                        avg = square_avg.addcmul(grad_avg, grad_avg, value=-1).sqrt_().add_(group['eps'])
                    else:
                        avg = square_avg.sqrt().add_(group['eps'])
    
                    if group['momentum'] > 0:
                        buf = state['momentum_buffer']
                        buf.mul_(group['momentum']).addcdiv_(grad, avg)
                        p.add_(buf, alpha=-group['lr'])
                    else:
                        p.addcdiv_(grad, avg, value=-group['lr'])
    
            return loss

> 这里通过 grad = p.grad 得到每个参数的梯度，也就是1式的。
> 
> 如果使用 weight\_decay 的话，那么相当于目标函数加上，所以相当于是梯度相当于要再加上，故使用了 grad = grad.add(p, alpha=group\['weight\_decay'\])。
> 
> square\_avg.mul\_(alpha).addcmul\_(grad, grad, value=1 - alpha) 对应10式，计算当前步的。
> 
> centered 这一项是 False 的话直接 square\_avg.sqrt().add\_(group\['eps'\]) 对开根号。  
> centered 这一项是 True 的话就把方差使用梯度作归一化。
> 
> 最后通过 p.addcdiv\_(grad, avg, value=-group\['lr'\]) 更新梯度，相当于是上面的 3 式。  
> RMSprop算是Adagrad的一种发展，和Adadelta的变体，效果趋于二者之间

### AdaDelta ###

> **定义优化器：**

CLASS torch.optim.Adadelta(params, lr=1.0, rho=0.9, eps=1e-06, weight_decay=0)

> **参数：**

*  **params** (iterable) – 优化器作用的模型参数。
 *  **lr** (float) – learning rate – 相当于是统一框架中的。
 *  **rho**(float,optional) – 计算梯度平方的滑动平均超参数 (默认值：0.9)
 *  **weight\_decay** (float, optional) – 权重衰减系数 weight decay (L2 penalty) (默认值：0)
 *  **eps**(float,optional)：防止分母为0的一个小数 (默认值：1e-10)

> **源码解读：**

import torch
    
    from .optimizer import Optimizer
    
    
    [docs]class Adadelta(Optimizer):
        """Implements Adadelta algorithm.
    
        It has been proposed in `ADADELTA: An Adaptive Learning Rate Method`__.
    
        Arguments:
            params (iterable): iterable of parameters to optimize or dicts defining
                parameter groups
            rho (float, optional): coefficient used for computing a running average
                of squared gradients (default: 0.9)
            eps (float, optional): term added to the denominator to improve
                numerical stability (default: 1e-6)
            lr (float, optional): coefficient that scale delta before it is applied
                to the parameters (default: 1.0)
            weight_decay (float, optional): weight decay (L2 penalty) (default: 0)
    
        __ https://arxiv.org/abs/1212.5701
        """
    
        def __init__(self, params, lr=1.0, rho=0.9, eps=1e-6, weight_decay=0):
            if not 0.0 <= lr:
                raise ValueError("Invalid learning rate: {}".format(lr))
            if not 0.0 <= rho <= 1.0:
                raise ValueError("Invalid rho value: {}".format(rho))
            if not 0.0 <= eps:
                raise ValueError("Invalid epsilon value: {}".format(eps))
            if not 0.0 <= weight_decay:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
    
            defaults = dict(lr=lr, rho=rho, eps=eps, weight_decay=weight_decay)
            super(Adadelta, self).__init__(params, defaults)
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                for p in group['params']:
                    if p.grad is None:
                        continue
                    grad = p.grad
                    if grad.is_sparse:
                        raise RuntimeError('Adadelta does not support sparse gradients')
                    state = self.state[p]
    
                    # State initialization
                    if len(state) == 0:
                        state['step'] = 0
                        state['square_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        state['acc_delta'] = torch.zeros_like(p, memory_format=torch.preserve_format)
    
                    square_avg, acc_delta = state['square_avg'], state['acc_delta']
                    rho, eps = group['rho'], group['eps']
    
                    state['step'] += 1
    
                    if group['weight_decay'] != 0:
                        grad = grad.add(p, alpha=group['weight_decay'])
    
                    square_avg.mul_(rho).addcmul_(grad, grad, value=1 - rho)
                    std = square_avg.add(eps).sqrt_()
                    delta = acc_delta.add(eps).sqrt_().div_(std).mul_(grad)
                    p.add_(delta, alpha=-group['lr'])
                    acc_delta.mul_(rho).addcmul_(delta, delta, value=1 - rho)
    
            return loss

> 这里通过 grad = p.grad 得到每个参数的梯度，也就是1式的。
> 
> 如果使用 weight\_decay 的话，那么相当于目标函数加上，所以相当于是梯度相当于要再加上，故使用了 grad = grad.add(p, alpha=group\['weight\_decay'\])。
> 
> square\_avg.mul\_(rho).addcmul\_(grad, grad, value=1 - rho) 对应10式，计算当前步的。std = square\_avg.add(eps).sqrt\_() 对开根号。
> 
> 最后通过 p.add\_(delta, alpha=-group\['lr'\]) 更新梯度，相当于是上面的 3 式。  
> delta 的分子项是，分母项是开根号。acc\_delta 是对 delta 的滑动平均。

## Adam ##

谈到这里，Adam和Nadam的出现就很自然而然了——它们是前述方法的集大成者。我们看到，SGD-M在SGD基础上增加了一阶动量，AdaGrad和AdaDelta在SGD基础上增加了二阶动量。把一阶动量和二阶动量都用起来，就是Adam了——Adaptive + Momentum。

SGD的一阶动量：

![69432e20bd7f2177c73a90b00d7bc48b.png][]

加上AdaDelta的二阶动量：

优化算法里最常见的两个超参数就都在这里了，前者控制一阶动量，后者控制二阶动量。

## Nadam ##

最后是Nadam。我们说Adam是集大成者，但它居然遗漏了Nesterov，这还能忍？必须给它加上，按照NAG的步骤1：

![c4d28992113d10915a3ff608afd56ba6.png][]

这就是Nesterov + Adam = Nadam了。

> **定义优化器：**

CLASS torch.optim.Adam(params, lr=0.001, betas=(0.9, 0.999), eps=1e-08, weight_decay=0, amsgrad=False)

> **参数：**

*  **params** (iterable) – 优化器作用的模型参数。
 *  **lr** (float) – learning rate – 相当于是统一框架中的。
 *  **betas**(Tuple\[float,float\],optional) – coefficients used for computing running averages of gradient and its square ((默认值：(0.9, 0.999))
 *  **weight\_decay** (float, optional) – 权重衰减系数 weight decay (L2 penalty) (默认值：0)
 *  **eps**(float,optional)：防止分母为0的一个小数 (默认值：1e-10)

> **源码解读：**

import math
    import torch
    from .optimizer import Optimizer
    
    
    [docs]class Adam(Optimizer):
        r"""Implements Adam algorithm.
    
        It has been proposed in `Adam: A Method for Stochastic Optimization`_.
    
        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)
    
        .. _Adam\: A Method for Stochastic Optimization:
            https://arxiv.org/abs/1412.6980
        .. _On the Convergence of Adam and Beyond:
            https://openreview.net/forum?id=ryQu7f-RZ
        """
    
        def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8,
                     weight_decay=0, amsgrad=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]))
            if not 0.0 <= weight_decay:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
            defaults = dict(lr=lr, betas=betas, eps=eps,
                            weight_decay=weight_decay, amsgrad=amsgrad)
            super(Adam, self).__init__(params, defaults)
    
        def __setstate__(self, state):
            super(Adam, self).__setstate__(state)
            for group in self.param_groups:
                group.setdefault('amsgrad', False)
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                for p in group['params']:
                    if p.grad is None:
                        continue
                    grad = p.grad
                    if grad.is_sparse:
                        raise RuntimeError('Adam does not support sparse gradients, please consider SparseAdam instead')
                    amsgrad = group['amsgrad']
    
                    state = self.state[p]
    
                    # State initialization
                    if len(state) == 0:
                        state['step'] = 0
                        # Exponential moving average of gradient values
                        state['exp_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        # Exponential moving average of squared gradient values
                        state['exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        if amsgrad:
                            # Maintains max of all exp. moving avg. of sq. grad. values
                            state['max_exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
    
                    exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
                    if amsgrad:
                        max_exp_avg_sq = state['max_exp_avg_sq']
                    beta1, beta2 = group['betas']
    
                    state['step'] += 1
                    bias_correction1 = 1 - beta1 ** state['step']
                    bias_correction2 = 1 - beta2 ** state['step']
    
                    if group['weight_decay'] != 0:
                        grad = grad.add(p, alpha=group['weight_decay'])
    
                    # Decay the first and second moment running average coefficient
                    exp_avg.mul_(beta1).add_(grad, alpha=1 - beta1)
                    exp_avg_sq.mul_(beta2).addcmul_(grad, grad, value=1 - beta2)
                    if amsgrad:
                        # Maintains the maximum of all 2nd moment running avg. till now
                        torch.max(max_exp_avg_sq, exp_avg_sq, out=max_exp_avg_sq)
                        # Use the max. for normalizing running avg. of gradient
                        denom = (max_exp_avg_sq.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
                    else:
                        denom = (exp_avg_sq.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
    
                    step_size = group['lr'] / bias_correction1
    
                    p.addcdiv_(exp_avg, denom, value=-step_size)
    
            return loss

> 这里通过 grad = p.grad 得到每个参数的梯度，也就是1式的。
> 
> 如果使用 weight\_decay 的话，那么相当于目标函数加上，所以相当于是梯度相当于要再加上，故使用了 grad = grad.add(p, alpha=group\['weight\_decay'\])。
> 
> exp\_avg.mul\_(beta1).add\_(grad, alpha=1 - beta1) 计算12式。  
> exp\_avg\_sq.mul\_(beta2).addcmul\_(grad, grad, value=1 - beta2) 计算13式。  
> 因为15式的缘故，要给分母除以 math\*\*.\*\*sqrt(bias\_correction2)。  
> 因为14式的缘故，要给分子除以 bias\_correction1。  
> 最后通过 p.addcdiv\_(exp\_avg, denom, value=-step\_size) 更新梯度，相当于是上面的 3 式。

## AdamW ##

下图1所示为Adam的另一个改进版：AdamW。

简单来说，AdamW就是Adam优化器加上L2正则，来限制参数值不可太大，这一点属于机器学习入门知识了。以往的L2正则是直接加在损失函数上，比如这样子：加入正则，损失函数就会变成这样子：

![62c19c543cf8666ef8060e87307f204c.png][]

所以在计算梯度时要加上粉色的这一项。

但AdamW稍有不同，如下图所示，将正则加在了绿色位置。

![2598e93bde87138cfe942c1ed87f5b83.png][]

图1：AdamW

至于为何这么做？直接摘录BERT里面的原话看看：

> **Just** adding the square of the weights to the loss function is \*not\* the correct way of using L2 regularization/weight decay with Adam, since that will interact with the m and v parameters in strange ways. Instead we want to decay the weights in a manner that doesn't interact with the m/v parameters. This is equivalent to adding the square of the weights to the loss with plain (non-momentum) SGD. Add weight decay at the end (fixed version).

这段话意思是说，如果直接将L2正则加到loss上去，由于Adam优化器的后序操作，该正则项将会与和产生奇怪的作用。因而，AdamW选择将正则项加在了Adam的和等参数被计算完之后、在与学习率相乘之前，所以这也表明了weight\_decay和正则虽目的一致、公式一致，但用法还是不同，二者有着明显的差别。以 PyTorch1.7.0 中的AdamW代码为例：

> **定义优化器：**

CLASS torch.optim.AdamW(params, lr=0.001, betas=(0.9, 0.999), eps=1e-08, weight_decay=0.01, amsgrad=False)

> **参数：**

> **源码解读：**

import math
    import torch
    from .optimizer import Optimizer
    
    
    [docs]class AdamW(Optimizer):
        r"""Implements AdamW algorithm.
    
        The original Adam algorithm was proposed in `Adam: A Method for Stochastic Optimization`_.
        The AdamW variant was proposed in `Decoupled Weight Decay Regularization`_.
    
        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 coefficient (default: 1e-2)
            amsgrad (boolean, optional): whether to use the AMSGrad variant of this
                algorithm from the paper `On the Convergence of Adam and Beyond`_
                (default: False)
    
        .. _Adam\: A Method for Stochastic Optimization:
            https://arxiv.org/abs/1412.6980
        .. _Decoupled Weight Decay Regularization:
            https://arxiv.org/abs/1711.05101
        .. _On the Convergence of Adam and Beyond:
            https://openreview.net/forum?id=ryQu7f-RZ
        """
    
        def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8,
                     weight_decay=1e-2, amsgrad=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]))
            if not 0.0 <= weight_decay:
                raise ValueError("Invalid weight_decay value: {}".format(weight_decay))
            defaults = dict(lr=lr, betas=betas, eps=eps,
                            weight_decay=weight_decay, amsgrad=amsgrad)
            super(AdamW, self).__init__(params, defaults)
    
        def __setstate__(self, state):
            super(AdamW, self).__setstate__(state)
            for group in self.param_groups:
                group.setdefault('amsgrad', False)
    
    [docs]    @torch.no_grad()
        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:
                with torch.enable_grad():
                    loss = closure()
    
            for group in self.param_groups:
                for p in group['params']:
                    if p.grad is None:
                        continue
    
                    # Perform stepweight decay
                    p.mul_(1 - group['lr'] * group['weight_decay'])
    
                    # Perform optimization step
                    grad = p.grad
                    if grad.is_sparse:
                        raise RuntimeError('Adam does not support sparse gradients, please consider SparseAdam instead')
                    amsgrad = group['amsgrad']
    
                    state = self.state[p]
    
                    # State initialization
                    if len(state) == 0:
                        state['step'] = 0
                        # Exponential moving average of gradient values
                        state['exp_avg'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        # Exponential moving average of squared gradient values
                        state['exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
                        if amsgrad:
                            # Maintains max of all exp. moving avg. of sq. grad. values
                            state['max_exp_avg_sq'] = torch.zeros_like(p, memory_format=torch.preserve_format)
    
                    exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
                    if amsgrad:
                        max_exp_avg_sq = state['max_exp_avg_sq']
                    beta1, beta2 = group['betas']
    
                    state['step'] += 1
                    bias_correction1 = 1 - beta1 ** state['step']
                    bias_correction2 = 1 - beta2 ** state['step']
    
                    # Decay the first and second moment running average coefficient
                    exp_avg.mul_(beta1).add_(grad, alpha=1 - beta1)
                    exp_avg_sq.mul_(beta2).addcmul_(grad, grad, value=1 - beta2)
                    if amsgrad:
                        # Maintains the maximum of all 2nd moment running avg. till now
                        torch.max(max_exp_avg_sq, exp_avg_sq, out=max_exp_avg_sq)
                        # Use the max. for normalizing running avg. of gradient
                        denom = (max_exp_avg_sq.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
                    else:
                        denom = (exp_avg_sq.sqrt() / math.sqrt(bias_correction2)).add_(group['eps'])
    
                    step_size = group['lr'] / bias_correction1
    
                    p.addcdiv_(exp_avg, denom, value=-step_size)
    
            return loss

> 与 Adam 不一样的地方是：  
> Adam 如果使用 weight\_decay 的话，那么相当于目标函数加上，所以相当于是梯度相当于要再加上，故使用了 grad = grad.add(p, alpha=group\['weight\_decay'\])。
> 
> 而 AdamW 是 p.mul\_(1 - group\['lr'\] \* group\['weight\_decay'\]) 直接让参数：  
>   
> 这样才能和绿色框一致。

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[b7adf2d44416ff29051afca0c37e25b2.png]: /images/20220829/4a6fe9d395694d238b21f2df30674526.png
[695da8188d7244cb3cd059f1941f0549.png]: /images/20220829/6490b10b260b49fba7955a6ed2ba8af7.png
[864c38c0d6405c66b72c63650397cf3a.png]: /images/20220829/bb81137de17341f7be4e8f553aba86a8.png
[72bb40f59c0f098bd090919bb891d7b0.png]: /images/20220829/71f8b31c02aa4a33b6b16dba30a96f4b.png
[69432e20bd7f2177c73a90b00d7bc48b.png]: /images/20220829/f7e1286ec99d4262af9cee4894fe3aad.png
[c4d28992113d10915a3ff608afd56ba6.png]: /images/20220829/2fb707da7e234b64a10db5947826543f.png
[62c19c543cf8666ef8060e87307f204c.png]: /images/20220829/1aed9e61ca1d49be8ebbf8d173a69745.png
[2598e93bde87138cfe942c1ed87f5b83.png]: /images/20220829/c722e667f683480c981b26138aeeaf4f.png
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[CVPR 2021_GAN]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe2a4289c95dcb9f047c18e63b3be9d78e5e75f1d94dc80be34686ce6b4f22cdee2d7cc7f2ae&idx=1&mid=2247491634&scene=21&sn=427cb6bfd6868662f2b5ae1ac37f6072#wechat_redirect
[CVPR 2021_GAN 1]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29bfe4c95e36f275332b31b993532b98d52acc76d26c15e2321de4cf52141510bee3ff96af&idx=1&mid=2247490911&scene=21&sn=671926492c3b36e3350db4963f7fcb16#wechat_redirect
[GAN_StyleGAN]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29bedbc95e37cd709f583372918478ad8d3cff5309a85f3a940e7c0356050278f89d7e1c79&idx=2&mid=2247490656&scene=21&sn=bbe156fef077f23263b57e506ad43314#wechat_redirect
[20_ StyleGAN]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29b979c95e306f03b134809b16009d06aa0694b03da3368f84625de3700e6a56221d49d1bf&idx=1&mid=2247490498&scene=21&sn=324cfbbde362a728aa5336f4b200f7a3#wechat_redirect
[100_CVPR 2020_GAN]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29a94cc95e205ae880cb9518cb16a54c0abde69e5a8654f7fedf70a0cc990775265bf28a1c&idx=1&mid=2247486455&scene=21&sn=94156f53845cdf07e859e343862be745#wechat_redirect
[_ _Python]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29abb6c95e22a04d1eb019f964c8ae97b2f3af98c22f730f5d5ed2aea9fdb320b00c688967&idx=2&mid=2247485709&scene=21&sn=55a51c98ab48f393ba305c082ba0e4fe#wechat_redirect
[_ _Think Python]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29a418c95e2d0e9a3de46e0e4d1b8d4947154d8639418a511a3091300aa31699932046556e&idx=2&mid=2247485091&scene=21&sn=f569d5d70c2e6c9cc567fbc5147787a4#wechat_redirect
[_ _Pytorch]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29aa78c95e236e380c9fd2ae79917d7451ccdd4803c5e6047a16f921d1b61d18efc79ada7c&idx=2&mid=2247485635&scene=21&sn=90608bce32fe353cf3c64b39e3406bc3#wechat_redirect
[_ _2020]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29a4a4c95e2db2aab16bdcefa78795bbb7efbc70b533bb197d73f617558f2f8a523120c6e3&idx=2&mid=2247484959&scene=21&sn=563999cc8d48498beed00270844ceec4#wechat_redirect
[Link 1]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29a4bac95e2dac758c4aac10a3aac49eb062bc0829d4ef2ebc8e6d5069d246385c5084a2b4&idx=1&mid=2247484929&scene=21&sn=523220858287ed3e6105919703c8e833#wechat_redirect
[_TensorFlow 2.0]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29a5b0c95e2ca6599d4a91d5d40900566b7525de3dd85259e7a6c173aa26aa7ea3641d1aaf&idx=2&mid=2247485195&scene=21&sn=e3387067db29d1759755c6c2ea96c870#wechat_redirect
[Link 2]: http://mp.weixin.qq.com/s?__biz=MzU5MTgzNzE0MA%3D%3D&chksm=fe29aa8fc95e23993e057fb0dbe09f2810ea07184a439c173f404a7ab657cadcde3d24efa52a&idx=2&mid=2247485492&scene=21&sn=c4fb135de02cd52f06bc61b0e79476e4#wechat_redirect
[68a62c7f59b7dc9cbb82fdd63b19ffc9.png]: /images/20220829/82fbff4eec6549a0b972be6df7f5ca98.png