PyTorch－Adam优化算法原理，公式，应用

　　　　概念：Adam 是一种可以替代传统随机梯度下降过程的一阶优化算法，它能基于训练数据迭代地更新神经网络权重。Adam 最开始是由 OpenAI 的 Diederik Kingma 和多伦多大学的 Jimmy Ba 在提交到 2015 年 ICLR 论文（Adam: A Method for Stochastic Optimization）中提出的．该算法名为「Adam」，其并不是首字母缩写，也不是人名。它的名称来源于适应性矩估计（adaptive moment estimation）

　　Adam(Adaptive Moment Estimation)本质上是带有动量项的RMSprop，它利用梯度的一阶矩估计和二阶矩估计动态调整每个参数的学习率。它的优点主要在于经过偏置校正后，每一次迭代学习率都有个确定范围，使得参数比较平稳。其公式如下：

　　其中，前两个公式分别是对梯度的一阶矩估计和二阶矩估计，可以看作是对期望E|gt|，E|gt^2|的估计;
公式3，4是对一阶二阶矩估计的校正，这样可以近似为对期望的无偏估计。可以看出，直接对梯度的矩估计对内存没有额外的要求，而且可以根据梯度进行动态调整。最后一项前面部分是对学习率n形成的一个动态约束，而且有明确的范围。

　　优点：

1、结合了Adagrad善于处理稀疏梯度和RMSprop善于处理非平稳目标的优点;
2、对内存需求较小;
3、为不同的参数计算不同的自适应学习率;
4、也适用于大多非凸优化-适用于大数据集和高维空间。

　　应用和源码：

　　参数实例：

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

　　参数含义：

　　params(iterable)：可用于迭代优化的参数或者定义参数组的dicts。

　　lr (float, optional) ：学习率(默认: 1e-3) betas (Tuple[float, float], optional)：

　　用于计算梯度的平均和平方的系数(默认: (0.9, 0.999)) eps (float, optional)：

　　为了提高数值稳定性而添加到分母的一个项(默认: 1e-8) weight_decay (float, optional)：权重衰减(如L2惩罚)(默认: 0)

　　torch.optim.adam源码：

 import math
 from .optimizer import Optimizer class Adam(Optimizer):
     def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8，weight_decay=0):
         defaults = dict(lr=lr, betas=betas, eps=eps,weight_decay=weight_decay)
         super(Adam, self).__init__(params, defaults)     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
                 state = self.state[p]                 # State initialization
                 if len(state) == 0:
                     state['step'] = 0
                     # Exponential moving average of gradient values
                     state['exp_avg'] = grad.new().resize_as_(grad).zero_()
                     # Exponential moving average of squared gradient values
                     state['exp_avg_sq'] = grad.new().resize_as_(grad).zero_()                 exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq']
                 beta1, beta2 = group['betas']                 state['step'] += 1                 if group['weight_decay'] != 0:
                     grad = grad.add(group['weight_decay'], p.data)                 # 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)                 denom = exp_avg_sq.sqrt().add_(group['eps'])                 bias_correction1 = 1 - beta1 ** state['step']
                 bias_correction2 = 1 - beta2 ** state['step']
                 step_size = group['lr'] * math.sqrt(bias_correction2) / bias_correction1                 p.data.addcdiv_(-step_size, exp_avg, denom)         return loss

　　使用例子：

 import torch # N is batch size; D_in is input dimension;
 # H is hidden dimension; D_out is output dimension.
 N, D_in, H, D_out = 64, 1000, 100, 10 # Create random Tensors to hold inputs and outputs
 x = torch.randn(N, D_in)
 y = torch.randn(N, D_out) # Use the nn package to define our model and loss function.
 model = torch.nn.Sequential(
     torch.nn.Linear(D_in, H),
     torch.nn.ReLU(),
     torch.nn.Linear(H, D_out),
 )
 loss_fn = torch.nn.MSELoss(reduction='sum') # Use the optim package to define an Optimizer that will update the weights of
 # the model for us. Here we will use Adam; the optim package contains many other
 # optimization algoriths. The first argument to the Adam constructor tells the
 # optimizer which Tensors it should update.
 learning_rate = 1e-4
 optimizer = torch.optim.Adam(model.parameters(), lr=learning_rate)
 for t in range(500):
     # Forward pass: compute predicted y by passing x to the model.
     y_pred = model(x)     # Compute and print loss.
     loss = loss_fn(y_pred, y)
     print(t, loss.item())     # Before the backward pass, use the optimizer object to zero all of the
     # gradients for the variables it will update (which are the learnable
     # weights of the model). This is because by default, gradients are
     # accumulated in buffers( i.e, not overwritten) whenever .backward()
     # is called. Checkout docs of torch.autograd.backward for more details.
     optimizer.zero_grad()     # Backward pass: compute gradient of the loss with respect to model
     # parameters
     loss.backward()     # Calling the step function on an Optimizer makes an update to its
     # parameters
     optimizer.step()