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：
1 import math 2 from .optimizer import Optimizer 3 4 class Adam(Optimizer): 5 def __init__(self, params, lr=1e-3, betas=(0.9, 0.999), eps=1e-8，weight_decay=0): 6 defaults = dict(lr=lr, betas=betas, eps=eps,weight_decay=weight_decay) 7 super(Adam, self).__init__(params, defaults) 8 9 def step(self, closure=None): 10 loss = None 11 if closure is not None: 12 loss = closure() 13 14 for group in self.param_groups: 15 for p in group['params']: 16 if p.grad is None: 17 continue 18 grad = p.grad.data 19 state = self.state[p] 20 21 # State initialization 22 if len(state) == 0: 23 state['step'] = 0 24 # Exponential moving average of gradient values 25 state['exp_avg'] = grad.new().resize_as_(grad).zero_() 26 # Exponential moving average of squared gradient values 27 state['exp_avg_sq'] = grad.new().resize_as_(grad).zero_() 28 29 exp_avg, exp_avg_sq = state['exp_avg'], state['exp_avg_sq'] 30 beta1, beta2 = group['betas'] 31 32 state['step'] += 1 33 34 if group['weight_decay'] != 0: 35 grad = grad.add(group['weight_decay'], p.data) 36 37 # Decay the first and second moment running average coefficient 38 exp_avg.mul_(beta1).add_(1 - beta1, grad) 39 exp_avg_sq.mul_(beta2).addcmul_(1 - beta2, grad, grad) 40 41 denom = exp_avg_sq.sqrt().add_(group['eps']) 42 43 bias_correction1 = 1 - beta1 ** state['step'] 44 bias_correction2 = 1 - beta2 ** state['step'] 45 step_size = group['lr'] * math.sqrt(bias_correction2) / bias_correction1 46 47 p.data.addcdiv_(-step_size, exp_avg, denom) 48 49 return loss
使用例子：
1 import torch 2 3 # N is batch size; D_in is input dimension; 4 # H is hidden dimension; D_out is output dimension. 5 N, D_in, H, D_out = 64, 1000, 100, 10 6 7 # Create random Tensors to hold inputs and outputs 8 x = torch.randn(N, D_in) 9 y = torch.randn(N, D_out) 10 11 # Use the nn package to define our model and loss function. 12 model = torch.nn.Sequential( 13 torch.nn.Linear(D_in, H), 14 torch.nn.ReLU(), 15 torch.nn.Linear(H, D_out), 16 ) 17 loss_fn = torch.nn.MSELoss(reduction='sum') 18 19 # Use the optim package to define an Optimizer that will update the weights of 20 # the model for us. Here we will use Adam; the optim package contains many other 21 # optimization algoriths. The first argument to the Adam constructor tells the 22 # optimizer which Tensors it should update. 23 learning_rate = 1e-4 24 optimizer = torch.optim.Adam(model.parameters(), lr=learning_rate) 25 for t in range(500): 26 # Forward pass: compute predicted y by passing x to the model. 27 y_pred = model(x) 28 29 # Compute and print loss. 30 loss = loss_fn(y_pred, y) 31 print(t, loss.item()) 32 33 # Before the backward pass, use the optimizer object to zero all of the 34 # gradients for the variables it will update (which are the learnable 35 # weights of the model). This is because by default, gradients are 36 # accumulated in buffers( i.e, not overwritten) whenever .backward() 37 # is called. Checkout docs of torch.autograd.backward for more details. 38 optimizer.zero_grad() 39 40 # Backward pass: compute gradient of the loss with respect to model 41 # parameters 42 loss.backward() 43 44 # Calling the step function on an Optimizer makes an update to its 45 # parameters 46 optimizer.step()
到这里，相信对付绝大多数的应用是可以的了．我的目的也就基本完成了．接下来就要在应用中加深理解了．
参考文档：
１ https://blog.csdn.net/kgzhang/article/details/77479737
２ https://pytorch.org/tutorials/beginner/examples_nn/two_layer_net_optim.html
