We shall make use of Adam optimization to briefly explain the epsilon coefficient. For the Adam optimizer, we know that the first and second moments are calculated via;

$$V_{dw}=\beta_1 \cdot V_{dw}+(1-\beta_1)\cdot \partial w\S_{dw}=\beta_2 S_{dw}+(1-\beta_2)\cdot \partial w^2$$

$\partial w$ is the derivative of the loss function with respect to a parameter.

$V{dw}$ is the running average of the decaying gradients(momentum term) and $S{dw}$ is the decaying average of the gradients.

$$V_{dw}=\beta_1 \cdot V_{dw}+(1-\beta_1)$$

And the parameter updates are done as follows;

$$theta_{k+1}=\theta_k-\eta \cdot \frac{V_{dw}^{corrected}}{\sqrt{S_{dw}^{corrected}}+\epsilon}$$

The epsilon in the aforementioned update is the epsilon coefficient.

Note that when the bias-corrected $S_{dw}$ gets close to zero, the denominator is undefined. Hence, the update is arbitrary. To rectify this, we use a small epsilon such that it stabilizes this numeric.

The standard value of the epsilon is 1e-08.
      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.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 algorithms. 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,amsgrad=true,eps=1e-08)
#setting the amsgrad to be true
#setting the epsilon to be 1e-08
#note that we are using Adam in our example
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 Tensors it will update (which are the learnable weights
  # of the model)

  # Backward pass: compute gradient of the loss with respect to model parameters

  # Calling the step function on an Optimizer makes an update to its parameters

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