Epsilon Coefficient

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;

$$V_{dw}=\beta1 \cdot V{dw}+(1-\beta1)\cdot \partial w\S{dw}=\beta2 S{dw}+(1-\beta_2)\cdot \partial w^2$$

$$\partial w$$ 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.

And the parameter updates are done as;

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

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.

Epsilon in code example

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 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)
  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()

Further Resources

• AMSgrad Variant
• Adam
• Base Learning Rate