Adam can be understood as updating weights inversely proportional to the scaled L2 norm (squared) of past gradients. AdaMax extends this to the so-called infinite norm (max) of past gradients.

The calculation of the infinite norm exhibits a stable behavior. Mathematically, the infinite norm can be viewed as,

$$u_t=\beta2^\infty v{t-1} + (1-\beta2^\infty v{t-1})|g_t|^\infty=max(\beta2 \cdot v{t-1},|g_t|)$$

We see the mathematical equivalence of the calculation of the infinite norm with the maximum of the parameters calculated up to t. Now the update would be similar to Adam,

$$\theta_{t+1}=\theta_t- \eta \cdot \frac{m_t}{u_t}$$

Again, here $\eta$ is the base learning rate and $m_t$ is the momentum similar to as discussed in Adam.

The exponential moving average and the infinite norm are calculated in Adamax. Mathematically, given by the formula,

$$V_{dw}=\beta1 \cdot V{dw}+(1-\beta_1)\cdot \partial w\ u_t=\beta2^\infty v{t-1} + (1-\beta2^\infty v{t-1})|g_t|^\infty$$

Here $\beta_1$ and $\beta_2$ are the betas. They are the exponential decay rates of the first moment and the exponentially weighted infinity norm.

      # importing the library
import torch
import torch.nn as nn

x = torch.randn(10, 3)
y = torch.randn(10, 2)

# Build a fully connected layer.
linear = nn.Linear(3, 2)

# Build MSE loss function and optimizer.
criterion = nn.MSELoss()

# Optimization method using Adamax
optimizer = torch.optim.Adamax(linear.parameters(), lr=0.002, 
betas=(0.9, 0.999), eps=1e-08, weight_decay=0)

# Forward pass.
pred = linear(x)

# Compute loss.
loss = criterion(pred, y)
print('loss:', loss.item())


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