AdaMax is an Adam-family optimizer that uses a running infinity norm of the gradients to scale updates. A minimal implementation needs two state values for each parameter: an exponentially averaged gradient and an infinity-norm accumulator. The equations below follow the documented AdaMax formulation in the original Adam paper and PyTorch’s Adamax API documentation.
How AdaMax differs from Adam
Adam scales its exponentially averaged gradient using a running second-moment estimate. AdaMax replaces that scaling estimate with an infinity-norm accumulator: element by element, it tracks the larger of the decayed previous accumulator and the current gradient magnitude (with epsilon added). The first-moment estimate still supplies the update direction. Kingma and Ba introduced AdaMax as an Adam variant based on the infinity norm in Adam: A Method for Stochastic Optimization.
AdaMax update equations
For a minimization objective, let θₜ be the parameter vector at step t and gₜ its gradient at the current parameters. Keep the following optimizer state:
- mₜ: the exponentially averaged gradient, or first moment.
- uₜ: the running infinity-norm accumulator.
- β₁ and β₂: decay factors for the first moment and infinity accumulator.
- γ: the learning rate; ε: a small stabilizing constant.
Initialize m₀ and u₀ to zero, with shapes matching the parameters. At each step, calculate the gradient and update the states elementwise:
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- First moment:
mₜ = β₁mₜ₋₁ + (1 − β₁)gₜ - Infinity accumulator:
uₜ = max(β₂uₜ₋₁, |gₜ| + ε) - Parameter update:
θₜ = θₜ₋₁ − γmₜ / ((1 − β₁ᵗ)uₜ)
The maximum and absolute value are elementwise operations. In this formulation, the bias correction is applied to the first moment through 1 − β₁ᵗ; the update does not apply a corresponding second-moment correction to uₜ.
Implement the optimizer step
This framework-neutral pseudocode expresses the documented equations. It assumes the caller computes gradients and invokes the step once per batch. Tensor operations such as abs, maximum, and division must act elementwise, and state must persist across calls.
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initialize m = zeros_like(parameters)
initialize u = zeros_like(parameters)
t = 0
for each batch:
gradients = gradient_of_loss(parameters, batch)
t = t + 1
m = beta1 * m + (1 - beta1) * gradients
u = elementwise_max(beta2 * u, elementwise_abs(gradients) + epsilon)
parameters = parameters - learning_rate * m / ((1 - beta1 ** t) * u)For multiple parameter tensors, keep separate m and u tensors for each one, increment a consistent step counter, and update each tensor using its own gradient and state. The counter should advance once per optimizer step, not once per parameter tensor. Do not reinitialize the state between batches; doing so discards the accumulated history that defines the optimizer.
Weight decay and implementation details
PyTorch’s documented pseudocode supports optional coupled weight decay by adding λθ to the gradient before the moment and accumulator updates. In that convention, use the modified gradient in both equations:
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g = gradient + weight_decay * parameters
m = beta1 * m + (1 - beta1) * g
u = elementwise_max(beta2 * u, elementwise_abs(g) + epsilon)Do not assume every optimizer’s weight-decay option has the same meaning. Epsilon placement also matters: the PyTorch equation adds ε to the absolute gradient inside the maximum, rather than adding it to the final denominator. Match the chosen reference implementation if compatibility is important.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Reference settings and library differences
PyTorch documents these Adamax API defaults (accessed 2026-10-04). They are library defaults, not universal recommendations or proof of optimal settings.
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| Setting | PyTorch documented default |
|---|---|
| Learning rate | 0.002 |
| Betas (β₁, β₂) | (0.9, 0.999) |
| Epsilon | 1e-08 |
| Weight decay | 0 |
The PyTorch API also exposes options including foreach, maximize, differentiable, and capturable; these are library-interface features rather than requirements for the educational algorithm above. Apple’s MLX Adamax documentation describes AdaMax as an infinity-norm Adam variant. Its note that the MLX Adam implementation omits bias correction for first and second moments concerns that Adam implementation; it should not be taken as a general statement about all AdaMax implementations. When matching a framework, check its actual equations, epsilon placement, bias-correction convention, weight-decay semantics, and state handling.
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Checklist for a from-scratch implementation
- Initialize m and u to zero for every parameter tensor.
- Compute gradients at the current parameter values, then update the states and parameters in order.
- Use elementwise absolute value and maximum for tensor gradients.
- Increment the step count consistently and retain all optimizer state between batches.
- Choose and document the precise conventions for epsilon and weight decay when comparing against a library.
- Treat framework defaults as reference values to evaluate for your task, not as guaranteed best-performing hyperparameters.
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