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Nadam is an adaptive gradient optimizer that combines Adam’s first- and second-moment estimates with a Nesterov-style adjustment to the momentum contribution. To implement it from scratch, keep a first-moment tensor and a second-moment tensor for each parameter, apply the chosen variant’s bias corrections consistently, then subtract the adjusted update during minimization.
What Nadam changes about Adam
Adam scales updates using an exponential moving average of gradients and another of squared gradients. Nadam retains both estimates but changes how the first moment contributes: its update combines a current-gradient term with a momentum term, in a Nesterov-style arrangement. The second moment still supplies coordinate-wise adaptive scaling.
That adjustment is the key to implementing Nadam correctly. There are multiple implementation conventions, so the momentum coefficients and their bias corrections must come from the same formulation. The equations below follow the Nadam variant documented by PyTorch; they are not a universal set of constants or conventions.
PyTorch-style Nadam update rule
For minimization, let θt−1 be the parameter vector before step t and let gt = ∇ft(θt−1) be the current minibatch gradient. The square and division operations below are elementwise. Initialize the moment tensors m0 and v0 to zero.
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- Update the first and second moments:
m_t = β₁ m_(t−1) + (1 − β₁) g_tv_t = β₂ v_(t−1) + (1 − β₂) g_t² - Compute the time-dependent momentum coefficients used by this variant:
μ_t = β₁ (1 − ½ · 0.96^(tψ))μ_(t+1) = β₁ (1 − ½ · 0.96^((t+1)ψ))
Here ψ is the momentum-decay parameter. - Form the adjusted first moment and corrected second moment:
m̂_t = μ_(t+1) m_t / (1 − ∏_(i=1)^(t+1) μ_i) + (1 − μ_t) g_t / (1 − ∏_(i=1)^t μ_i)v̂_t = v_t / (1 − β₂^t) - Update the parameters:
θ_t = θ_(t−1) − γ_t m̂_t / (√v̂_t + ε)
γt is the learning rate at step t. Add ε after taking the square root, as shown.
The adjusted first moment includes both the current gradient and momentum estimate, each with its corresponding correction. Dozat’s derivation presents the same central idea—Nesterov momentum incorporated into Adam—using its own notation and bias-correction presentation. Do not splice coefficients from one source into another source’s correction formula without verifying the resulting recurrence. Dozat’s Nadam paper and PyTorch’s NAdam documentation describe the respective formulations.
Implementation details that affect correctness
Keep state and timestep aligned
Each parameter needs matching m and v tensors, initialized to zero. The displayed PyTorch pseudocode starts at t = 1. If an implementation counts from zero, adjust all exponents and products to preserve the same first update; changing only the displayed t while leaving the corrections unchanged gives a different algorithm.
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Use the gradient and denominator as specified
For minimization, calculate the ordinary gradient and subtract the final update. Square g elementwise when updating v, and use the square root of the corrected v̂ in the denominator. Epsilon helps numerical stability, but its value is an implementation choice rather than a universal Nadam constant.
Separate optimizer recurrence from training-system features
Weight decay, gradient clipping, gradient accumulation, mixed precision, and learning-rate schedules are additional choices, not part of the core recurrence above. PyTorch documents both coupled weight decay, which adds a decay term to the gradient, and an optional decoupled form it identifies with NAdamW behavior. If you add any of these features, document and match them when comparing implementations. PyTorch’s NAdam API lists its options; availability can depend on framework version.
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Framework defaults are variant-specific
Published defaults differ, so report the framework and release or documentation branch when reproducing a result. The following values are documented defaults, not canonical Nadam constants:
| Documented variant | Learning rate | β₁, β₂ | ε | Momentum decay |
|---|---|---|---|---|
| TensorFlow v2.16.1 Nadam API | 0.001 | 0.9, 0.999 | 1e-7 | not stated (TensorFlow v2.16.1 API) |
| PyTorch current main NAdam documentation | 0.002 | 0.9, 0.999 | 1e-8 | 0.004 |
TensorFlow describes Nadam as Adam with Nesterov momentum. The values above come from its versioned v2.16.1 API and PyTorch’s current main documentation. The cited PyTorch page is a moving documentation branch, so it should not be treated as a pinned release reference.
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How to evaluate Nadam against Adam
Nadam is not guaranteed to outperform Adam. Dozat evaluated nine optimizers on word2vec, MNIST classification, and a Penn TreeBank LSTM language-model task, and described mixed, task-dependent results. In the paper’s language-model test results, Adam’s test perplexity was 111.0 and Nadam’s was 105.5; this is a result for that task and setup, not a general performance statistic. In the MNIST discussion, RMSProp exceeded Nadam on the test set even though Nadam performed best on the development set. Dozat’s paper reports these task-specific findings.
For a fair comparison, hold the important experimental conditions in view:
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- Objective, dataset, model, and initialization.
- Learning rate, moment hyperparameters, and tuning budget.
- Regularization, including whether weight decay is coupled or decoupled.
- Training budget and stopping rule.
- The exact optimizer implementation and framework version.
Framework API pages specify behavior and options; they are not independent benchmark evidence. Treat reported results as evidence for their named tasks, not as a promise about a different model or dataset.
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