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What Adam does during training
Adam stands for Adaptive Moment Estimation. It is a first-order method: it uses gradients of the objective with respect to the model’s parameters, rather than calculating a full Hessian or a full covariance matrix. For each parameter coordinate, it keeps estimates based on the current gradient and the history of earlier gradients.
At step t, let gt be the gradient of the current objective at the current parameters. The minimizing form of Adam’s update is:
- mt = β1mt−1 + (1 − β1)gt
- vt = β2vt−1 + (1 − β2)gt2
- m̂t = mt / (1 − β1t) and v̂t = vt / (1 − β2t)
- θt = θt−1 − α · m̂t / (√v̂t + ε)
Here, θ denotes the parameters and α is the learning rate. The square and division in the final expression are coordinate-wise: a parameter’s update is scaled using its own estimated squared-gradient magnitude.
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The first average: a smoothed direction
mt is an exponentially weighted moving average of gradients, often called the first moment estimate. It smooths the direction across steps, so the update reflects recent gradient history rather than only the latest noisy estimate. β1 controls how much history is retained: a larger value gives the past more influence.
The second average: a smoothed scale
vt is an exponentially weighted moving average of squared gradients, or second raw moment estimate. Squaring removes the sign, so this tracks the recent magnitude of gradients for each coordinate. β2 controls how quickly this scale estimate responds to new values. Adam divides by its square root, reducing the step for coordinates whose recent squared gradients have been relatively large.
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Why Adam corrects bias at the start
Both running averages are initialized to zero. Early on, they therefore tend to be smaller in magnitude than the histories they are meant to estimate. Adam corrects this startup bias by dividing each average by 1 − βt for its corresponding beta value. The correction is strongest at early steps and diminishes as the estimates accumulate history.
What epsilon does
ε is added to the denominator to help numerical stability. It is not another gradient history or a replacement for the learning rate; it affects the scale of the normalized update, especially when the denominator is small.
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How Adam’s design is motivated
In their paper “Adam: A Method for Stochastic Optimization”, Diederik P. Kingma and Jimmy Ba introduce Adam as a gradient-based method for stochastic objectives that uses adaptive estimates of lower-order moments. Their abstract describes the method as computationally efficient, light on memory, invariant to diagonal rescaling of gradients, and suited to non-stationary objectives and noisy or sparse gradients. These are the paper’s stated properties and motivation, not a promise that Adam will be best for every task or training setup.
What PyTorch’s documented defaults mean
The current PyTorch main documentation lists the following defaults for its torch.optim.Adam API. They describe that API, not universal settings for all frameworks or a recipe guaranteed to work best for every model and dataset.
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| Setting | PyTorch main default | Role |
|---|---|---|
lr |
0.001 |
The learning rate, or base scale of the parameter update. |
betas |
(0.9, 0.999) |
The coefficients for the running averages of the gradient and squared gradient, respectively. |
eps |
1e-8 |
The numerical-stability term added in the denominator. |
weight_decay |
0 |
The documented default weight-decay value. |
amsgrad |
False |
Whether to use the optional AMSGrad variant. |
PyTorch’s documentation also exposes choices such as foreach, fused, maximize, capturable, differentiable, and decoupled_weight_decay. These are implementation or API options, and their availability and behavior can vary by version; consult the documentation for the version in use. The PyTorch Adam documentation describes coupled weight decay as the default behavior. Setting decoupled_weight_decay=True makes the optimizer equivalent to AdamW according to that documentation, so Adam and AdamW should not be treated as interchangeable names when discussing regularization.
Why Adam is not guaranteed to converge
Adam’s practical popularity does not amount to a general convergence guarantee. In “On the Convergence of Adam and Beyond,” Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar give an explicit example of a simple convex optimization setting where Adam does not converge to the optimum. They identify an issue with earlier analysis and propose methods with longer-term memory, including AMSGrad.
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This result shows that convergence claims depend on the assumptions and the algorithm variant being analyzed. It does not show that Adam routinely fails on deep-learning workloads. PyTorch provides AMSGrad as an optional variant; choosing it is not, by itself, evidence that it will improve a given training run.
How to decide whether Adam is working for your model
Use the documented defaults as a starting point only when appropriate for your framework and task. The sources do not establish a universally optimal learning rate or an across-task winner, so evaluate choices against the objective that matters for your work.
Quick Recap
- Track the training objective and the validation measure relevant to the intended use; improvement on training data alone does not establish good validation performance.
- Compare optimizers, including Adam and SGD with momentum where relevant, under a fixed compute or training budget rather than comparing runs with unequal opportunity to train.
- Check stability across multiple seeds, convergence speed, memory use, sensitivity to learning rate and learning-rate schedules, and generalization. These are comparison criteria, not results established by the cited papers or API documentation.
- Change settings deliberately and record the framework version and optimizer options, especially when comparing weight-decay behavior or implementation variants.
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