Why architecture matters here
Adam/AdamW matters because it's the optimizer that trains most transformers (and much of deep learning) -- its adaptive per-parameter learning rates and momentum making it robust and effective -- so understanding it is understanding how these models are trained. Training neural networks requires an optimizer (adjusting the parameters via the gradients to minimize the loss). Adam (and AdamW -- the transformer variant) is the workhorse (training most transformers and much of deep learning) -- because it's robust and effective (its adaptivity -- per-parameter learning rates -- and momentum handling the varied, noisy gradients of deep networks well -- versus SGD's single learning rate, which is harder to tune and less robust). So Adam/AdamW is central to how these models are trained (the optimizer doing the actual parameter updates -- the learning). For understanding deep learning training (especially transformers), understanding Adam/AdamW (how it adaptively updates the parameters) is essential -- it's the optimizer behind the models, and its hyperparameters and memory cost are practical training concerns.
The momentum-plus-adaptive-scaling insight is the core of Adam, combining two powerful ideas. Adam combines two ideas that each address a limitation of plain SGD. Momentum (the first moment): instead of using the raw, noisy gradient (which fluctuates -- the stochastic gradient noisy), Adam uses a smoothed gradient (an exponential moving average of the gradients -- the first moment) -- reducing the noise (the average smoother than the individual noisy gradients) and accelerating in consistent directions (the momentum building up in directions where the gradient is consistent -- like a ball rolling downhill, accelerating). Adaptive per-parameter scaling (the second moment): Adam tracks each parameter's gradient magnitude/variance (the second moment -- the exponential average of the squared gradients) and scales each parameter's update inversely to it (dividing by the square root of the second moment) -- so a parameter with large/noisy gradients (large second moment) takes smaller steps (scaled down -- cautious), and a parameter with small/consistent gradients (small second moment) takes larger steps (scaled up) -- an adaptive per-parameter learning rate (each parameter's effective step size adapted to its gradient magnitude). Combining these -- momentum (smoothed direction) and adaptive scaling (per-parameter step size) -- gives Adam its robustness (handling varied, noisy gradients -- smoothing the noise, adapting the step per parameter). This momentum-plus-adaptive-scaling combination is the core of Adam. Understanding the momentum-plus-adaptive-scaling core (smoothed gradient direction + per-parameter adaptive step size) is understanding what makes Adam effective.
And the AdamW-decoupled-weight-decay insight is a crucial practical detail for transformers, because how weight decay interacts with the adaptivity matters. Weight decay (a regularization -- shrinking the weights toward zero -- to prevent overfitting) is important for training. The original Adam handled weight decay by adding it to the gradient (L2 regularization -- the weight decay going through the adaptive scaling) -- but this interacts badly with Adam's adaptive scaling (the weight decay scaled per-parameter by the second moment -- so the effective weight decay varies per parameter -- not the intended uniform decay -- weakening the regularization). AdamW fixes this by decoupling the weight decay from the adaptive scaling: the weight decay is applied directly to the weights (separately -- not through the gradient/adaptive scaling) -- so it's a proper, uniform weight decay (as intended -- not distorted by the adaptivity). This decoupled weight decay (AdamW) improves the generalization (the proper regularization -- better than the original Adam's distorted weight decay) -- which is why AdamW (not the original Adam) is used for transformers (the proper weight decay mattering for the generalization). So AdamW's decoupled weight decay (applying it directly, not through the adaptive scaling -- proper regularization) is a crucial practical detail (the reason AdamW is preferred for transformers). Understanding the AdamW-decoupled-weight-decay insight (decoupling the weight decay from the adaptivity -- proper regularization -- why AdamW is used) is understanding a crucial practical detail.
The architecture: every piece explained
Top row: the problem and moments. The problem: a single learning rate (SGD) isn't enough -- different parameters need different step sizes (varied gradients). Momentum (1st moment): a smoothed (exponentially-averaged) gradient -- reducing noise, accelerating in consistent directions. Adaptive (2nd moment): tracking each parameter's gradient variance/magnitude -- for per-parameter scaling (dividing the update by the square root of it -- adaptive step sizes). Bias correction: correcting the moment estimates' initialization bias (the moments start at zero -- biased toward zero early -- corrected -- so the early updates aren't distorted).
Middle row: the rule and variant. The update rule: the update is (roughly) the first moment (momentum) divided by the square root of the second moment (variance) -- plus epsilon for stability -- momentum for direction, adaptive scaling for step size. AdamW: the variant with decoupled weight decay (applying the weight decay directly to the weights -- not through the adaptive scaling -- proper regularization -- used for transformers). Hyperparameters: the learning rate (the base step size), the betas (beta1, beta2 -- the moments' exponential averaging rates -- how much history), and epsilon (a small constant for numerical stability -- avoiding division by zero) -- the tuning knobs. Memory cost: storing the two moment estimates per parameter (~2x the parameters' memory -- the optimizer states) -- a notable memory cost (relevant for large models).
Bottom rows: comparison and schedules. vs SGD: Adam (adaptive per-parameter learning rates, momentum -- robust, effective, easier to tune) vs SGD (a single learning rate, optionally with momentum -- simpler, sometimes better final generalization with careful tuning, but harder to tune) -- Adam's adaptivity making it the default for transformers. Schedules + warmup: the learning rate over time -- a schedule (e.g., decaying the learning rate) and warmup (starting with a small learning rate and ramping up -- important for transformer training stability -- avoiding early instability) -- managing the learning rate over training. The ops strip: hyperparameter tuning (tuning the learning rate -- the most important -- the betas, epsilon, weight decay -- for effective, stable training), memory (the optimizer's memory cost -- ~2x the parameters for the moment states -- a consideration for large models -- and why memory-efficient optimizers/techniques exist), and stability (training stability -- the warmup, the learning rate, gradient clipping -- avoiding divergence/instability).
End-to-end flow
Trace an Adam update. During training, the gradients are computed (for each parameter). Adam updates its estimates: the first moment (momentum -- the exponential average of the gradients -- updated with the new gradient -- a smoothed gradient) and the second moment (the exponential average of the squared gradients -- the gradient magnitude/variance -- updated). Bias correction is applied (correcting the moments' early bias toward zero). Then the update is computed: (roughly) the bias-corrected first moment (momentum) divided by the square root of the bias-corrected second moment (plus epsilon) -- so the direction is the smoothed gradient (momentum) and the per-parameter step size is scaled by the inverse gradient magnitude (adaptive -- large-gradient parameters smaller steps, small-gradient larger). The parameters are updated (by the learning rate times this) -- and, for AdamW, the weight decay is applied directly (decoupled -- shrinking the weights). So Adam updated the parameters adaptively (momentum for direction, per-parameter adaptive scaling for step size) -- the robust, effective update. The Adam update (momentum divided by the root of the variance, with bias correction and decoupled weight decay) adaptively updated the parameters.
The adaptivity and AdamW vignettes show the key ideas. An adaptivity case: two parameters have different gradient characteristics -- one with large, noisy gradients (a large second moment) and one with small, consistent gradients (a small second moment). Adam scales their updates adaptively -- the large-gradient parameter takes smaller steps (scaled down by its large second moment -- cautious -- avoiding overshooting on the large/noisy gradient), and the small-gradient parameter takes larger steps (scaled up -- making progress despite the small gradient) -- the adaptive per-parameter learning rate (versus SGD's single rate -- which would be too large for the noisy parameter or too small for the small-gradient one). The adaptivity handled the varied gradients. An AdamW case: the team uses AdamW (not the original Adam) for their transformer -- because AdamW's decoupled weight decay (applied directly -- proper regularization) improves the generalization (versus the original Adam's distorted weight decay -- through the adaptive scaling) -- so the transformer generalizes better (the proper weight decay). The AdamW gave the proper weight decay.
The memory and schedule vignettes complete it. A memory case: the team notes Adam's memory cost -- storing the two moment estimates per parameter (~2x the parameters' memory -- the optimizer states) -- which for a large model is significant (2x the parameter memory just for the optimizer) -- a consideration (and why techniques like memory-efficient optimizers, or QLoRA's paged optimizers, exist) -- managing the optimizer memory. The memory cost was considered. A schedule case: the team uses a learning-rate schedule with warmup -- starting with a small learning rate and ramping up (warmup -- avoiding early instability -- important for transformers) then decaying it over training (the schedule) -- managing the learning rate over time (for stable, effective training). The warmup and schedule stabilized the training. The consolidated discipline the team documents: use Adam/AdamW to train transformers (its adaptive per-parameter learning rates -- from the second moment -- and momentum -- the first moment -- making it robust and effective -- versus SGD's single learning rate), understand the two ideas (momentum -- smoothed gradient direction; adaptive scaling -- per-parameter step size) and the update rule (momentum divided by the root of the variance, with bias correction), use AdamW (the decoupled weight decay -- proper regularization -- for transformers), tune the hyperparameters (the learning rate especially -- plus betas, epsilon, weight decay), manage the memory cost (~2x the parameters for the moment states -- a consideration for large models), use a learning-rate schedule with warmup (for stable transformer training), and ensure stability (warmup, learning rate, gradient clipping) -- because Adam/AdamW is the workhorse optimizer that trains most transformers (adaptive per-parameter learning rates with momentum -- robust and effective), and understanding it (the moments, the update rule, AdamW's decoupled weight decay) is understanding how these models are trained.