AdamW Optimizer Introduction and Explanation | Generated by AI

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Question: What is a comprehensive introduction to the AdamW optimizer?

Answer:

Comprehensive Introduction to the AdamW Optimizer

The AdamW optimizer is a popular variant of the Adam optimizer widely used in deep learning for training neural networks. It improves upon standard Adam by properly handling weight decay (a form of regularization), leading to better generalization, especially in large models like transformers.

AdamW was introduced in the 2017 paper “Decoupled Weight Decay Regularization” by Ilya Loshchilov and Frank Hutter (published at ICLR 2019). The “W” stands for “Weight decay,” and the key innovation is decoupling weight decay from the adaptive learning rate updates.

Background: The Adam Optimizer

Adam (Adaptive Moment Estimation), proposed by Kingma and Ba in 2014, is an adaptive gradient algorithm that combines ideas from Momentum and RMSProp.

It maintains two moving averages:

First moment (mean of gradients, like momentum): \( m_t \)
Second moment (uncentered variance of gradients): \( v_t \)

The update rules for Adam are:

Compute gradient: \( g_t = \nabla_\theta f_t(\theta_{t-1}) \)
Update biased first moment:
\( m_t = \beta_1 m_{t-1} + (1 - \beta_1) g_t \)
Update biased second moment:
\( v_t = \beta_2 v_{t-1} + (1 - \beta_2) g_t^2 \)
Bias correction:
\( \hat{m}_t = \frac{m_t}{1 - \beta_1^t} \)
\( \hat{v}_t = \frac{v_t}{1 - \beta_2^t} \)
Parameter update:
\( \theta_t = \theta_{t-1} - \eta \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} \)

Default hyperparameters:

Learning rate \( \eta = 0.001 \)
\( \beta_1 = 0.9 \)
\( \beta_2 = 0.999 \)
\( \epsilon = 10^{-8} \)

Adam is efficient and robust but sometimes generalizes worse than SGD with momentum when strong regularization is used.

The Problem with Weight Decay in Standard Adam

Weight decay is a regularization technique that penalizes large weights by adding a term like \( \frac{\lambda}{2} |\theta|^2 \) to the loss (L2 regularization) or directly decaying weights.

In SGD, L2 regularization and true weight decay are equivalent (up to rescaling). But in adaptive optimizers like Adam, they are not.

Most implementations of Adam (e.g., early PyTorch, TensorFlow) apply “weight decay” by adding \( \lambda \theta \) to the gradient (treating it as L2 regularization). This couples weight decay with the adaptive mechanism, causing:

Uneven regularization across parameters
Interference with adaptive learning rates
Poorer generalization compared to true weight decay

AdamW: Decoupled Weight Decay

AdamW fixes this by decoupling weight decay: it applies the Adam update first (using only the loss gradient), then separately applies weight decay.

The update rules for AdamW are the same as Adam for moments (steps 1–4 above), but the parameter update becomes:

\( \theta_t = \theta_{t-1} - \eta \left( \frac{\hat{m}t}{\sqrt{\hat{v}_t} + \epsilon} + \lambda \theta{t-1} \right) \)

Or equivalently (common implementation):

First, Adam step: \( \theta_t’ = \theta_{t-1} - \eta \cdot \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} \)
Then, decay: \( \theta_t = \theta_t’ - \eta \lambda \theta_{t-1} \) (or approximate with \( (1 - \eta \lambda) \theta_t’ \))

This ensures:

Weight decay is applied consistently (proportional to current weights, independent of gradient magnitude)
Adaptive learning rates remain unaffected by regularization
Hyperparameters (learning rate and weight decay) are independent

Advantages of AdamW

Better generalization: Often outperforms standard Adam, especially with learning rate schedules (e.g., cosine annealing, warmup)
Improved performance: Widely used in state-of-the-art models (BERT, ViT, GPT, etc.)
Stability: More reliable convergence in large-scale training
Standard in modern libraries: PyTorch’s torch.optim.AdamW implements the correct decoupled version; Keras and TensorFlow also have dedicated AdamW

Typical weight decay values: 0.01–0.1 (often 0.01 or 0.05 for transformers)

When to Use AdamW

Training transformers or large vision models
When regularization is important for generalization
Fine-tuning pre-trained models
Most modern deep learning tasks (it’s often the default choice now)

In summary, AdamW is a small but crucial improvement over Adam that makes weight decay work as intended in adaptive optimizers, leading to superior performance in practice.

References:

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