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Weight Decay & AdamW: Decoupled Regularization

AdamW is the standard optimizer for LLM training—GPT, LLaMA, etc. all use it

Concept 58 of 100OptimizationPhase 3
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key equation
θt+1=(1−ηλ)θt−η⋅Adam_step\theta_{t+1} = (1 - \eta\lambda)\theta_t - \eta \cdot \text{Adam\_step}
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Concept 58 of 100AdamWOptimization / Phase 3: Optimization & generalization
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Adam rescales gradients, so L2 regularization gets rescaled too—breaking the intended effect

θt+1=(1−ηλ)θt−η⋅Adam_step\theta_{t+1} = (1 - \eta\lambda)\theta_t - \eta \cdot \text{Adam\_step}
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Why It Matters for Modern Models

  • AdamW is the standard optimizer for LLM training—GPT, LLaMA, etc. all use it
  • The distinction between L2 and weight decay is a common source of bugs in training
  • Weight decay strength is one of the most important hyperparameters for generalization

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • Adam rescales gradients, so L2 regularization gets rescaled too—breaking the intended effect
  • AdamW applies weight decay after the Adam update, preserving the regularization strength
  • Weight decay = "prefer simpler models"—it keeps weights small unless data strongly supports them

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Key Equation
θt+1=(1−ηλ)θt−η⋅Adam_step\theta_{t+1} = (1 - \eta\lambda)\theta_t - \eta \cdot \text{Adam\_step}

L2 regularization adds penalty to loss:

Lreg=L+λ2∥θ∥2L_{reg} = L + \frac{\lambda}{2} \|\theta\|^2
∇Lreg=∇L+λθ\nabla L_{reg} = \nabla L + \lambda \theta

Weight decay directly shrinks weights:

θt+1=(1−ηλ)θt−η∇L\theta_{t+1} = (1 - \eta \lambda) \theta_t - \eta \nabla L

For SGD: L2 regularization = weight decay. For Adam: they differ!

AdamW decouples weight decay from gradient updates:

mt=β1mt−1+(1−β1)∇Lm_t = \beta_1 m_{t-1} + (1-\beta_1) \nabla L
vt=β2vt−1+(1−β2)(∇L)2v_t = \beta_2 v_{t-1} + (1-\beta_2) (\nabla L)^2
θt+1=θt−η(m^tv^t+ϵ+λθt)\theta_{t+1} = \theta_t - \eta \left( \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} + \lambda \theta_t \right)

Canonical Papers

Decoupled Weight Decay Regularization

Loshchilov & Hutter2019ICLR
Read paper →

Fixing Weight Decay Regularization in Adam

Loshchilov & Hutter2018arXiv
Read paper →

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Prerequisites

∇AdamℒML/CE/KL

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