Weight decay ($\lambda$)

Weight decay supplies a parameter-shrinkage contribution to an optimizer update, with a coefficient commonly denoted $\lambda$. Two implementations must be distinguished. Coupled L2 regularization adds the penalty $\lambda \Vert \theta {\Vert }^2/2$ to the objective function, replacing its data gradient $g$ by $g+\lambda \theta$. Decoupled weight decay applies parameter shrinkage separately from the optimizer's processing of the data gradient. These operations are equivalent for plain gradient descent with matched coefficients, but not in general for an adaptive gradient method.^[ref]

Use the PyTorch convention here: a decoupled update shrinks parameters by $1-\eta \lambda$, where $\eta$ is the learning rate. A source writing $1-\lambda$ instead has absorbed the step-size factor into its decay coefficient. For an unmodified gradient step,In AdamW, the decay contribution does not enter the stored first- or second-moment estimates of the gradient.^[ref]

An original fixed-scaling example exposes the difference. Let $\theta =(2,2{)}^T$, $g=(1,1{)}^T$, $\eta =0.1$, $\lambda =0.2$ and diagonal matrix $A=\mathrm{diag}(1,10)$. Applying the same scaling to the entire coupled gradient givesSeparating uniform parameter shrinkage givesThe second coordinate receives different shrinkage. This is an illustrative fixed matrix, not a full AdamW simulation. With history-dependent scaling, coupling the penalty into the gradient can also alter the optimizer state. State the update equation and coefficient convention when reporting a decay setting; the label alone does not specify either.

Decoupled weight decay applies a parameter-shrinkage contribution separately from the optimizer's processing of the data gradient. Write $u_t$ for the processed data-gradient direction, including any algorithm-specific optimizer state. Using the coefficient convention documented for PyTorch AdamW,The shrinkage term uses the pre-update parameter and is not inserted into the gradient moments.^[ref]

This differs from coupled L2 regularization, which processes $g_t+\lambda {\theta }_{t-1}$ as a single gradient input. Decoupled weight decay was introduced for adaptive optimization in the AdamW work; the original paper also writes decay coefficients with the learning-rate factor absorbed differently, so its coefficient must be translated before comparing equations.^[ref]

An original limiting case isolates the shrinkage. Suppose the processed direction is zero at every step and $\eta ,\lambda$ are constant. Repeated substitution givesWith $\eta =0.1$, $\lambda =0.2$ and ${\theta }_0=1$, one update gives 0.98 and 50 updates give approximately 0.36417. For varying learning rates, the corresponding factor isThus the same numerical decay coefficient can have a different accumulated effect under a different learning rate schedule. In this zero-direction model, $0<\eta \lambda <1$ gives sign-preserving shrinkage; other factors have different behaviour.

The update order is also part of the definition. For $\theta =1$, $u=2$, $\eta =0.1$ and $\lambda =0.2$, the stated formula gives $0.98-0.20=0.78$. Applying shrinkage after first taking the gradient step would instead give $(1-0.20)0.98=0.784$. Both operations may be described informally as decay plus a step, but they are different recurrences. An optimization checkpoint must retain the relevant state and settings to reproduce the chosen one.

Coupled L2 regularization adds a quadratic parameter penalty to the objective function and sends its gradient through the same optimizer update as the data gradient. With coefficient $\lambda \ge 0$, defineSince the gradient of the squared Euclidean norm divided by two is $\theta$, a supplied data gradient $g=\nabla f(\theta )$ becomesThe factor $1/2$ fixes the convention: a penalty written as $\lambda \Vert \theta {\Vert }_2^2$ instead contributes $2\lambda \theta$.

For plain gradient descent, this substitution givesIt is therefore equivalent to decoupled weight decay with the matched coefficient in this particular update. The equivalence need not survive coordinate-dependent processing of the gradient.^[ref]

For an original fixed-scaling example, let $\theta =(1,1{)}^T$, $g=0$, $\eta =0.1$, $\lambda =0.2$ and diagonal matrix $D=\mathrm{diag}(1,0.1)$. Applying $D$ to the whole coupled gradient givesThe same quadratic penalty produces unequal coordinate shrinkage after this scaling. Uniform decoupled weight decay would instead give $(0.98,0.98{)}^T$. The example isolates a fixed scaling operation, rather than claiming to simulate a complete adaptive gradient method.

Coupling can also change the optimizer state. For example, a momentum buffer formed from $\tilde{g}$ stores contributions from $\lambda \theta$ along with the data gradient. If an adaptive accumulator uses squared gradient entries, it receives $(g_i+\lambda {\theta }_i{)}^2$, including a cross term $2\lambda g_i{\theta }_i$. Subsequent updates can therefore differ even when the next data gradient is the same. State both the penalty convention and the optimizer's gradient-processing rule when describing an L2 setting.