Optimization

Weight Decay & AdamW: Decoupled Regularization

Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

status: reviewimportance: importantdifficulty 3/5math: undergraduateread: 13mprediction probe

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Concept Structure

Weight Decay & AdamW: Decoupled Regularization

01Intuition

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02Math

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03Code

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04Interactive Demo

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2prerequisites

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Learner Contract

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You are here becauseWhy shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

This Optimization concept is the current object: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

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Adam Optimizer Maximum Likelihood

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Updatedpage 2026-06-29

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ConceptWeight Decay & AdamW: Decoupled RegularizationOptimization EquationWeight Decay & AdamW: Decoupled Regularization equation 1Exact equation object CodeWeight Decay & AdamW: Decoupled Regularization code witness 1Exact code witness ClaimWeight Decay & AdamW: Decoupled Regularization central claimMechanism claim EquationWeight Decay & AdamW: Decoupled Regularization math objectsEquation and notation CodeWeight Decay & AdamW: Decoupled Regularization code witnessRunnable code object

ConceptWeight Decay & AdamW: Decoupled RegularizationOptimization

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concept:optimization/weight-decay-adamw

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01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoCommit a prediction against the code witness.

Intuition

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Section prompt

Regularization often starts with a simple preference: all else equal, prefer smaller weights.

There are two closely related ways to express that idea:

add an $L_2$ penalty to the loss,
directly shrink parameters a little bit every step.

For plain SGD, those are effectively the same update. That is why many people casually treat "L2 regularization" and "weight decay" as synonyms.

For Adam, they are not the same. Adam rescales coordinates using running estimates of gradient magnitude, so an $L_2$ term added to the gradient gets rescaled too. That means different parameters can experience very different effective regularization strengths.

AdamW fixes this by decoupling weight decay from the adaptive gradient step. First do the Adam update. Then shrink the parameters directly.

Math

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Section prompt

Equation 1 $L_{reg}(\theta) = L(\theta) + \frac{\lambda}{2} \lVert \theta \rVert_2^2,$ Equation 2 $\nabla L_{reg}(\theta) = \nabla L(\theta) + \lambda \theta.$

Let $L(\theta)$ be the task loss and $\lambda > 0$ the regularization strength.

With an $L_2$ penalty, the regularized objective is

L_{reg}(\theta) = L(\theta) + \frac{\lambda}{2} \lVert \theta \rVert_2^2,

so the gradient becomes

\nabla L_{reg}(\theta) = \nabla L(\theta) + \lambda \theta.

For SGD, this leads to

\theta_{t+1} = \theta_t - \eta \nabla L(\theta_t) - \eta \lambda \theta_t = (1 - \eta \lambda)\theta_t - \eta \nabla L(\theta_t),

which is exactly a weight-decay step.

For Adam, the adaptive preconditioner changes things. AdamW writes the update as

m_t = \beta_1 m_{t-1} + (1-\beta_1) g_t,

v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2,

\theta_{t+1} = \theta_t - \eta \frac{\hat m_t}{\sqrt{\hat v_t} + \epsilon} - \eta \lambda \theta_t.

The key point is that the shrinkage term $-\eta \lambda \theta_t$ is not divided by $\sqrt{\hat v_t} + \epsilon$ . Regularization stays regularization instead of getting entangled with Adam's coordinatewise scaling.

Code

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Section prompt

Code witness 1import numpy as np theta = np.array([1.0, 1.0]) g = np.array([0.1, 0.1]) vhat = np.array([1e-...python

import numpy as np

theta = np.array([1.0, 1.0])
g = np.array([0.1, 0.1])
vhat = np.array([1e-4, 1.0])  # Adam sees very different curvature/noise
lr = 1e-2
wd = 0.1
eps = 1e-8

adam_with_l2 = theta - lr * ((g + wd * theta) / (np.sqrt(vhat) + eps))
adamw = theta - lr * (g / (np.sqrt(vhat) + eps)) - lr * wd * theta

print("Adam + L2 :", np.round(adam_with_l2, 4))
print("AdamW     :", np.round(adamw, 4))

In the first coordinate, the effective shrinkage under "Adam + L2" becomes much larger because Adam divides by a tiny sqrt(vhat). AdamW avoids that distortion.

Interactive Demo

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Section prompt

Prediction check: the first coordinate has vhat=1e-4, so Adam's denominator is tiny there. Before running the code, predict which method shrinks coordinate 1 more aggressively, then change vhat[0] and watch the coupling disappear for AdamW.

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Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

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Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

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Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

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Mechanism witnesses

Equation 1

L_{reg}(\theta) = L(\theta) + \frac{\lambda}{2} \lVert \theta \rVert_2^2,

Equation 2

\nabla L_{reg}(\theta) = \nabla L(\theta) + \lambda \theta.

Code witness 1import numpy as np theta = np.array([1.0, 1.0]) g = np.array([0.1, 0.1]) vhat = np.array([1e-...

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Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

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ConceptWeight Decay & AdamW: Decoupled RegularizationOptimization

Code witness comparisonWeight Decay & AdamW: Decoupled Regularization code witness 1theta = np.array([1.0, 1.0])Prediction before revealWeight Decay & AdamW: Decoupled Regularization predictionUse the prediction probe to commit to the mechanism before moving on.

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conceptOptimization

Weight Decay & AdamW: Decoupled Regularization

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What is the smallest example that makes Weight Decay & AdamW: Decoupled Regularization click without losing the math?

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Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Weight Decay & AdamW: Decoupled Regularization Object key: concept:optimization/weight-decay-adamw Context: Optimization Anchor id: concept/concept-notebook/optimization/weight-decay-adamw Open question: What is the smallest example that makes Weight Decay & AdamW: Decoupled Regularization click without losing the math? Evidence to inspect: - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

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concept/concept-notebook/optimization/weight-decay-adamw
concept:optimization/weight-decay-adamw

Learning Map

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BeforeAdam Optimizer

NowIntuition → Math → Code → Demo

TryUse the prediction probe to commit to the mechanism before moving on.

NextScaling Laws & Emergent Abilities

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Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior.

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You are my AI learning companion for Continuous Function. Current context: Optimization concept. Learning surface: Weight Decay & AdamW: Decoupled Regularization. What this page says: Why shrinking weights is not the same as adding an L2 penalty inside Adam, and how AdamW restores the intended regularization behavior. Current section: Intuition, math, code, and interactive demo. Suggested next step: Use the prediction probe to commit to the mechanism before moving on.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Optimization

optimizationregularizationadamwweight-decay

Prerequisites

Adam Optimizer Maximum Likelihood

Leads To

Scaling Laws & Emergent Abilities

SGD & Momentum: The Workhorses of Optimization (review)Label Smoothing & Soft Targets (review)

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