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Adam Optimizer

Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

published · difficulty 3/5 · 16 min read

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Training is noisy: mini-batch gradients bounce around, and different parameters can have very different natural scales.

Adam combines two simple stabilizers:

  1. Momentum: smooth the gradient over time, so you don’t overreact to one noisy batch.
  2. RMS normalization: keep an exponential moving average of squared gradients so each parameter gets a step size that matches its typical gradient scale.

A useful mental model is: Adam maintains a per-parameter “velocity” (direction) and a per-parameter “uncertainty/scale” (how big gradients usually are), then divides one by the other.

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InspectTrack the same object through the notation and check each symbol.Leave with the invariant the equations preserve.
Equation 1Inspect equation objectm_t = \beta_1 m_{t-1} + (1-\beta_1) g_tEquation 2Inspect equation objectv_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2Equation 3Inspect equation object\hat m_t = (m_t)/(1-\beta_1^t), \qquad \hat v_t = (v_t)/(1-\beta_2^t)

Let gt=∇θLt(θt)g_t = \nabla_\theta \mathcal{L}_t(\theta_t) be the (stochastic) gradient at step tt.

Adam keeps exponential moving averages:

mt=β1mt−1+(1−β1)gtm_t = \beta_1 m_{t-1} + (1-\beta_1) g_t
vt=β2vt−1+(1−β2)gt2v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2

Because mtm_t and vtv_t start at zero, Adam uses bias correction:

m^t=mt1−β1t,v^t=vt1−β2t\hat m_t = \frac{m_t}{1-\beta_1^t}, \qquad \hat v_t = \frac{v_t}{1-\beta_2^t}

Update rule:

θt+1=θt−αm^tv^t+ε.\theta_{t+1} = \theta_t - \alpha \frac{\hat m_t}{\sqrt{\hat v_t} + \varepsilon}.

The division is elementwise, so Adam acts like a diagonal preconditioner: coordinates with consistently large gradients get smaller effective steps, while coordinates with consistently small gradients get larger relative steps. Bias correction matters most early in training because the moving averages are initialized at zero. The small ε\varepsilon is not a learning signal; it prevents division by zero and can affect stability when gradients are tiny.

Typical defaults: β1=0.9\beta_1=0.9, β2=0.999\beta_2=0.999, ε=10−8\varepsilon=10^{-8}.

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Code witness 1import torch alpha = 0.1 beta1, beta2 = 0.9, 0.999 eps = 1e-8 theta = torch.tensor([3.0], req...python
import torch

alpha = 0.1
beta1, beta2 = 0.9, 0.999
eps = 1e-8

theta = torch.tensor([3.0], requires_grad=True)
m = torch.zeros_like(theta)
v = torch.zeros_like(theta)

for t in range(1, 51):
    loss = (theta ** 2).sum()
    loss.backward()

    g = theta.grad.detach()
    m = beta1 * m + (1 - beta1) * g
    v = beta2 * v + (1 - beta2) * (g * g)

    mhat = m / (1 - beta1 ** t)
    vhat = v / (1 - beta2 ** t)

    theta = (theta - alpha * mhat / (vhat.sqrt() + eps)).detach().requires_grad_(True)

    if t in [1, 5, 10, 50]:
        print(t, "theta=", theta.item(), "loss=", loss.item())
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Live Concept Demo

Explore Adam Optimizer

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difficulty 3/5undergraduatecode-aligned
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01Choose lensTrace a quantity
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04CarryNext: Weight Decay & AdamW: Decoupled Regularization

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Use the demo to compare Adam to SGD variants and see how the moving averages change the effective step size.

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Concept: Adam Optimizer

What is the smallest example that makes Adam Optimizer click without losing the math?

BeforeDerivativesNow4/4 sections readyTryManipulate one control and predict the visible change.NextWeight Decay & AdamW: Decoupled Regularization
Object contextOptimization
StateOptimization
1 source attachedChecking local snapshot
concept:optimization/adam
Math3 equation objectsm_t = \beta_1 m_{t-1} + (1-\beta_1) g_tCodeAdam Optimizer code witness 1alpha = 0.1DemoAdam Optimizer predictionManipulate one control and predict the visible change.
ConceptLearner lens

Adam Optimizer

What is the smallest example that makes Adam Optimizer click without losing the math?

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Carry inDerivatives

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Work hereAdam Optimizer

Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

Carry outWeight Decay & AdamW: Decoupled Regularization

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Test the linkManipulate one control and predict the visible change.Then continue to Weight Decay & AdamW: Decoupled Regularization

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ConceptAdam OptimizerOptimization
01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary1 source link attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

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Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

Demo notes open01 / Intuition
Editorial optimization illustration of adaptive Adam steps using momentum and second-moment scaling across a loss surface.
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Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

4/4 stages readyDemo notes connected
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Object - ConceptAdam OptimizerQuestion

What is the smallest example that makes Adam Optimizer click without losing the math?

concept:optimization/adam
Boundary

sources: kingma-2014-adam

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selected object source · paper · 2014Adam: A Method for Stochastic OptimizationKingma and Ba
Located CF editorial boundary

Grounds Adam's first- and second-moment estimates, bias correction, and adaptive step-size mechanics.

Used here as

Kingma and Ba's Algorithm 1 initializes m0 and v0 to zero, updates biased first-moment and second raw-moment estimates from gt and gt^2, computes bias-corrected estimates, and updates par...

Caveat

This checks Adam's original adaptive-moment update mechanics, not convergence guarantees, AdamW decoupled weight decay, AMSGrad, optimizer generalization debates, sparse-gradient variants...

Open source

Claim Review

Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

Object - ConceptAdam OptimizerQuestion

What is the smallest example that makes Adam Optimizer click without losing the math?

concept:optimization/adam
Boundary

sources: kingma-2014-adam

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1 CF editorial source-scope review recorded

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Adam maintains exponential moving averages of gradients and squared gradients, bias-corrects both from zero initialization, and takes elementwise adaptive steps by subtracting alpha*mhat_t/(sqrt(vhat_t)+epsilon).
Registered CF editorial boundary
Adam: A Method for Stochastic OptimizationGrounds Adam's first- and second-moment estimates, bias correction, and adaptive step-size mechanics.
Used here as

Kingma and Ba's Algorithm 1 initializes m0 and v0 to zero, updates biased first-moment and second raw-moment estimates from gt and gt^2, computes bias-corrected estimates, and updates parameters with alpha*m...

Local witness
source-span-kingma-2014-adammath-object-1math-object-2code-witness-1interactive-demo
Equation 1
mt=β1mt−1+(1−β1)gtm_t = \beta_1 m_{t-1} + (1-\beta_1) g_t
Equation 2
vt=β2vt−1+(1−β2)gt2v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2
Caveat

This checks Adam's original adaptive-moment update mechanics, not convergence guarantees, AdamW decoupled weight decay, AMSGrad, optimizer generalization debates, sparse-gradient variants, or framework-speci...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Checked Kingma and Ba Algorithm 1 plus sections 2 and 3: the paper initializes m0 and v0 as zero vectors, updates biased first and second raw moment estimates from gt and gt^2, divides by 1-beta1^t and 1-beta2^t for bias correction, states vector operations are element-wise, and updates theta with alpha*mhat/(sqrt(vhat)+epsilon). Scope is original Adam update mechanics only.

Reviewer: codex+oracle; reviewed 2026-05-06

Practice · Adam Optimizer

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Adam is an adaptive optimizer that combines momentum (EMA of gradients) with per-parameter RMS normalization (EMA of squared gradients).

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Adam Optimizer

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Optimization

concept:optimization/adam
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    1. ObjectConceptAdam Optimizer
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    ConceptAdam OptimizerOptimization
    Code witness comparisonAdam Optimizer code witness 1alpha = 0.1Prediction before revealAdam Optimizer predictionManipulate one control and predict the visible change.
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    conceptOptimization

    Adam Optimizer

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    I am working in Continuous Function's research reading room. Object: concept - Adam Optimizer Object key: concept:optimization/adam Context: Optimization Anchor id: concept/concept-notebook/optimization/adam Open question: What is the smallest example that makes Adam Optimizer click without losing the math? Evidence to inspect: - Source ids to inspect: kingma-2014-adam - 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 Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids kingma-2014-adam must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Adam Optimizer" feel predictable rather than familiar." | assumption: Source ids kingma-2014-adam must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: kingma-2014-adam" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. 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. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Adam Optimizer" feel predictable rather than familiar. - Assumption to keep visible: Source ids kingma-2014-adam must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

    concept/concept-notebook/optimization/adam concept:optimization/adam