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Optimization & training / 2014

Adam: understand the update, not just the defaults

Why two moving averages can change the geometry of a training problem. Work through the first step and inspect the effect of bias correction.

Adam: A Method for Stochastic Optimization · Kingma & Ba · arXiv:1412.6980

Original editorial analysis by Token Lab · Source checked 2026-10-25 · Paper claims and synthetic examples are separated.

Detailed technical articles are available in English and Chinese. This locale shows the English article with localized navigation.

01 · What the paper establishes

Adam maintains first- and second-moment estimates and corrects their initialization bias. The paper compares optimization behavior on several learning tasks. This is evidence about those experiments, not a guarantee that one optimizer wins everywhere.

Primary source: paper abstract ↗
Study guide & lesson outlineLEARNING CONTRACT

Understand → implement → investigate

Goal: explain the mechanism in this lesson, reproduce one example and test a failure case. Reading alone does not mark mastery.

Prerequisite route →Related implementation practice →
  • Understand: state assumptions, shapes/units and a counterexample.
  • Implement: reproduce a numerical example and inspect boundaries.
  • Investigate: compare one alternative under a matched budget.

02 · The mechanism

Text / 原始公式
mₜ = β₁mₜ₋₁ + (1−β₁)gₜ
vₜ = β₂vₜ₋₁ + (1−β₂)gₜ²
m̂ₜ = mₜ/(1−β₁ᵗ), v̂ₜ = vₜ/(1−β₂ᵗ)
θₜ = θₜ₋₁ − αm̂ₜ/(√v̂ₜ + ε)

Think of each coordinate as having a direction estimate and a scale estimate. Momentum averages direction; the squared-gradient average supplies the denominator. The denominator is not a Hessian estimate, so calling Adam Newton’s method would be misleading. With zero initialization, early averages are too small; dividing by the missing geometric-series mass corrects that effect under the moment-estimation assumptions.

Notation and intuition are teaching explanations. Consult the original method for complete assumptions and implementation details.

Mechanism map & worked derivation

Gradient g → Update m and v → Correct startup bias → Scale and update θ
Original schematic; read the steps from top to bottom. SVG ↓
Symbols and assumptions
SymbolMeaning
gₜCurrent gradient; same shape as parameters
mₜ, vₜExponential averages of g and g²; v is not centered variance
α, β₁, β₂, εStep size, two decay rates and numerical stabilizer
  1. Expand the recurrence: β₌₁ᵗβ₁ᵗ⁻ʲgⱼ. If E[gⱼ]=μ is constant, E[mₜ]=(1−β₁ᵗ)μ. This identifies the bias term rather than introducing correction by convention.
  2. Repeat the argument for the raw second moment. Divide by the corresponding geometric mass, then normalize direction by the square root of the corrected second moment.
  3. For constant , the second-step uncorrected first- and second-moment estimates are and Correcting yields 2 and 4 again. For a sign-changing gradient, that equality no longer holds: the averages encode history.
Experiment checkpoint

Log both raw and corrected moments. If a constant-gradient test does not recover the same moments at every step, inspect indexing and initialization.

LOCAL / SYNTHETIC / INTERACTIVE

Change one variable. Explain what moves.

Original teaching calculation, not a reproduction of the paper’s benchmark. Bars are relative within this panel; numeric values and units are authoritative.

04 · Work through an example

Original worked example: let , β, β and α Then and ; corrected moments are 2 and 4. Ignoring ε, the step is −0.001. Without correction it is approximately −0.003162. This calculation isolates the first step; it does not predict an entire loss curve. If all gradients are multiplied by a positive constant, the ratio approximately cancels that scale, except where ε matters.

05 · Design your experiment

Implement scalar SGD and Adam on using the same start and equal gradient-evaluation budgets. Log loss, gradient norm, moment estimates and actual step norm. Sweep learning rates for both methods with identical trial counts. Add a second test where the gradient changes sign to reveal momentum lag. Preserve failed runs rather than plotting only the best trajectory.

Research deliverable

State one hypothesis; save the configuration and raw results; identify a competing explanation; report the narrowest supported conclusion. Use the linked lab as a local starting point, not a full paper reproduction.

Open related lesson / lab →

06 · Limits & reviewer questions

Editorial caution: rapid training-loss reduction does not establish better generalization. Compare held-out loss at matched compute and inspect mixed-precision behavior, ε placement and optimizer-state memory. The original convergence discussion should not be read as a universal nonconvex convergence guarantee.

Why can a tiny gradient still produce a noticeable first update?

For nonzero g and negligible ε, bias correction gives g/√g²=sign(g). The learning rate sets the first-step magnitude. Near zero, ε breaks that cancellation; a zero gradient produces zero movement.

07 · Read the original

Algorithm 1 and the bias-correction discussion. Trace the state variables before reading the convergence analysis.

arXiv ↗PDF ↗

Year is the first arXiv submission year. Unversioned links follow revisions; record the exact version in your reproduction report. DeepSeek-R1 here is explicitly pinned to v1.

Continue this topic

MATHEMATICAL REVIEW · 2026-10-26

Assumptions, implementation & acceptance

Initialize =0; t≥1 and 0≤β₁,β₂<1. Operations are elementwise. Bias correction of v assumes a stationary raw second moment. The demo reports absolute update magnitudes.

Displayed formula, bilingual explanation and declared slider range. This code reproduces the scalar teaching calculation, not the paper training system.

Before running: predict a boundary case

Implement scalar SGD and Adam on using the same start and equal gradient-evaluation budgets. Log loss, gradient norm, moment estimates and actual step norm. Sweep learning rates for both methods with identical trial counts. Add a second test where the gradient changes sign to reveal momentum lag. Preserve failed runs rather than plotting only the best trajectory.

Hint: isolate the assumption

Keep the baseline input fixed, change one parameter or boundary, and compare both the numerical result and the claimed units. A passing example verifies this fixture only.

Minimal executable example

Save the example and linked modules in one folder. Use Node.js 22 or newer, then run node adam.mjs. No third-party runtime dependencies.

import {paperCalc} from './paper-core.mjs';
const result=paperCalc("adam",2);
console.log(result);
Download this examplepaper-core.mjs
Reference output — reveal after predicting
{
  "labels": [
    "Bias corrected step",
    "Uncorrected step"
  ],
  "values": [
    0.000999999995,
    0.003162277160168459
  ],
  "unit": "absolute step"
}

Test and transfer

Implement scalar SGD and Adam on using the same start and equal gradient-evaluation budgets. Log loss, gradient norm, moment estimates and actual step norm. Sweep learning rates for both methods with identical trial counts. Add a second test where the gradient changes sign to reveal momentum lag. Preserve failed runs rather than plotting only the best trajectory.

Record input, code version, expected result and observed result in the notebook below. Then construct a counterexample that violates the assumption above and explain why the original conclusion no longer follows.

Download complete kit (.tar.gz) · Complete reference kit and regression checks

Predict, test and keep evidence

Before revealing an answer, predict one result. After the exercise, record the discrepancy and a transfer test. Saved locally across language editions.

Open notebook / export / clear →

The mathematical review scope is recorded per lesson. It does not establish empirical paper reproduction; report suspected errors in your notes. Review scope