Continuous Function

Optimization

Learning Rate Schedules: Warmup, Decay & Cycling

Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

status: publishedimportance: importantdifficulty 3/5math: undergraduateread: 14mlive demo
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Editorial optimization illustration of warmup, decay, and cycling learning-rate curves over training steps.

Concept Structure

Learning Rate Schedules: Warmup, Decay & Cycling

01Intuition

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02Math

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03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

2prerequisites
1next concepts
2related links

Learning map

Learning Rate Schedules: Warmup, Decay & Cycling
BeforeAdam OptimizerNow4/4 sections readyTryManipulate one control and predict the visible change.NextScaling Laws & Emergent Abilities

Object flow

4/4 sections readyAsk about thisResearch room
ConceptLearning Rate Schedules: Warmup, Decay & CyclingOptimizationEquationLearning Rate Schedules: Warmup, Decay & Cycling equation 1Exact equation objectCodeLearning Rate Schedules: Warmup, Decay & Cycling code witness 1Exact code witnessDemoLearning Rate Schedules: Warmup, Decay & Cycling interactive demoVisualization objectClaimLearning-rate schedules change the scalar learning rate over training...Exact claim checkSourceCyclical Learning Rates for Training Neural NetworksExact source object
ConceptLearning Rate Schedules: Warmup, Decay & CyclingOptimization
2 sources attachedLocal snapshot ready
concept:optimization/learning-rate-schedules
Codewitness nearbyPredictbefore revealRoomobject handoff

Conceptual Bridge

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Carry inAdam Optimizer

Bring the mental model from Adam Optimizer; this page will reuse it instead of restarting from zero.

Work hereLearning Rate Schedules: Warmup, Decay & Cycling

Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

Carry outScaling Laws & Emergent Abilities

The next edge should feel earned: use the demo prediction here before following Scaling Laws & Emergent Abilities.

Test the linkManipulate one control and predict the visible change.Then continue to Scaling Laws & Emergent Abilities
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 DemoManipulate the mechanism and watch the idea respond.
01

01

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

The learning rate is the step size of training. It is also one of the most sensitive knobs in deep learning: the same model can diverge, crawl, or converge depending on the schedule.

Two page-local teaching patterns, treated here as extensions beyond the listed Smith/SGDR sources:

  • Warmup: start small, then ramp up. This page uses warmup as a teaching pattern for smaller early update scale, but the listed Smith/SGDR sources do not by themselves source large-model warmup practice or Adam warmup rationale.
  • Decay: reduce the learning-rate multiplier later so the scalar update scale is smaller late in training.

This page includes warmup followed by decay as a teaching shape, but this source-checked claim focuses only on learning-rate range tests, cyclical policies, and cosine annealing/restarts.

02

02

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt
Equation 1\eta_t = \eta_{\max}\cdot \frac{t}{T_w},\qquad t\le T_w.Equation 2\eta_t = \eta_{\min} + \tfrac12(\eta_{\max}-\eta_{\min})\left(1+\cos\left(\pi\,\frac{t-T_w}{T...

Let ηt\eta_t be the learning rate at step tt.

Linear warmup to a peak ηmax⁡\eta_{\max} over TwT_w steps:

ηt=ηmax⁡⋅tTw,t≤Tw.\eta_t = \eta_{\max}\cdot \frac{t}{T_w},\qquad t\le T_w.

Cosine decay from ηmax⁡\eta_{\max} to ηmin⁡\eta_{\min} over TT total steps (after warmup):

ηt=ηmin⁡+12(ηmax⁡−ηmin⁡)(1+cos⁡(π t−TwT−Tw)),t≥Tw.\eta_t = \eta_{\min} + \tfrac12(\eta_{\max}-\eta_{\min})\left(1+\cos\left(\pi\,\frac{t-T_w}{T-T_w}\right)\right),\qquad t\ge T_w.

A simple classical alternative is:

ηt=η0t+1.\eta_t = \frac{\eta_0}{\sqrt{t+1}}.

Schedules can interact with curvature and sharpness: if the learning rate is too high relative to local curvature, training can become unstable. The edge-of-stability connection is background context, not checked by the Smith/SGDR claim here.

03

03

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt
Code witness 1import numpy as np def warmup_cosine(T, Tw, eta_max, eta_min): lr = np.zeros(T) for t in rang...python
import numpy as np

def warmup_cosine(T, Tw, eta_max, eta_min):
    lr = np.zeros(T)
    for t in range(T):
        if t < Tw:
            lr[t] = eta_max * (t + 1) / max(1, Tw)
        else:
            u = (t - Tw) / max(1, T - Tw - 1)
            lr[t] = eta_min + 0.5 * (eta_max - eta_min) * (1 + np.cos(np.pi * u))
    return lr

lr = warmup_cosine(T=10000, Tw=500, eta_max=3e-4, eta_min=3e-5)
print("lr[0], lr[Tw], lr[-1]:", lr[0], lr[500], lr[-1])
print("avg lr:", float(lr.mean()))
04

04

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Use the demo to tune warmup length and decay style, and build intuition for how "big early steps" vs "small late steps" show up as different curves.

Live Concept Demo

Explore Learning Rate Schedules: Warmup, Decay & Cycling

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 3/5undergraduatecode-aligned
Demo Prediction Checkpoint

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After The First Pass

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Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

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Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

Prediction open01 / Intuition
Editorial optimization illustration of warmup, decay, and cycling learning-rate curves over training steps.
Prediction lens

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Commit first

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Visual Inquiry

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Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

4/4 stages readyLive demo connected
Prediction

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Commit first

Pick the cue that should make Learning Rate Schedules: Warmup, Decay & Cycling easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

paper · 2015Cyclical Learning Rates for Training Neural NetworksSmith

Grounds LR range tests and cyclical policies that vary the global learning rate between lower and upper bounds.

Open source
paper · 2016SGDR: Stochastic Gradient Descent with Warm RestartsLoshchilov and Hutter

Grounds cosine annealing within SGD warm-restart runs, where restarts are emulated by increasing the learning rate while keeping the current parameters.

Open source

Claim Review

Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

Status1 substantive review recorded

Claims without a substantive review badge still need exact source-support review.

Sources2 references

smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr

Witnesses4 local objects

Use equation, code, and demo objects to check whether the source support is operational.

Substantively reviewedLearning-rate schedules change the scalar learning rate over training. Smith supports LR range tests and cyclical policies that vary LR between bounds; SGDR supports cosine annealing with warm restarts. Warmup, inverse-sqrt decay, and LLM-oriented framing here are teaching extensions, not sourced by these papers.Claim metadata: source checked

Smith treats LR as a key hyperparameter, describes CLR varying between boundary values, and gives an LR range test for bounds. SGDR defines warm restarts by increasing LR after each run and decaying LR inside a run by cosine annealing. Page equations/code/demo render schedule curves but do not source warmup or LLM practice.

Sources: Cyclical Learning Rates for Training Neural Networks, SGDR: Stochastic Gradient Descent with Warm RestartsDoes not check large-model warmup practice, Adam bias-correction warmup rationale, edge-of-stability claims, inverse-sqrt schedules, convergence guarantees, or universal schedule superiority; support is limited to LR range tests/CLR and SGDR cosine warm restarts.A bounded review summary is present; still check caveats and exact source scope.

Smith supports scalar LR as a key hyperparameter, CLR between min/max bounds, and an LR range test that linearly increases LR to choose bounds. Loshchilov and Hutter support SGDR restarts as LR increases while reusing parameters, plus cosine annealing from eta_max to eta_min within a run. Warmup, inverse-sqrt, LLM practice, Adam warmup, edge-of-stability, convergence, and universal-superiority material remains caveated teaching context.

Reviewer: codex+oracle; reviewed 2026-05-07
source-span-smith-2015-cyclical-learning-ratessource-span-loshchilov-2016-sgdrmath-object-1math-object-2code-witness-1interactive-demo

Source support candidates

paper 2015Cyclical Learning Rates for Training Neural Networks

Grounds LR range tests and cyclical policies that vary the global learning rate between lower and upper bounds.

paper 2016SGDR: Stochastic Gradient Descent with Warm Restarts

Grounds cosine annealing within SGD warm-restart runs, where restarts are emulated by increasing the learning rate while keeping the current parameters.

Mechanism witnesses

Equation 1
ηt=ηmax⁡⋅tTw,t≤Tw.\eta_t = \eta_{\max}\cdot \frac{t}{T_w},\qquad t\le T_w.
Equation 2
ηt=ηmin⁡+12(ηmax⁡−ηmin⁡)(1+cos⁡(π t−TwT−Tw)),t≥Tw.\eta_t = \eta_{\min} + \tfrac12(\eta_{\max}-\eta_{\min})\left(1+\cos\left(\pi\,\frac{t-T_w}{T-T_w}\right)\right),\qquad t\ge T_w.
Code witness 1import numpy as np def warmup_cosine(T, Tw, eta_max, eta_min): lr = np.zeros(T) for t in rang...Demo stateLive mechanism probe

Practice Loop

Try the idea before it explains itself

Schedule shapes that change the scalar learning-rate scale over training, with sourced CLR/range-test and SGDR cosine-restart examples plus caveated warmup/decay teaching patterns.

Readiness0/3 checks ready
Predict

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Hint 1

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Hint 2

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Hint 3

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Object research drawerClose
ConceptLearning Rate Schedules: Warmup, Decay & CyclingOptimization
Code witness comparisonLearning Rate Schedules: Warmup, Decay & Cycling code witness 1lr = np.zeros(T)Prediction before revealLearning Rate Schedules: Warmup, Decay & Cycling interactive demoManipulate one control and predict the visible change.
Grounded room questionWhat is the smallest example that makes Learning Rate Schedules: Warmup, Decay & Cycling click without losing the math?Local snapshot ready

Research Room

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conceptOptimization

Learning Rate Schedules: Warmup, Decay & Cycling

Anchored question

What is the smallest example that makes Learning Rate Schedules: Warmup, Decay & Cycling click without losing the math?

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Evidence to inspect
  • Source ids to inspect: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr
  • 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
Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Learning Rate Schedules: Warmup, Decay & Cycling Object key: concept:optimization/learning-rate-schedules Context: Optimization Anchor id: concept/concept-notebook/optimization/learning-rate-schedules Open question: What is the smallest example that makes Learning Rate Schedules: Warmup, Decay & Cycling click without losing the math? Evidence to inspect: - Source ids to inspect: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr - 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.

Open source object
concept/concept-notebook/optimization/learning-rate-schedules concept:optimization/learning-rate-schedules