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.

published · difficulty 3/5 · 14 min read

Intuition

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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.

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Math

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Equation 1Inspect equation object\eta_t = \eta_{\max}\cdot (t)/(T_w),\qquad t<= T_w.Equation 2Inspect equation object\eta_t = \eta_{\min} + \tfrac12(\eta_{\max}-\eta_{\min})(1+\cos(\pi (t-T_w)/(T-T_w))),\qquad...Equation 3Inspect equation object\eta_t = (\eta_0)/(sqrt(t+1)).

Let $\eta_t$ be the learning rate at step $t$ .

Linear warmup to a peak $\eta_{\max}$ over $T_w$ steps:

\eta_t = \eta_{\max}\cdot \frac{t}{T_w},\qquad t\le T_w.

Cosine decay from $\eta_{\max}$ to $\eta_{\min}$ over $T$ total steps (after warmup):

\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:

\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.

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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()))

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Live Concept Demo

Explore Learning Rate Schedules: Warmup, Decay & Cycling

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difficulty 3/5undergraduatecode-aligned

Demo inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity

02ObserveDemo state pending

03GroundName the equation, invariant, or control that explains it.

04CarryNext: Scaling Laws & Emergent Abilities

Choose what to inspect in Learning Rate Schedules: Warmup, Decay & Cycling. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

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.

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Object

4/4 sections ready

Concept: Learning Rate Schedules: Warmup, Decay & Cycling

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

BeforeAdam OptimizerNow4/4 sections readyTryManipulate one control and predict the visible change.NextScaling Laws & Emergent Abilities

Object contextOptimization

StateOptimization

2 sources attachedChecking local snapshot

concept:optimization/learning-rate-schedules

Math3 equation objects\eta_t = \eta_{\max}\cdot (t)/(T_w),\qquad t<= T_w.CodeLearning Rate Schedules: Warmup, Decay & Cycling code witness 1lr = np.zeros(T)DemoLearning Rate Schedules: Warmup, Decay & Cycling predictionManipulate one control and predict the visible change.

ConceptLearner lens

Learning Rate Schedules: Warmup, Decay & Cycling

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

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01PredictCommit before reveal.02ExplainName the invariant.03Ground2 source links nearby.04ExperimentChange one knob.05CarrySave the next question.

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Carry inAdam Optimizer

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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

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ConceptLearning Rate Schedules: Warmup, Decay & CyclingOptimization

01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary2 source links 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.

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.

Demo notes open01 / Intuition

Editorial optimization illustration of warmup, decay, and cycling learning-rate curves over training steps.

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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 readyDemo notes connected

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Object - ConceptLearning Rate Schedules: Warmup, Decay & CyclingQuestion

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

concept:optimization/learning-rate-schedules

Boundary

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

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Evidence

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selected object source · paper · 2015Cyclical Learning Rates for Training Neural NetworksSmith

Located CF editorial boundary

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

Used here as

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 a...

Caveat

Does not check large-model warmup practice, Adam bias-correction warmup rationale, edge-of-stability claims, inverse-sqrt schedules, convergence guarantees, or universal schedule superior...

Open source

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

Located CF editorial boundary

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

Used here as

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 a...

Caveat

Does not check large-model warmup practice, Adam bias-correction warmup rationale, edge-of-stability claims, inverse-sqrt schedules, convergence guarantees, or universal schedule superior...

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.

Object - ConceptLearning Rate Schedules: Warmup, Decay & CyclingQuestion

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

concept:optimization/learning-rate-schedules

Boundary

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

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1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 2 references and 3 local witnesses are available for inspection.

Learning-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.

Registered CF editorial boundary

Cyclical Learning Rates for Training Neural NetworksGrounds LR range tests and cyclical policies that vary the global learning rate between lower and upper bounds.SGDR: Stochastic Gradient Descent with Warm RestartsGrounds cosine annealing within SGD warm-restart runs, where restarts are emulated by increasing the learning rate while keeping the current parame...

Used here as

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 insid...

Local witness

source-span-smith-2015-cyclical-learning-rates source-span-loshchilov-2016-sgdr math-object-1 math-object-2 code-witness-1 interactive-demo

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-T_w}\right)\right),\qquad t\ge T_w.

Caveat

Does 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 limi...

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

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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.

AttemptNo learning claim inferred

Object - ConceptLearning Rate Schedules: Warmup, Decay & CyclingQuestion

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

concept:optimization/learning-rate-schedules

Boundary

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

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Selected object routeAsk from this object; carry one invariant back.sources: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr

ObjectConceptLearning Rate Schedules: Warmup, Decay & Cycling
PredictBefore revealLearning Rate Schedules: Warmup, Decay & Cycling prediction
WitnessCompare codeLearning Rate Schedules: Warmup, Decay & Cycling code witness 1
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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 predictionManipulate 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?Checking local snapshot

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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?

Source boundaryInspect source ids: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdrStable content-object key attached

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The learner can state the mechanism in their own words

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Source ids to inspect: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr
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What would resolve this

The learner can state the mechanism in their own words
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The learner can predict how the mechanism changes under one perturbation

Object-attached 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 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 smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr 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 "Learning Rate Schedules: Warmup, Decay & Cycling" feel predictable rather than familiar." | assumption: Source ids smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr 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: smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr" | 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 "Learning Rate Schedules: Warmup, Decay & Cycling" feel predictable rather than familiar. - Assumption to keep visible: Source ids smith-2015-cyclical-learning-rates, loshchilov-2016-sgdr 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/learning-rate-schedules
concept:optimization/learning-rate-schedules

Learning Map

Before / Now / Try / Next

BeforeAdam Optimizer

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextScaling Laws & Emergent Abilities

Learnerpicture first Researchersource check Experimentertest the demo Professorname invariant

01Predict 02Ground 03Code 04Carry

01Intuitionready
02Mathready
03Codeready
04Interactive Demoready

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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.

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Context prompt

You are my AI learning companion for Continuous Function. Current context: Optimization concept. Learning surface: Learning Rate Schedules: Warmup, Decay & Cycling. What this page says: 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. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. 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.

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Domain

Optimization

optimizationlearning-ratewarmupcosine-decay

Prerequisites

Adam Optimizer Loss Landscapes, Sharpness & Flat Minima

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Scaling Laws & Emergent Abilities

Weight Initialization: Xavier, He & muP Layer Normalization & RMSNorm

Within this domain