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

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The learning rate is a scalar multiplier in an optimizer update. In plain gradient descent it directly scales the gradient step; adaptive optimizers also transform the gradient before applying that multiplier. Changing the multiplier over a finite run changes how large the optimizer's applied updates can be at different times.

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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Equation 1Inspect equation objectD_{linear}(u)=(1-u)\eta_{\max}+u\eta_{\min}, \qquad D_{cosine}(u)=\eta_{\min}+\tfrac12(\eta_{...Equation 2Inspect equation object\eta_t=D ((t)/(T-1)),\qquad 0<= t<T.Equation 3Inspect equation object\eta_t= \displaystyle \eta_{\max}(t+1)/(T_w), 0<= t<T_w; [6pt] \displaystyle D ((t-T_w+1)/(T-...

Indexing note. This page defines exactly $T$ learning rates applied to optimizer updates $t=0,\ldots,T-1$ . Assume integers $T\ge 2$ and $0\le T_w<T$ , with $0\le \eta_{\min}\le \eta_{\max}$ . This finite, endpoint-inclusive definition is a page-local teaching convention, not a universal scheduler API convention; libraries differ over whether a boundary value is an applied update or a scheduler state before or after one. The warmup omits the conceptual zero endpoint, so its first applied rate is positive when $\eta_{\max}>0$ ; $\eta_{\max}=0$ admits the degenerate all-zero schedule.

For a decay coordinate $u\in[0,1]$ , define

D_{\mathrm{linear}}(u)=(1-u)\eta_{\max}+u\eta_{\min}, \qquad D_{\mathrm{cosine}}(u)=\eta_{\min}+\tfrac12(\eta_{\max}-\eta_{\min})\left(1+\cos(\pi u)\right).

Choose $D=D_{\mathrm{linear}}$ or $D=D_{\mathrm{cosine}}$ . When $T_w=0$ , there is no warmup phase, and $D$ spans all $T$ applied updates:

\eta_t=D\!\left(\frac{t}{T-1}\right),\qquad 0\le t<T.

When $T_w>0$ , the first $T_w$ applied updates form a linear warmup that omits the conceptual zero endpoint, followed by decay:

\eta_t= \begin{cases} \displaystyle \eta_{\max}\frac{t+1}{T_w}, & 0\le t<T_w,\\[6pt] \displaystyle D\!\left(\frac{t-T_w+1}{T-T_w}\right), & T_w\le t<T. \end{cases}

Update $T_w-1$ reaches $\eta_{\max}$ and is the shared $u=0$ endpoint of warmup and decay. When $\eta_{\min}<\eta_{\max}$ , update $T_w$ uses $u=1/(T-T_w)$ and is already below the peak, so the peak is not repeated. When $\eta_{\min}=\eta_{\max}$ , the post-warmup segment instead remains constant at $\eta_{\max}$ . The final applied update, $t=T-1$ , uses $u=1$ and reaches $\eta_{\min}$ . Here $\eta_{\min}$ is the decay endpoint, not necessarily the minimum over the warmup-and-decay sequence: an early positive warmup rate can be smaller.

A separate page-local teaching alternative is inverse-square-root decay, where $\eta_0$ is the rate at update $t=0$ :

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

For every displayed family, the demo's reported mean is the discrete mean over all applied updates,

\bar\eta=\frac{1}{T}\sum_{t=0}^{T-1}\eta_t,

not the mean of the smaller set of points used to draw the chart. SGDR supplies the within-cycle half-cosine shape and its maximum/minimum endpoints; it does not supply this page's linear warmup or finite-update indexing.

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Code witness 1import math def warmup_cosine(T, Tw, eta_max, eta_min): if not isinstance(T, int) or isinstan...python

import math

def warmup_cosine(T, Tw, eta_max, eta_min):
    if not isinstance(T, int) or isinstance(T, bool) or T < 2:
        raise ValueError("T must be an integer with T >= 2")
    if not isinstance(Tw, int) or isinstance(Tw, bool) or not 0 <= Tw < T:
        raise ValueError("Tw must be an integer with 0 <= Tw < T")
    if not all(math.isfinite(v) for v in (eta_max, eta_min)):
        raise ValueError("learning-rate bounds must be finite")
    if not 0 <= eta_min <= eta_max:
        raise ValueError("require 0 <= eta_min <= eta_max")

    lr = [0.0] * T
    for t in range(T):
        if Tw > 0 and t < Tw:
            lr[t] = eta_max * (t + 1) / Tw
        else:
            u = t / (T - 1) if Tw == 0 else (t - Tw + 1) / (T - Tw)
            lr[t] = eta_min + 0.5 * (eta_max - eta_min) * (1 + math.cos(math.pi * u))
    return lr

T, Tw = 10_000, 500
lr = warmup_cosine(T=T, Tw=Tw, eta_max=3e-4, eta_min=3e-5)
print("lr[0], lr[Tw-1], lr[Tw], lr[-1]:", lr[0], lr[Tw - 1], lr[Tw], lr[-1])
print("avg lr over all applied updates:", sum(lr) / len(lr))

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Live Concept Demo

Explore Learning Rate Schedules: Warmup, Decay & Cycling

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difficulty 3/5undergraduatecode-aligned

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Manipulate one control and predict the visible change.

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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...

Predict the schedule's behavior, then reveal the finite applied-update sequence. Tune warmup length and decay style while watching the exact first rate, mean over all $T$ updates, final rate, and the boundary between the last warmup update and the first decay update. The chart uses a smaller anchored sample only for drawing.

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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 objectsD_{linear}(u)=(1-u)\eta_{\max}+u\eta_{\min}, \qquad D_{cosine}(u)=\eta_{\min}+\tfrac12(\eta_{...CodeLearning Rate Schedules: Warmup, Decay & Cycling code witness 1raise ValueError("T must be an integer with T >= 2")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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4/4 sections ready

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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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 supports LR range tests and CLR between boundary values. SGDR supports cosine annealing between maximum and minimum endpoints within a restart cycle. The finite-update indexing, sha...

Caveat

Does not check warmup practice or rationale, inverse-sqrt schedules, a universal scheduler-library indexing convention, convergence guarantees, or schedule superiority. Source support cov...

Open source

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

Located CF editorial boundary

Grounds the endpoint-inclusive half-cosine shape within SGD warm-restart cycles. It does not define this page's linear warmup or its enumeration of exactly T applied updates.

Used here as

Smith supports LR range tests and CLR between boundary values. SGDR supports cosine annealing between maximum and minimum endpoints within a restart cycle. The finite-update indexing, sha...

Caveat

Does not check warmup practice or rationale, inverse-sqrt schedules, a universal scheduler-library indexing convention, convergence guarantees, or schedule superiority. Source support cov...

Open source

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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.

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 the endpoint-inclusive half-cosine shape within SGD warm-restart cycles. It does not define this page's linear warmup or its enumeration of...

Used here as

Smith supports LR range tests and CLR between boundary values. SGDR supports cosine annealing between maximum and minimum endpoints within a restart cycle. The finite-update indexing, shared boundary, and ex...

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

D_{\mathrm{linear}}(u)=(1-u)\eta_{\max}+u\eta_{\min}, \qquad D_{\mathrm{cosine}}(u)=\eta_{\min}+\tfrac12(\eta_{\max}-\eta_{\min})\left(1+\cos(\pi u)\right).

Equation 2

\eta_t=D\!\left(\frac{t}{T-1}\right),\qquad 0\le t<T.

Caveat

Does not check warmup practice or rationale, inverse-sqrt schedules, a universal scheduler-library indexing convention, convergence guarantees, or schedule superiority. Source support covers only LR range te...

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 bounds, and an LR range test for selecting bounds. Loshchilov and Hutter support warm restarts and half-cosine annealing from eta_max to eta_min within a cycle. Neither paper defines this page's linear warmup, inverse-sqrt family, finite applied-update indexing, convergence behavior, or schedule superiority; those claims are excluded or labeled as page-local teaching definitions.

Reviewer: codex+gpt-pro; reviewed 2026-08-21

Practice · Learning Rate Schedules: Warmup, Decay & Cycling

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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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Learning Rate Schedules: Warmup, Decay & Cycling

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

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Optimization

concept:optimization/learning-rate-schedules

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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 1raise ValueError("T must be an integer with T >= 2")Prediction before revealLearning Rate Schedules: Warmup, Decay & Cycling predictionManipulate one control and predict the visible change.

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conceptOptimization

Learning Rate Schedules: Warmup, Decay & Cycling

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

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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

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Optimization

optimization
learning-rate
warmup
cosine-decay

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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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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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