FA-99101 / Learning rate schedules / Open access

Cosine restarts are misaligned with the end of warmup · case 01

Cosine restarts are misaligned with the end of warmup.

Verified by executionVariant 1 · 6 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The restart remainder is calculated from the global step, not elapsed post-warmup steps.

THE FAILURE

The restart remainder is calculated from the global step, not elapsed post-warmup steps.

Unsuccessful approach: Offsetting the corrected phase by one step creates a discontinuity at the warmup boundary.

Case contract

Warm up linearly, then apply a repeating cosine over a fixed post-warmup period.

Why this case matters

A deterministic scalar schedule makes step-to-rate mapping explicit for a small training loop.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(step, base, warmup, period):
    import math
    return base*step/warmup if step<warmup else base*(1+math.cos(2*math.pi*(step%period)/period))/2
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('warmup',solve(1,8,4,3),2); check('boundary',solve(4,8,4,3),8); check('midcycle',solve(6,8,4,4),0); check('quarter cycle',solve(5,8,4,4),4); check('restart',solve(8,8,4,4),8); check('next peak',solve(12,6,4,4),6)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
warmup	`2.0`	`2`	Passed
boundary	`2.000000000000001`	`8`	Failed
midcycle	`0.0`	`0`	Passed
quarter cycle	`4.0`	`4`	Passed
restart	`8.0`	`8`	Passed
next peak	`6.0`	`6`	Passed

SHA-256 / 0066ea9891e121b4d5fc4de613090e907261bdcd768d9ed76c7331292220a82e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(step, base, warmup, period):
    import math
    phase=(step-warmup)%period
    return base*(step/warmup) if step<warmup else base*(1+math.cos(2*math.pi*(phase+1)/period))/2
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('warmup',solve(1,8,4,3),2); check('boundary',solve(4,8,4,3),8); check('midcycle',solve(6,8,4,4),0); check('quarter cycle',solve(5,8,4,4),4); check('restart',solve(8,8,4,4),8); check('next peak',solve(12,6,4,4),6)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
warmup	`2.0`	`2`	Passed
boundary	`2.000000000000001`	`8`	Failed
midcycle	`3.999999999999999`	`0`	Failed
quarter cycle	`0.0`	`4`	Failed
restart	`4.0`	`8`	Failed
next peak	`3.0`	`6`	Failed

SHA-256 / 8720ea168f61d87ba6e7634f3956ed75eee7cae12b1e5ad47b4e4dd6af60c08d

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 6 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

These are toy scalar contracts with integer steps and finite parameters; they do not model framework defaults or optimizer state. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-30T16:48:20.351688+00:00.

Case digest / 8420f94bfe5961f462a11fbf4f3e4495dc2f2ca63247b7ed158121931b70e2ae