FA-99071 / Learning rate schedules / Open access

Inverse square root decay falls below base at warmup completion · case 01

Inverse square root decay falls below base at warmup completion.

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

ROOT CAUSE

The schedule omits the square-root warmup scale needed to normalize its value at warmup.

THE FAILURE

The schedule omits the square-root warmup scale needed to normalize its value at warmup.

Unsuccessful approach: Using a linear denominator instead of a root still produces the wrong decay curve.

Case contract

For positive step and warmup, return base*sqrt(warmup)/sqrt(max(step,warmup)).

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):
    import math
    return base/math.sqrt(max(step,warmup))
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('past warmup',solve(16,8,4),4); check('at warmup',solve(4,6,4),6); check('before warmup',solve(1,10,9),10); check('one step',solve(1,9,1),9); check('large step',solve(25,5,1),1); check('fractional scale',solve(8,12,2),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
past warmup	`2.0`	`4`	Failed
at warmup	`3.0`	`6`	Failed
before warmup	`3.3333333333333335`	`10`	Failed
one step	`9.0`	`9`	Passed
large step	`1.0`	`1`	Passed
fractional scale	`4.242640687119285`	`6`	Failed

SHA-256 / 1ec0013845e7d1fb881306e1f56cdc9af08301a92574d414fc7c06b7dd70e748

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(step, base, warmup):
    import math
    return base*math.sqrt(warmup)/max(step,warmup)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('past warmup',solve(16,8,4),4); check('at warmup',solve(4,6,4),6); check('before warmup',solve(1,10,9),10); check('one step',solve(1,9,1),9); check('large step',solve(25,5,1),1); check('fractional scale',solve(8,12,2),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
past warmup	`1.0`	`4`	Failed
at warmup	`3.0`	`6`	Failed
before warmup	`3.3333333333333335`	`10`	Failed
one step	`9.0`	`9`	Passed
large step	`0.2`	`1`	Failed
fractional scale	`2.121320343559643`	`6`	Failed

SHA-256 / c8f338b325698c5c0e8b4eb74287c22375750ae39278a43077e7b6104fd6479f

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 2026-09-30T16:48:19.910349+00:00.

Case digest / 30f8361931eb765a67f31156db76d4548ea038a182ab810c029473e56eb74485