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Adaptive hysteresis can fail quietly when the active label controls which residuals estimate noise and that estimate then controls the label threshold. The loop can freeze a wrong manifold. False-lock diagnostics and guarded escape rules break the feedback while keeping the globally safe method in charge.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. The estimator and detector can form a loop
Suppose the detector labels a candidate manifold active. It then estimates noise only from residuals accepted under that label. A narrow residual window lowers the estimated noise, which tightens thresholds and makes the same label even harder to leave.
The reverse loop can also occur: contaminated residuals inflate noise, widen the dead zone and preserve an obsolete manifold. Adaptation must be audited as a dynamical system, not only as a sequence of formulas.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Define a false lock operationally
A false lock is not merely a long residence time. It is a persistent label accompanied by evidence that the local model no longer explains accepted global progress, feasibility, stationarity or held-out residuals.
Use at least one diagnostic not computed from the same filtered window that sets the threshold. This separation reduces self-confirming feedback.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A threshold-feedback ledger
Consider an active-set detector with normalised margin z and estimated noise sigma. The label controls the window used to update sigma.
| Iteration | Label | z | Estimated sigma | Entry/exit width | Independent gate |
|---|---|---|---|---|---|
| 80 | Active | 0.018 | 0.006 | 0.018 | Pass |
| 100 | Active | 0.010 | 0.004 | 0.012 | Pass |
| 120 | Active | 0.004 | 0.002 | 0.006 | Fail |
| 140 | Active | 0.003 | 0.001 | 0.006 floor | Fail |
The floor prevents thresholds collapsing to zero, but it does not by itself unlock a wrong state. A separate escape trigger is required.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked escape-and-reset rule
Use an illustrative rule: if the independent gate fails for two accepted global steps, suspend local acceleration, clear the in-label window, inflate the provisional noise floor and require three fresh global steps before re-entry.
| Event | State action | Memory action | Next evidence requirement |
|---|---|---|---|
| Second gate failure | Exit active mode | Quarantine old window | Three accepted global steps |
| First recovery step | Global method only | Start fresh buffer | No re-entry |
| Third recovery step | Evaluate candidate | Estimate sigma from fresh buffer | Margin and gate must pass |
| Immediate repeat failure | Extend cooldown | Reset buffer again | Escalate to conservative fallback |
The counts are tunable design values. Validate them against genuine switches, noise bursts and degenerate cases rather than treating them as universal constants.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Detect drift before lock-in
Track the ratio of in-label noise to an external or longer-window estimate, threshold movement per iteration, gate disagreement and cumulative accepted progress. Sudden divergence identifies feedback earlier than a final failure.
Freeze threshold adaptation during rejected steps, line-search backtracking or detected distribution shifts. Otherwise the noise estimator can learn from algorithmic transients that it helped create.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Keep the global method authoritative
Local manifold acceleration is a privilege granted by evidence. The globally convergent base method should remain available when gates fail, thresholds saturate or repeated escapes occur.
A restart can remove momentum and local memory, but it does not repair an invalid model. Log whether improvement came from leaving a false lock, changing step dynamics or merely waiting longer.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: rank identification
Transfer the design to low-rank optimisation. A rank label selects a tangent space, while spectral residuals inside that space estimate noise. Repeated singular values can make the selected rank self-reinforcing.
Use a held-out spectral gap or full-space stationarity measure as the independent gate. After escape, recompute the spectrum under the global step before reactivating rank-specific acceleration.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the identification branch
The adaptive-hysteresis guide defines normalised widths, floors and ceilings. The degenerate-manifold guide explains when local identification assumptions fail.
Use the identification-time guide to distinguish candidate detection from a certificate strong enough for acceleration.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
An escape rule can create chatter if it is too sensitive and long stalls if it is too conservative. Independent gates may still share data or model errors with the detector.
Restarting after a false lock is an algorithmic safeguard, not a proof of finite identification. The proof still depends on convergence, partial smoothness, nondegeneracy and the chosen method.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive raw and normalised margins, noise windows, external estimators, floors, ceilings, label states, gate results, threshold changes, accepted and rejected steps, restarts, cooldowns and memory resets.
Test stable manifolds, genuine transitions, repeated singular values, scale changes, burst noise and deliberate window contamination. Compare objective progress, feasibility and wall-clock cost with and without the escape mechanism.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not use one noise estimate everywhere?
A single estimator can still be contaminated by transitions or label-dependent filtering. Comparing differently sourced estimates helps expose feedback.
Should every gate failure force an exit?
Usually not. Persistence or severity rules reduce reactions to one transient, but critical feasibility failures may warrant immediate fallback.
Can memory be kept after escape?
Only if it cannot recreate the false lock. Quarantine and staged reintroduction are safer when the failure mechanism is uncertain.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Sun and colleagues provide finite-time manifold-identification conditions and bounds for proximal methods. O’Donoghue and Candès study adaptive restart for accelerated gradient schemes. Wang and colleagues develop accelerated active-manifold identification for low-rank optimisation. The feedback ledger and escape state machine above are original safeguards and do not replace those convergence conditions.
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