Find your next learning step
A greedy audit guarantee depends on adaptive diminishing returns. Remote-sensing evidence can violate that property when one label unlocks the value of another: a cloud cue clarifies a thermal anomaly, a sensor seam reveals a calibration branch, or two scenes jointly identify a rare event. Stress tests turn those possibilities into explicit counterexamples and failure maps.
Open the chapter contents · Remote Sensing root · Gold Standard of Education · How X Works
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. A guarantee must survive conditional histories
Adaptive submodularity compares the expected marginal value of the same candidate under two compatible information states, one containing more observations than the other. The candidate’s gain should not increase as information accumulates.
Testing only average batch utility is insufficient. The violation may appear in a rare history that is precisely where an audit policy faces its most important choice.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Construct minimal complementarity counterexamples
Begin with two candidate scenes and two possible states. Define the observation likelihoods and utility completely, then calculate the marginal gain before and after one outcome.
A minimal example is valuable because it identifies the mechanism of failure. Large simulations may show greedy underperformance without revealing whether the cause is complementarity, constraints, posterior updating or approximate inference.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A conditional-gain stress ledger
An audit utility rewards correctly resolving incident type. Values below are conditional expected gains for scene B before and after observing scene A.
| History | Gain of B | Diminishing-return check | Interpretation |
|---|---|---|---|
| No labels | 0.10 | Baseline | B is weak alone |
| A shows smoke | 0.31 | Fails | Thermal B becomes diagnostic |
| A shows cloud | 0.04 | Passes | B becomes redundant |
| A is missing | 0.16 | Fails | B replaces lost evidence |
Weight the histories for expected policy evaluation, but do not erase the formal violation. A theorem’s assumptions are logical conditions, not majority votes.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked greedy-versus-lookahead counterexample
A two-audit budget offers scenes A, B and C. Greedy selects C for immediate gain 0.24. A then B has gains 0.18 and 0.35 when A reveals the enabling state.
| Policy | First gain | Second gain | Total in enabling branch |
|---|---|---|---|
| Greedy C then A | 0.24 | 0.12 | 0.36 |
| Lookahead A then B | 0.18 | 0.35 | 0.53 |
| Fixed A and C | 0.18 | 0.20 | 0.38 |
The example does not prove lookahead always wins. It shows why the adaptive-greedy approximation theorem cannot be invoked when marginal gains increase.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Build a guarantee failure map
Index violations by biome, sensor, season, label history, missingness pattern and utility component. Mark whether each branch violates adaptive monotonicity, adaptive submodularity or only an implementation approximation.
A failure map can still support local guarantees. If diminishing returns holds inside clear strata, use greedy there and route violating branches to rollout, paired selection or expert review.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Check inference and modelling approximations
Monte Carlo error can fabricate small marginal-gain reversals. Attach uncertainty intervals, common random numbers and repeated calculations before declaring a violation.
The opposite risk also matters: approximate posterior updates may smooth away real complementarity. Validate stress cases with exact enumeration or high-accuracy computation on reduced instances.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: wildfire confirmation
Transfer the method to wildfire confirmation where optical smoke, thermal hotspots and wind context interact. A thermal audit may be uninformative before smoke detection and highly valuable after it.
The transfer check distinguishes evidence complementarity from duplicate sensing, incorporates cloud-driven missingness and evaluates detection delay as well as final accuracy.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the adaptive-audit branch
The adaptive-submodularity guide states the positive guarantee. The lookahead guide values enabling observations.
Use the sequential audit guide when the objective is a stable downstream decision rather than coverage alone.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Failure of adaptive submodularity does not mean greedy is unusable. It means the familiar near-optimality guarantee is unavailable unless a different theorem applies.
One counterexample is enough to disprove a global property, but operational importance still depends on how often the history occurs and what decision loss it creates.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the state model, utility, conditional histories, gain estimates, numerical error, counterexamples, constraint family, baseline policies and failure-map strata.
Report the smallest reproducible violation and the largest decision-relevant one. Define which branches remain eligible for greedy selection and which require lookahead or paired audits.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can empirical checks prove adaptive submodularity?
Finite checks can establish the property only for the enumerated finite model. In larger models they are diagnostics, not universal proof.
What is complementarity?
Two observations are complementary when one raises the value of obtaining the other, contrary to diminishing returns.
Should every violation trigger lookahead?
No. Use decision impact, frequency and computational cost to route material violations while keeping simple policies where they remain adequate.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Golovin and Krause introduce adaptive submodularity and its greedy guarantees. Fujii and Sakaue develop approximation results beyond adaptive submodularity. Krause and colleagues study robust observation selection under submodular structure. The remote-sensing counterexamples above are original teaching examples.
Continue through the Remote Sensing root, the Gold Standard of Education and How X Works.
Top · Contents · Previous · How X Works