# Conformal Calibration of Escalation Thresholds

## Problem

The model cascade router uses a heuristic threshold: route to the cheapest tier where P(success) >= 0.65. This threshold was hand-tuned. There is no guarantee that it provides adequate coverage — we might be escalating too often (wasting cost) or not often enough (accepting too many failures).

## Solution

Conformal risk control (Angelopoulos et al., 2022, arxiv:2208.02814) provides distribution-free coverage guarantees. Applied to our escalation problem following RouteNLP (arxiv:2604.23577):

**Guarantee**: P(failure AND no escalation) ≤ α

This means: if the router decides NOT to escalate (i.e., it trusts the cheap model), the probability that the cheap model actually fails is at most α.

## Method

### 1. Nonconformity Score

For each calibration example i at tier t:
- Compute the calibrated P(success) from the XGBoost + isotonic model
- If the model failed (y_i = 0), the nonconformity score is s_i = P(success)_i
- If the model succeeded (y_i = 1), the nonconformity score is s_i = -∞ (always safe)

### 2. Conformal Threshold

Set the escalation threshold as the ⌈(1-α)(n+1)/n⌉-th quantile of nonconformity scores among **failed examples only** (following RouteNLP's correctly-handled subset approach).

This gives threshold λ̂_t for each tier t.

### 3. Routing Decision

```
if P(success at tier t) >= λ̂_t:
    use tier t  (no escalation needed)
else:
    escalate to tier t+1
```

### 4. Coverage Verification

On a held-out test set, verify that:
- P(y=fail | P(success) >= λ̂_t) ≤ α  for each tier t
- This is the violation rate — should be ≤ α

## Expected Impact

From RouteNLP's results:
- With 500 calibration examples per tier, violation rate is 4.2% at α=0.05
- With 100 examples, violation rate is 7.2%
- With 1000 examples, violation rate is 3.9%

Our SWE-Router dataset has 500 tasks × 8 models = 4000 total outcomes, giving us ~800 per tier. Expected violation rate: ~4%.

## Implementation

The module is in `aco/conformal.py`:

```python
from aco.conformal import ConformalEscalationCalibrator

# Calibrate
cal = ConformalEscalationCalibrator(alpha=0.05)
thresholds = cal.calibrate(psuccess, outcomes)

# Use in routing
if cal.should_escalate(tier=2, psuccess=0.62):
    # Escalate to tier 3
    ...
```

## Integration with v10 Router

The conformal calibrator replaces the hardcoded 0.65 threshold in `route_v10()`:

```python
# Before
for t in range(1, 6):
    if tier_probs[t] >= 0.65:  # heuristic
        return t, tier_probs[t], tier_probs

# After
for t in range(1, 6):
    if not cal.should_escalate(t, tier_probs[t]):  # conformal
        return t, tier_probs[t], tier_probs
```

## Sensitivity Analysis

| α | Expected Violation Rate | Expected Escalation Rate | Cost Impact |
|---|------------------------|-------------------------|-------------|
| 0.01 | ~1% | High (conservative) | +10-15% cost |
| 0.05 | ~4% | Medium | Baseline |
| 0.10 | ~8% | Low (aggressive) | -5-10% cost |
| 0.20 | ~15% | Very low | -15-20% cost |

For production use, α=0.05 is recommended. For high-risk domains (legal, medical), use α=0.01.

## Caveats

1. **Exchangeability assumption**: Conformal guarantees require that calibration and test data are exchangeable. Distribution shift (new task types, new models) invalidates the guarantee.
2. **Sample size**: With only 500 tasks, per-tier calibration has ~100 examples per tier. More data improves calibration.
3. **Conditional vs marginal**: The guarantee is marginal (averaged over all inputs), not conditional (for a specific input type). Conditional coverage requires stronger assumptions.

## References

- Angelopoulos, A.N., et al. "Conformal Risk Control." NeurIPS 2022. arxiv:2208.02814
- RouteNLP: "Closed-Loop LLM Routing with Conformal Cascading and Distillation Co-Optimization." arxiv:2604.23577
- CP-Router: "An Uncertainty-Aware Router Between LLM and LRM." arxiv:2505.19970
- CAP: "Learning Conformal Abstention Policies for Adaptive Risk Management." arxiv:2502.06884