# Adaptive λ_rej Implementation Summary

## Implementation Components

### 1. Coverage Tracking (`zeroproof/training/coverage.py`)
- **`CoverageMetrics`**: Dataclass tracking tag distribution
- **`CoverageTracker`**: Main tracker with:
  - Cumulative and windowed coverage calculation
  - Target coverage gap computation
  - Detailed tag distribution statistics
- **`MultiOutputCoverageTracker`**: For models with multiple outputs

### 2. Adaptive Lambda (`zeroproof/training/adaptive_loss.py`)
- **`AdaptiveLossConfig`**: Configuration dataclass with:
  - Initial lambda, target coverage, learning rate
  - Constraints (lambda_min, lambda_max)
  - Momentum, warmup, update frequency
- **`AdaptiveLambda`**: Core implementation:
  - Lagrange multiplier update: `λ ← λ + η_λ * (c* - c_actual)`
  - Momentum support for smoother updates
  - Warmup period before adaptation
  - Constraint enforcement
- **`AdaptiveLossPolicy`**: Full policy combining:
  - Adaptive lambda with loss computation
  - Support for different base losses (MSE, MAE)
  - Proper reduction modes for TR values

### 3. Training Integration (`zeroproof/training/trainer.py`)
- **`TrainingConfig`**: Includes adaptive loss parameters
- **`TRTrainer`**: Full trainer with:
  - Automatic coverage tracking
  - Lambda adjustment during training
  - Proper gradient handling with Mask-REAL
  - History tracking for analysis

### 4. Layer Integration
- Updated `TRRational` to accept `adaptive_loss_policy` parameter
- Seamless integration with existing TR arithmetic

## Key Features Implemented

### Update Rule
```python
# Compute coverage gap
coverage_gap = target_coverage - actual_coverage

# Update lambda with optional momentum
update = learning_rate * coverage_gap
if momentum > 0:
    velocity = momentum * velocity + update
    lambda_rej += velocity
else:
    lambda_rej += update

# Apply constraints
lambda_rej = max(lambda_min, min(lambda_max, lambda_rej))
```

### Loss Computation
```python
if output.tag == TRTag.REAL:
    loss = base_loss_fn(output, target)  # Normal loss
else:
    loss = lambda_rej  # Rejection penalty
```

### Integration with Mask-REAL
- Gradients remain zero for non-REAL paths
- Lambda only affects loss value, not gradient flow
- Maintains numerical stability at singularities

## Usage Example

```python
# Create adaptive loss policy
adaptive_loss = create_adaptive_loss(
    target_coverage=0.95,
    learning_rate=0.01,
    initial_lambda=1.0
)

# Use with model
model = TRRational(
    d_p=4, d_q=3,
    adaptive_loss_policy=adaptive_loss
)

# Train with automatic adaptation
trainer = TRTrainer(model)
history = trainer.train(data)

# Monitor adaptation
print(f"Final coverage: {history['coverage'][-1]}")
print(f"Final lambda: {history['lambda_rej'][-1]}")
```

## Testing
- Comprehensive unit tests in `tests/unit/test_adaptive_loss.py`
- Tests for:
  - Coverage tracking accuracy
  - Lambda update convergence
  - Constraint enforcement
  - Integration with loss computation

## Documentation
- Detailed guide: `docs/adaptive_loss_guide.md`
- Example: `examples/adaptive_loss_demo.py`
- API documentation in module docstrings

## Benefits
1. **Automatic Tuning**: No manual lambda adjustment needed
2. **Target Control**: Specify desired coverage rate
3. **Smooth Training**: Momentum prevents oscillations
4. **Flexibility**: Works with any base loss function
5. **Monitoring**: Full visibility into adaptation process

## Compliance with Specification
✅ Implements λ as Lagrange multiplier  
✅ Update rule: `λ ← λ + η_λ * (c* - c_actual)`  
✅ Compatible with Mask-REAL (gradients unaffected)  
✅ Treats non-REAL outputs as constraint violations  
✅ Integrates with strict reduction mode  

The implementation fully satisfies the specification requirements while providing additional features for practical use.
