model-equivariance-auditor（模型等方差审核员）

Model Equivariance Auditor

Even with equivariant libraries, implementation bugs can break equivariance. A model that claims equivariance but lacks it will train poorly and give inconsistent predictions. Catching these bugs early saves significant debugging time.

Workflow

Copy this checklist and track your progress:

Equivariance Audit Progress:
- [ ] Step 1: Gather model and symmetry specification
- [ ] Step 2: Run numerical equivariance tests
- [ ] Step 3: Test individual layers
- [ ] Step 4: Check gradient equivariance
- [ ] Step 5: Identify and diagnose failures
- [ ] Step 6: Document audit results

Step 1: Gather model and symmetry specification

Collect: the implemented model, the intended symmetry group, whether each output should be invariant or equivariant, the transformation functions for input and output spaces. Review the architecture specification from design phase. Clarify ambiguities with user before testing.

Step 2: Run numerical equivariance tests

Execute end-to-end equivariance tests using Test Implementation. For invariance: verify ||f(T(x)) - f(x)|| < ε. For equivariance: verify ||f(T(x)) - T'(f(x))|| < ε. Use multiple random inputs and transformations. Record error statistics. See Error Interpretation for thresholds. For ready-to-use test code, see Test Code Templates.

Step 3: Test individual layers

If end-to-end test fails, isolate the problem by testing layers individually. For each layer: freeze other layers, test equivariance of that layer alone. This identifies which layer breaks equivariance. Use Layer-wise Testing protocol. Check nonlinearities, normalizations, and custom operations especially carefully.

Step 4: Check gradient equivariance

Verify that gradients also respect equivariance (important for training). Compute gradients at x and T(x). Check that gradients transform appropriately. Gradient bugs can cause training to "unlearn" equivariance. See Gradient Testing.

Step 5: Identify and diagnose failures

If tests fail, use Common Failure Modes to diagnose. Check: non-equivariant nonlinearities, batch normalization issues, incorrect output transformation, numerical precision problems, implementation bugs in custom layers. Provide specific fix recommendations. For step-by-step troubleshooting, consult Debugging Guide.

Step 6: Document audit results

Create audit report using Output Template. Include: pass/fail for each test, error magnitudes, identified issues, and recommendations. Distinguish between: exact equivariance (numerical precision), approximate equivariance (acceptable error), and broken equivariance (needs fixing). For detailed audit methodology, see Methodology Details. Quality criteria for this output are defined in Quality Rubric.

Test Implementation

End-to-End Equivariance Test

import torch

def test_model_equivariance(model, x, input_transform, output_transform,
                            n_tests=100, tol=1e-5):
    """
    Test if model is equivariant: f(T(x)) ≈ T'(f(x))

    Args:
        model: The neural network to test
        x: Sample input tensor
        input_transform: Function that transforms input
        output_transform: Function that transforms output
        n_tests: Number of random transformations to test
        tol: Error tolerance

    Returns:
        dict with test results
    """
    model.eval()
    errors = []

    with torch.no_grad():
        for _ in range(n_tests):
            # Generate random transformation
            T = sample_random_transform()

            # Method 1: Transform input, then apply model
            x_transformed = input_transform(x, T)
            y1 = model(x_transformed)

            # Method 2: Apply model, then transform output
            y = model(x)
            y2 = output_transform(y, T)

            # Compute error
            error = torch.norm(y1 - y2).item()
            relative_error = error / (torch.norm(y2).item() + 1e-8)
            errors.append({
                'absolute': error,
                'relative': relative_error
            })

    return {
        'mean_absolute': np.mean([e['absolute'] for e in errors]),
        'max_absolute': np.max([e['absolute'] for e in errors]),
        'mean_relative': np.mean([e['relative'] for e in errors]),
        'max_relative': np.max([e['relative'] for e in errors]),
        'pass': all(e['relative'] < tol for e in errors)
    }

Invariance Test (Simpler Case)

def test_model_invariance(model, x, transform, n_tests=100, tol=1e-5):
    """Test if model output is invariant to transformations."""
    model.eval()
    errors = []

    with torch.no_grad():
        y_original = model(x)

        for _ in range(n_tests):
            T = sample_random_transform()
            x_transformed = transform(x, T)
            y_transformed = model(x_transformed)

            error = torch.norm(y_transformed - y_original).item()
            errors.append(error)

    return {
        'mean_error': np.mean(errors),
        'max_error': np.max(errors),
        'pass': max(errors) < tol
    }

Layer-wise Testing

Protocol

def test_layer_equivariance(layer, x, input_transform, output_transform):
    """Test a single layer for equivariance."""
    layer.eval()

    with torch.no_grad():
        T = sample_random_transform()

        # Transform then layer
        y1 = layer(input_transform(x, T))

        # Layer then transform
        y2 = output_transform(layer(x), T)

        error = torch.norm(y1 - y2).item()

    return {
        'layer': layer.__class__.__name__,
        'error': error,
        'pass': error < tolerance
    }

def audit_all_layers(model, x, transforms):
    """Test each layer individually."""
    results = []

    for name, layer in model.named_modules():
        if is_testable_layer(layer):
            result = test_layer_equivariance(layer, x, *transforms)
            result['name'] = name
            results.append(result)

    return results

What to Test Per Layer

Gradient Testing

Why Test Gradients?

Forward pass can be equivariant while backward pass is not. This causes:

• Training instability
• Model "unlearning" equivariance
• Inconsistent optimization

Gradient Equivariance Test

def test_gradient_equivariance(model, x, loss_fn, transform, tol=1e-4):
    """Test if gradients respect equivariance."""
    model.train()

    # Gradients at original input
    x1 = x.clone().requires_grad_(True)
    y1 = model(x1)
    loss1 = loss_fn(y1)
    loss1.backward()
    grad1 = x1.grad.clone()

    # Gradients at transformed input
    model.zero_grad()
    T = sample_random_transform()
    x2 = transform(x.clone(), T).requires_grad_(True)
    y2 = model(x2)
    loss2 = loss_fn(y2)
    loss2.backward()
    grad2 = x2.grad.clone()

    # Transform grad1 and compare to grad2
    grad1_transformed = transform_gradient(grad1, T)
    error = torch.norm(grad2 - grad1_transformed).item()

    return {'error': error, 'pass': error < tol}

Error Interpretation

Error Thresholds

Relative vs Absolute Error

• Absolute error: Raw difference magnitude
• Relative error: Normalized by output magnitude

Use relative error when output magnitudes vary. Use absolute when comparing to numerical precision.

Common Failure Modes

1. Non-Equivariant Nonlinearity

Symptom: Error increases after nonlinearity layers Cause: Using ReLU, sigmoid on equivariant features Fix: Use gated nonlinearities, norm-based, or restrict to invariant features

2. Batch Normalization Breaking Equivariance

Symptom: Error varies with batch composition Cause: BN computes different stats for different orientations Fix: Use LayerNorm, GroupNorm, or equivariant batch norm

3. Incorrect Output Transformation

Symptom: Test fails even for identity transform Cause: output_transform doesn't match model output type Fix: Verify output transformation matches layer output representation

4. Numerical Precision Issues

Symptom: Small but non-zero error everywhere Cause: Floating point accumulation, interpolation Fix: Use float64 for testing, accept small tolerance

5. Custom Layer Bug

Symptom: Error isolated to specific layer Cause: Implementation error in custom equivariant layer Fix: Review layer implementation against equivariance constraints

6. Padding/Boundary Effects

Symptom: Error higher near edges Cause: Padding doesn't respect symmetry Fix: Use circular padding or handle boundaries explicitly

Output Template

MODEL EQUIVARIANCE AUDIT REPORT
===============================

Model: [Model name/description]
Intended Symmetry: [Group]
Symmetry Type: [Invariant/Equivariant]

END-TO-END TESTS:
-----------------
Test samples: [N]
Transformations tested: [M]

Invariance/Equivariance Error:
- Mean absolute: [value]
- Max absolute: [value]
- Mean relative: [value]
- Max relative: [value]
- RESULT: [PASS/FAIL]

LAYER-WISE ANALYSIS:
--------------------
[For each layer]
- Layer: [name]
- Error: [value]
- Result: [PASS/FAIL]

GRADIENT TEST:
--------------
- Gradient equivariance error: [value]
- RESULT: [PASS/FAIL]

IDENTIFIED ISSUES:
------------------
1. [Issue description]
   - Location: [layer/component]
   - Severity: [High/Medium/Low]
   - Recommended fix: [description]

OVERALL VERDICT: [PASS/FAIL/NEEDS_ATTENTION]

Recommendations:
- [List of actions needed]