certified robustness provable defenses

# Certified Robustness & Provable Defenses

## Introduction & Motivation

Certified robustness: prove model robustness formally. Provide robustness guarantees. Applications: safety-critical systems, verified AI.

Motivation: Provide mathematical guarantees of robustness.

Applications: Autonomous driving, healthcare, security.

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## Core Concepts & Theory

### Robustness Certification

Formal verification.

### Randomized Smoothing

Statistical certification.

### Lipschitz Bounds

Constraint on gradients.

### Verification

Prove properties formally.

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## Mathematical Formulation

Certified Robustness:
$$||f(x+\delta) - f(x)|| \leq L ||\delta||$$

Randomized Smoothing:
$$f_c(x) = \arg\max_c P(f(x+N) = c)$$

Robustness Radius:
$$r(x) = \frac{\sigma}{2}(\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$$

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## Advanced Theory & Extensions

### Convex Relaxation

Tractable verification.

### Neural Network Verification

Abstract interpretation.

### Compositional Reasoning

Verify complex systems.

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## Computational Considerations

Verification: Polynomial for simple networks.

Sampling: Thousands of queries.

Scalability: Limited to modest networks.

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## Practical Implementation Strategies

### Smoothing Noise Scale

Choose sigma carefully.

### Sample Size

Balance accuracy and confidence.

### Verification Methods

Choose appropriate tool.

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## Benchmark Datasets & Evaluation

MNIST: Small networks.

CIFAR-10: Larger networks.

ImageNet: Large-scale challenges.

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## Key Challenges & Limitations

### Scalability

Limited to small networks.

### Loose Bounds

Conservative guarantees.

### Computational Cost

Expensive verification.

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## Hyperparameter Tuning

Sigma: 0.1-1.0.

Sample size: 100000+.

Verification epsilon: Small value.

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## Real-World Applications & Case Studies

Safety-Critical: Autonomous systems.

Healthcare: Medical imaging.

Security: Intrusion detection.

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## Integration with Other Methods

Certified robustness + adversarial training; + verification.

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## Summary & Key Takeaways

Certified robustness provides formal guarantees.

Principles:
1. Certification: Mathematical proof.
2. Randomization: Statistical guarantees.
3. Bounds: Lipschitz constraints.
4. Verification: Formal methods.
5. Scalability: Trade-off with guarantees.

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## Appendix: Practical Labs

### Lab 1: Randomized Smoothing

import numpy as np
from scipy.stats import norm

def randomized_smoothing_cert(base_classifier, x, num_samples=10000, sigma=0.25):
 """Certify robustness via randomized smoothing"""
 predictions = []
 
 for _ in range(num_samples):
 noise = np.random.normal(0, sigma, x.shape)
 x_noisy = x + noise
 pred = base_classifier.predict(x_noisy)
 predictions.append(pred)
 
 # Count predictions
 unique, counts = np.unique(predictions, return_counts=True)
 top_class = unique[np.argmax(counts)]
 second_class = unique[np.argsort(counts)[-2]]
 
 p_A = np.max(counts) / num_samples
 p_B = np.sort(counts)[-2] / num_samples
 
 # Certified radius
 if p_A > p_B:
 radius = (sigma/2) * (norm.ppf(p_A) - norm.ppf(p_B))
 else:
 radius = 0
 
 return radius

class DummyClassifier:
 def predict(self, x):
 return np.argmax(np.random.rand(10))

clf = DummyClassifier()
x = np.random.rand(10)
radius = randomized_smoothing_cert(clf, x)
assert radius >= 0
print(f"✓ Certified radius: {radius:.4f}")

### Lab 2: Lipschitz Constant Estimation

import numpy as np

def estimate_lipschitz_constant(model, test_points, num_samples=100):
 """Estimate Lipschitz constant of model"""
 max_ratio = 0
 
 for _ in range(num_samples):
 # Sample two close points
 x1 = test_points[np.random.randint(len(test_points))]
 noise = np.random.randn(*x1.shape) * 0.01
 x2 = x1 + noise
 
 # Compute ratio
 y1 = model.forward(x1)
 y2 = model.forward(x2)
 
 dist_x = np.linalg.norm(x2 - x1) + 1e-8
 dist_y = np.linalg.norm(y2 - y1)
 
 ratio = dist_y / dist_x
 max_ratio = max(max_ratio, ratio)
 
 return max_ratio

class SimpleModel:
 def forward(self, x):
 return x @ np.random.randn(len(x), 10)

test_data = np.random.rand(5, 10)
L = estimate_lipschitz_constant(SimpleModel(), test_data)
assert L >= 0
print(f"✓ Lipschitz constant: {L:.2f}")

### Lab 3: Verification via Interval Bound Propagation

import numpy as np

def ibp_verify(weights, input_bounds, true_label):
 """Interval Bound Propagation verification"""
 lower, upper = input_bounds
 
 # Forward pass with interval arithmetic
 for w in weights:
 # Simplified: linear layer
 lower = lower @ w.clip(min=0)
 upper = upper @ w.clip(max=0)
 
 # Check if true label has max lower bound
 if lower[true_label] > max(lower[:true_label] + lower[true_label+1:]):
 return True
 return False

weights = [np.random.randn(10, 10) for _ in range(2)]
bounds = (np.random.rand(10) - 0.1, np.random.rand(10) + 0.1)
verified = ibp_verify(weights, bounds, 0)
print(f"✓ IBP verification: {verified}")

### Lab 4: Trade-off Analysis

import numpy as np

def robustness_accuracy_tradeoff(model, data, labels, epsilons):
 """Analyze robustness-accuracy trade-off"""
 results = []
 
 for epsilon in epsilons:
 # Simulate certified training
 certified_acc = 0.95 - 0.2 * epsilon
 clean_acc = 0.98
 
 results.append({
 'epsilon': epsilon,
 'certified_acc': certified_acc,
 'clean_acc': clean_acc
 })
 
 return results

epsilons = np.linspace(0, 1, 5)
results = robustness_accuracy_tradeoff(None, None, None, epsilons)
assert len(results) == len(epsilons)
print("✓ Trade-off analysis complete")

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