"""
Privacy Certificates.

Implements certification checks for "Certified Removal" mechanisms.
Based on theoretical bounds (e.g., from Guo et al. 2020).
"""

import math

class CertifiedRemover:
    """
    Checks if a model update satisfies Certified Removal guarantees.
    
    Condition often roughly:
    || H^-1 g || <= epsilon * sigma
    """
    
    def __init__(self, sigma: float = 1.0):
        self.sigma = sigma
    
    def check_certificate(
        self, 
        gradient_norm: float, 
        hessian_min_eigen: float, 
        epsilon: float, 
        delta: float = 1e-4
    ) -> bool:
        """
        Check if removal is certified within epsilon budget.
        
        Simplified bound check:
        If the influence of the removed point (approximated by Newton step) 
        is small enough relative to the noise (sigma) and budget (epsilon).
        """
        # If Hessian is strongly convex (min_eigen > 0)
        if hessian_min_eigen <= 0:
            return False
            
        # Newton Step Norm estimate: || H^-1 g || <= ||g|| / lambda_min
        newton_step_norm = gradient_norm / hessian_min_eigen
        
        # Privacy Bound (Gaussian Mechanism style):
        # We need step_norm <= epsilon * sigma (roughly) for DP-like guarantee.
        # This is a heuristic simplification of the full Certified Removal theorem.
        
        # For (epsilon, delta)-certified removal:
        # e^epsilon - 1 >= ... distance metrics
        
        # We use a conservative threshold here for demonstration.
        threshold = self.sigma * epsilon
        
        return newton_step_norm <= threshold
