Non-Linear Trajectory Modeling for Multi-Step Gradient Inversion Attacks in Federated Learning

Federated Learning (FL) enables collaborative training while preserving privacy, yet Gradient Inversion Attacks (GIAs) pose severe threats by reconstructing private data from shared gradients. In realistic FedAvg scenarios with multi-step updates, existing surrogate methods like SME rely on linear interpolation to approximate client trajectories for privacy leakage. However, we demonstrate that linear assumptions fundamentally underestimate SGD's nonlinear complexity, encountering irreducible approximation barriers in non-convex landscapes with only one-dimensional expressiveness. We propose Non-Linear Surrogate Model Extension (NL-SME), the first framework introducing learnable quadratic Bézier curves for trajectory modeling in GIAs against FL. NL-SME leverages $|w|+1$-dimensional control point parameterization combined with dvec scaling and regularization mechanisms to achieve superior approximation accuracy. Extensive experiments on CIFAR-100 and FEMNIST demonstrate NL-SME significantly outperforms baselines across all metrics, achieving 94\%--98\% performance gaps and order-of-magnitude improvements in cosine similarity loss while maintaining computational efficiency. This work exposes critical privacy vulnerabilities in FL's multi-step paradigm and provides insights for robust defense development.