Data Reconstruction: Identifiability and Optimization with Sample Splitting

TL;DR

This paper investigates the conditions under which training data can be uniquely reconstructed from KKT conditions in neural networks and introduces a sample splitting technique to improve optimization-based reconstruction methods.

Contribution

It provides theoretical conditions for data identifiability in two-layer networks and proposes a sample splitting method to enhance reconstruction optimization.

Findings

Theoretical conditions for unique data reconstruction in two-layer networks.

Sample splitting improves the accuracy of existing reconstruction methods.

Experimental results show enhanced reconstruction performance with sample splitting.

Abstract

Training data reconstruction from KKT conditions has shown striking empirical success, yet it remains unclear when the resulting KKT equations have unique solutions and, even in identifiable regimes, how to reliably recover solutions by optimization. This work hereby focuses on these two complementary questions: identifiability and optimization. On the identifiability side, we discuss the sufficient conditions for KKT system of two-layer networks with polynomial activations to uniquely determine the training data, providing a theoretical explanation of when and why reconstruction is possible. On the optimization side, we introduce sample splitting, a curvature-aware refinement step applicable to general reconstruction objectives (not limited to KKT-based formulations): it creates additional descent directions to escape poor stationary points and refine solutions. Experiments demonstrate that augmenting several existing reconstruction methods with sample splitting consistently improves reconstruction performance.