Back to Square Roots: An Optimal Bound on the Matrix Factorization Error for Multi-Epoch Differentially Private SGD
This work addresses a theoretical gap in privacy-preserving machine learning for researchers and practitioners, though it is incremental as it builds on existing matrix factorization mechanisms.
The paper tackles the problem of multi-epoch matrix factorization error in differentially private SGD, introducing the Banded Inverse Square Root (BISR) method to derive a tight characterization and prove asymptotically optimal error, matching upper and lower bounds.
Matrix factorization mechanisms for differentially private training have emerged as a promising approach to improve model utility under privacy constraints. In practical settings, models are typically trained over multiple epochs, requiring matrix factorizations that account for repeated participation. Existing theoretical upper and lower bounds on multi-epoch factorization error leave a significant gap. In this work, we introduce a new explicit factorization method, Banded Inverse Square Root (BISR), which imposes a banded structure on the inverse correlation matrix. This factorization enables us to derive an explicit and tight characterization of the multi-epoch error. We further prove that BISR achieves asymptotically optimal error by matching the upper and lower bounds. Empirically, BISR performs on par with state-of-the-art factorization methods, while being simpler to implement, computationally efficient, and easier to analyze.