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This paper provides the first theoretical convergence analysis for machine learning training performed entirely within Fully Homomorphic Encryption (FHE). They introduce a differentially private (DP) training algorithm optimized for FHE, avoiding per-sample gradient clipping for efficiency. The analysis includes convergence proofs for approximate gradient descent using polynomial approximations of activation and loss functions necessary for FHE, along with data-independent hyperparameter selection.
Training machine learning models directly on encrypted data is now provably convergent, private, and more efficient thanks to a novel approach that avoids costly per-sample gradient clipping.
We present the first theoretical convergence analysis of machine learning training under fully homomorphic encryption (FHE), combined with a differentially private (DP) training algorithm tailored to encrypted computation. Our approach improves computational efficiency over standard differentially private gradient descent (DP-GD) while achieving comparable utility. In particular, we prove convergence of approximate gradient descent using polynomial approximations of activation and loss functions, which are required for FHE compatibility. To preserve privacy in downstream tasks, we integrate differential privacy without relying on costly per-sample gradient clipping, enabling scalable encrypted learning. We also provide data-independent hyperparameter selection and theoretically grounded strategies for polynomial approximation which can be of independent interest. Together, these contributions advance the feasibility of efficient, private, and secure machine learning on sensitive data.
