Differentially Private Clipped-SGD: High-Probability Convergence with Arbitrary Clipping Level

TL;DR

This paper provides the first high-probability convergence analysis for differentially private clipped SGD with a fixed clipping level, applicable to both convex and non-convex optimization under heavy-tailed noise, balancing convergence and privacy.

Contribution

It introduces a novel convergence analysis for DP-Clipped-SGD with fixed clipping, applicable to heavy-tailed noise, and offers a refined trade-off between convergence speed and privacy guarantees.

Findings

Converges to a neighborhood of the optimum with a faster rate.

Applicable to both convex and non-convex smooth optimization.

Balances convergence speed with differential privacy constraints.

Abstract

Gradient clipping is a fundamental tool in Deep Learning, improving the high-probability convergence of stochastic first-order methods like SGD, AdaGrad, and Adam under heavy-tailed noise, which is common in training large language models. It is also a crucial component of Differential Privacy (DP) mechanisms. However, existing high-probability convergence analyses typically require the clipping threshold to increase with the number of optimization steps, which is incompatible with standard DP mechanisms like the Gaussian mechanism. In this work, we close this gap by providing the first high-probability convergence analysis for DP-Clipped-SGD with a fixed clipping level, applicable to both convex and non-convex smooth optimization under heavy-tailed noise, characterized by a bounded central $\alpha$-th moment assumption, $\alpha \in (1,2]$. Our results show that, with a fixed clipping level, the method converges to a neighborhood of the optimal solution with a faster rate than the existing ones. The neighborhood can be balanced against the noise introduced by DP, providing a refined trade-off between convergence speed and privacy guarantees.