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

Saleh Vatan Khah, Savelii Chezhegov, Shahrokh Farahmand, Samuel Horváth, Eduard Gorbunov

Keywords: stochastic optimization, gradient clipping, high-probability convergence, SGD, Clipped-SGD, differential privacy, heavy-tailed noise

TL;DR: We provide the first high-probability analysis of DP-Clipped-SGD with a fixed clipping level for convex and non-convex smooth optimization under the heavy-tailed noise

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.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 10333