DIFF2: Differential Private Optimization via Gradient Differences for Nonconvex Distributed Learning

Abstract

Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is O( √ d/(nε DP )) in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where n is the sample size, d is the problem dimensionality and ε DP is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called DIFF2 (DIFFerential private optimization via gradient DIFFerences) that constructs a differential private global gradient estimator with possibly quite small variance based on communicated gradient differences rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of O(d 2/3 /(nε DP ) 4/3 ), which can be significantly better than the previous one in terms of the dependence on the sample size n. To the best of our knowledge, this is the first fundamental result to improve the standard utility O( √ d/(nε DP )) for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.