Clipped stochastic gradient descent (SGD) algorithms are among the most popular algorithms for privacy preserving optimization that reduces the leakage of users' identity in model training. This paper studies the convergence properties of these algorithms in a performative prediction setting, where the data distribution may shift due to the deployed prediction model. For example, the latter is caused by strategical users during the training of loan policy for banks. Our contributions are two-fold. First, we show that the straightforward implementation of a projected clipped SGD (PCSGD) algorithm may converge to a biased solution compared to the performative stable solution. We quantify the lower and upper bound for the magnitude of the bias and demonstrate a bias amplification phenomenon where the bias grows with the sensitivity of the data distribution. Second, we suggest two remedies to the bias amplification effect. The first one utilizes an optimal step size design for PCSGD that takes the privacy guarantee into account. The second one uses the recently proposed DiceSGD algorithm [Zhang et al., 2024]. We show that the latter can successfully remove the bias and converge to the performative stable solution. Numerical experiments verify our analysis.

本文研究了在预测性场景下剪裁随机梯度下降（SGD）算法的收敛性质，分析了其可能产生的偏差放大现象，并提出了两种解决方案和相应的实验验证。

隐私保护的剪切SGD算法：偏差放大和修正