标题： 关于加速优化在鲁棒和隐私估计中的好处 显示英文标题

标题： On the Benefits of Accelerated Optimization in Robust and Private Estimation

摘要： 我们研究了加速梯度方法的优势，特别是基于Frank-Wolfe方法和投影梯度下降法，用于隐私保护和重尾鲁棒性。 我们的方法如下：对于Frank-Wolfe方法，我们的技术基于定制的学习率以及约束集上$\ell_2$-范数梯度的均匀下界。 为了加速投影梯度下降，我们使用基于Nesterov动量的流行变体，并在$\mathbb{R}^p$上优化目标函数。 这些加速减少了迭代复杂度，转化为经验风险和总体风险最小化的更强统计保证。 我们的分析涵盖了三种设置：非随机数据、随机无模型数据以及参数模型（线性回归和广义线性模型）。 从方法论上讲，我们通过噪声梯度来实现隐私保护和鲁棒性。 我们通过高斯机制和高级组合确保差分隐私，并使用几何均值估计器实现重尾鲁棒性，这也改进了协变量维度上的依赖关系。 最后，我们将我们的速率与现有界限进行比较，并确定我们的方法达到最优收敛的场景。 显示英文摘要

摘要： We study the advantages of accelerated gradient methods, specifically based on the Frank-Wolfe method and projected gradient descent, for privacy and heavy-tailed robustness. Our approaches are as follows: For the Frank-Wolfe method, our technique is based on a tailored learning rate and a uniform lower bound on the gradient of the $\ell_2$-norm over the constraint set. For accelerating projected gradient descent, we use the popular variant based on Nesterov's momentum, and we optimize our objective over $\mathbb{R}^p$. These accelerations reduce iteration complexity, translating into stronger statistical guarantees for empirical and population risk minimization. Our analysis covers three settings: non-random data, random model-free data, and parametric models (linear regression and generalized linear models). Methodologically, we approach both privacy and robustness based on noisy gradients. We ensure differential privacy via the Gaussian mechanism and advanced composition, and we achieve heavy-tailed robustness using a geometric median-of-means estimator, which also sharpens the dependency on the dimension of the covariates. Finally, we compare our rates to existing bounds and identify scenarios where our methods attain optimal convergence.