Empirical Risk Minimization (ERM) is a standard technique in machine learning, where a model is selected by minimizing a loss function over constraint set. When the training dataset consists of private information, it is natural to use a differentially private ERM algorithm, and this problem has been the subject of a long line of work started with Chaudhuri and Monteleoni 2008. A private ERM algorithm outputs an approximate minimizer of the loss function and its error can be measured as the difference from the optimal value of the loss function. When the constraint set is arbitrary, the required error bounds are fairly well understood~\cite{BassilyST14}. In this work, we show that the geometric properties of the constraint set can be used to derive significantly better results. Specifically, we show that a differentially private version of Mirror Descent leads to error bounds of the form $\tilde{O}(G_{\mathcal{C}}/n)$ for a lipschitz loss function, improving on the $\tilde{O}(\sqrt{p}/n)$ bounds in Bassily, Smith and Thakurta 2014. Here $p$ is the dimensionality of the problem, $n$ is the number of data points in the training set, and $G_{\mathcal{C}}$ denotes the Gaussian width of the constraint set that we optimize over. We show similar improvements for strongly convex functions, and for smooth functions. In addition, we show that when the loss function is Lipschitz with respect to the $\ell_1$ norm and $\mathcal{C}$ is $\ell_1$-bounded, a differentially private version of the Frank-Wolfe algorithm gives error bounds of the form $\tilde{O}(n^{-2/3})$. This captures the important and common case of sparse linear regression (LASSO), when the data $x_i$ satisfies $|x_i|_{\infty} \leq 1$ and we optimize over the $\ell_1$ ball. We show new lower bounds for this setting, that together with known bounds, imply that all our upper bounds are tight.

本文介绍了隐私保护数据集下Empirical Risk Minimization（ERM）的改进算法——不同ially private ERM Algorithm。该算法通过利用限制条件的几何特性，在Lipschitz、强凸和光滑函数等情况下，提供了更严格的误差上界，并针对稀疏线性回归（LASSO）提出了新的下界。

超出最坏情况的私有经验风险最小化: 约束集几何形状的影响