论文标题
真实的广义线性模型
Truthful Generalized Linear Models
论文作者
论文摘要
在本文中，我们研究了代理人（个人）具有战略性或自我利益的情况，并且在报告数据时关注其隐私，估计广义线性模型（GLM）。与经典环境相比，我们的目标是设计机制，这些机制既可以激励大多数代理以真实地报告他们的数据并保留个人报告的隐私，而其输出也应接近基础参数。在本文的第一部分中，我们考虑了协变量是次高斯的情况，并且在他们只有有限的第四瞬间的情况下进行了重尾。首先，我们的动力是可能性函数的最大化器的固定条件，我们得出了一种新颖的私人和封闭形式估计器。根据估计器，我们提出了一种机制，该机制通过对几种规范模型的计算和付款方案进行一些适当的设计，例如线性回归，逻辑回归和泊松回归：（1）机制为$ O（1）$ - 共同差异私有（至少有可能是$ 1-o（1-o（1）$）; （2）这是一个$ o（\ frac {1} {n}）$ - 近似于$（1-o（1-o（1））$的$（1-o（1））$ - 代理的nash平衡真实地报告其数据，其中$ n $是代理的数量； （3）输出可以达到基础参数的$ O（1）$的错误； （4）对于机制中的$（1-o（1））$的代理分数是个人合理的； （5）分析师运行该机制所需的付款预算为$ O（1）$。在第二部分中，我们考虑了在更通用的环境下的线性回归模型，在该设置中，协变量和响应都是重尾，只有有限的第四次矩。通过使用$ \ ell_4 $ -norm收缩运算符，我们提出了一种私人估算器和付款方案，该方案具有与次高斯案例相似的属性。
In this paper we study estimating Generalized Linear Models (GLMs) in the case where the agents (individuals) are strategic or self-interested and they concern about their privacy when reporting data. Compared with the classical setting, here we aim to design mechanisms that can both incentivize most agents to truthfully report their data and preserve the privacy of individuals' reports, while their outputs should also close to the underlying parameter. In the first part of the paper, we consider the case where the covariates are sub-Gaussian and the responses are heavy-tailed where they only have the finite fourth moments. First, motivated by the stationary condition of the maximizer of the likelihood function, we derive a novel private and closed form estimator. Based on the estimator, we propose a mechanism which has the following properties via some appropriate design of the computation and payment scheme for several canonical models such as linear regression, logistic regression and Poisson regression: (1) the mechanism is $o(1)$-jointly differentially private (with probability at least $1-o(1)$); (2) it is an $o(\frac{1}{n})$-approximate Bayes Nash equilibrium for a $(1-o(1))$-fraction of agents to truthfully report their data, where $n$ is the number of agents; (3) the output could achieve an error of $o(1)$ to the underlying parameter; (4) it is individually rational for a $(1-o(1))$ fraction of agents in the mechanism ; (5) the payment budget required from the analyst to run the mechanism is $o(1)$. In the second part, we consider the linear regression model under more general setting where both covariates and responses are heavy-tailed and only have finite fourth moments. By using an $\ell_4$-norm shrinkage operator, we propose a private estimator and payment scheme which have similar properties as in the sub-Gaussian case.
