具有非披露性质的模型判别的近优方法
Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties
摘要：
令$θ_0,θ_1 \in \mathbb{R}^d$为与某些损失$\ell:\mathbb{R}^d\times \mathcal{Z}\to\mathbb{R}$和$\mathcal{Z}$上的两个分布$\mathbb{P}_0,\mathbb{P}_1$相关的总体风险最小化。模型$θ_0,θ_1$是未知的，可以通过从中绘制i.i.d样本来访问$\mathbb{P}_0,\mathbb{P}_1$。我们的工作是出于以下模型的歧视问题：“什么样的大小的样本从$\mathbb{P}_0$和$\mathbb{P}_1$允许区分两个假设$θ^*=θ_0$和$θ^*=θ_1$给定的$θ^*\in\{θ_0,θ_1\}$？“在回答这个问题的第一步，我们首先考虑一个具有平方损失的指定线性模型的情况。在这里，我们提供了匹配的上限和下限的样本复杂性，由$\min\{1/Δ^2,\sqrt{r}/Δ\}$给定一个常数因子;这里$Δ$是一个措施之间的分离$\mathbb{P}_0$和$\mathbb{P}_1$和$r$是秩的设计协方差矩阵。然后，我们将这个结果在两个方向上：（i）一般参数模型在渐近制度;（ii）广义线性模型在小样本（$n\le r$）弱矩假设。在这两种情况下，我们得出一个类似的形式，同时允许模型误指定样本的复杂性界限。事实上，我们的测试程序只通过一定的经验风险函数来访问$θ^*$。此外，允许我们达到统计置信度的观测值的数量不允许“解决”两个模型$-$，即恢复$θ_0,θ_1$到$O(Δ)$预测精度。这两个属性允许在应用任务中使用我们的框架，其中人们希望$\textit{identify}$预测模型，该模型可以是专有的，同时保证该模型实际上不能被识别代理$\textit{inferred}$。
英文摘要：
Let $θ_0,θ_1 \in \mathbb{R}^d$ be the population risk minimizers associated to some loss $\ell:\mathbb{R}^d\times \mathcal{Z}\to\mathbb{R}$ and two distributions $\mathbb{P}_0,\mathbb{P}_1$ on $\mathcal{Z}$. The models $θ_0,θ_1$ are unknown, and $\mathbb{P}_0,\mathbb{P}_1$ can be accessed by drawing i.i.d samples from them. Our work is motivated by the following model discrimination question: "What sizes of the samples from $\mathbb{P}_0$ and $\mathbb{P}_1$ allow to distinguish between the two hypotheses $θ^*=θ_0$ and $θ^*=θ_1$ for given $θ^*\in\{θ_0,θ_1\}$?" Making the first steps towards answering it in full generality, we first consider the case of a well-specified linear model with squared loss. Here we provide matching upper and lower bounds on the sample complexity as given by $\min\{1/Δ^2,\sqrt{r}/Δ\}$ up to a constant factor; here $Δ$ is a measure of separation between $\mathbb{P}_0$ and $\mathbb{P}_1$ and $r$ is the rank of the design covariance matrix. We then extend this result in two directions: (i) for general parametric models in asymptotic regime; (ii) for generalized linear models in small samples ($n\le r$) under weak moment assumptions. In both cases we derive sample complexity bounds of a similar form while allowing for model misspecification. In fact, our testing procedures only access $θ^*$ via a certain functional of empirical risk. In addition, the number of observations that allows us to reach statistical confidence does not allow to "resolve" the two models $-$ that is, recover $θ_0,θ_1$ up to $O(Δ)$ prediction accuracy. These two properties allow to use our framework in applied tasks where one would like to $\textit{identify}$ a prediction model, which can be proprietary, while guaranteeing that the model cannot be actually $\textit{inferred}$ by the identifying agent.
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