Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties

TL;DR

This paper develops near-optimal sample complexity bounds for model discrimination tasks, ensuring model identification without revealing detailed model parameters, applicable to various parametric and generalized linear models.

Contribution

It introduces a framework for model discrimination that guarantees non-disclosure of model details while achieving near-optimal sample complexity bounds.

Findings

Matching upper and lower bounds on sample complexity for linear models.

Extension of results to general parametric and generalized linear models.

Framework ensures model identification without revealing proprietary information.

Abstract

Let $\theta_0,\theta_1 \in \mathbb{R}^d$ 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 $\theta_0,\theta_1$ 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 $\theta^*=\theta_0$ and $\theta^*=\theta_1$ for given $\theta^*\in\{\theta_0,\theta_1\}$?" 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 $\min\{1/\Delta^2,\sqrt{r}/\Delta\}$ up to a constant factor; here $\Delta$ 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 $\theta^*$ 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 $\theta_0,\theta_1$ up to $O(\Delta)$ 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.