Mohamed Ndaoud

Mohamed Ndaoud, Peter Radchenko, Bradley Rava

Selective classification is a powerful tool for automated decision-making in high-risk scenarios, allowing classifiers to act only when confident and abstain when uncertainty is high. Given a target accuracy, our goal is to minimize indecisions, observations we do not automate. For difficult problems, the target accuracy may be unattainable without abstention. By using indecisions, we can control the misclassification rate to any user-specified level, even below the Bayes optimal error rate, while minimizing overall indecision mass. We provide a complete characterization of the minimax risk in selective classification, establishing continuity and monotonicity properties that enable optimal indecision selection. We revisit selective inference via the Neyman-Pearson testing framework, where indecision enables control of type 2 error given fixed type 1 error probability. For both classification and testing, we propose a finite-sample calibration method with non-asymptotic guarantees, proving plug-in classifiers remain consistent and that accuracy-based calibration effectively controls indecision mass. In the binary Gaussian mixture model, we uncover the first sharp phase transition in selective inference, showing minimal indecision can yield near-optimal accuracy even under poor class separation. Experiments on Gaussian mixtures and real datasets confirm that small indecision proportions yield substantial accuracy gains, making indecision a principled tool for risk control.

Stanislav Minsker, Mohamed Ndaoud, Yiqiu Shen

We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model in high dimensions and in the non-asymptotic setting. We first derive a lower and a matching upper bound for the minimax risk of clustering in this framework. We also show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense. Next, we characterize precisely the risk of $\ell_2$-regularized supervised least squares classifiers. We deduce the fact that the interpolating solution may outperform the regularized classifier, under mild assumptions on the covariance structure of the noise. Our analysis also shows that interpolation can be robust to corruption in the covariance of the noise when the signal is aligned with the "clean" part of the covariance, for the properly defined notion of alignment. To the best of our knowledge, this peculiar phenomenon has not yet been investigated in the rapidly growing literature related to interpolation. We conclude that interpolation is not only benign but can also be optimal, and in some cases robust.

Dmitrii M. Ostrovskii, Mohamed Ndaoud, Adel Javanmard et al.

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.