Neural Information Processing Systems http://nips.cc/

As a result, the demands for interval estimation, and consequently for its validity and precision, have experienced a sustained increase over time and are reflected in a number of recent studies. For example, in proteomics, confidence intervals are employed to assess the association between post-translational modifications and intrinsically disordered regions of proteins, validating hypotheses derived from predictive models and facilitating large-scale functional analyses (Tunyasuvunakool et al., 2021; Bludau et al., 2022). In genomic research, confidence intervals are leveraged to characterize the distribution of gene expression levels, enabling robust inferences about promoter sequence effects and genetic variability (Vaishnav et al., 2022). In the realm of environmental science, interval estimation can be used to monitor deforestation rates of forests, yielding uncertainty-aware insights critical for climate policy formulation (Bullock et al., 2020). As for social sciences, confidence intervals are utilized to evaluate relationships between socioeconomic factors, bolstering the robustness of conclusions drawn from census data (Ding et al., 2021).

Deep neural networks (DNNs) typically involve a large number of parameters and are trained to achieve zero or near-zero training error. Despite such interpolation, they often exhibit strong generalization performance on unseen data, a phenomenon that has motivated extensive theoretical investigations. Comforting results show that interpolation indeed may not affect the minimax rate of convergence under the squared error loss. In the mean time, DNNs are well known to be highly vulnerable to adversarial perturbations in future inputs. A natural question then arises: Can interpolation also escape from suboptimal performance under a future $X$-attack? In this paper, we investigate the adversarial robustness of interpolating estimators in a framework of nonparametric regression. A finding is that interpolating estimators must be suboptimal even under a subtle future $X$-attack, and achieving perfect fitting can substantially damage their robustness. An interesting phenomenon in the high interpolation regime, which we term the curse of simple size, is also revealed and discussed. Numerical experiments support our theoretical findings.

We study in-context learning for nonparametric regression with $α$-Hölder smooth regression functions, for some $α>0$. We prove that, with $n$ in-context examples and $d$-dimensional regression covariates, a pretrained transformer with $Θ(\log n)$ parameters and $Ω\bigl(n^{2α/(2α+d)}\log^3 n\bigr)$ pretraining sequences can achieve the minimax-optimal rate of convergence $O\bigl(n^{-2α/(2α+d)}\bigr)$ in mean squared error. Our result requires substantially fewer transformer parameters and pretraining sequences than previous results in the literature. This is achieved by showing that transformers are able to approximate local polynomial estimators efficiently by implementing a kernel-weighted polynomial basis and then running gradient descent.

We introduce the framework of {\em blind regression} motivated by {\em matrix completion} for recommendation systems: given $m$ users, $n$ movies, and a subset of user-movie ratings, the goal is to predict the unobserved user-movie ratings given the data, i.e., to complete the partially observed matrix. Following the framework of non-parametric statistics, we posit that user $u$ and movie $i$ have features $x_1(u)$ and $x_2(i)$ respectively, and their corresponding rating $y(u,i)$ is a noisy measurement of $f(x_1(u), x_2(i))$ for some unknown function $f$. In contrast with classical regression, the features $x = (x_1(u), x_2(i))$ are not observed, making it challenging to apply standard regression methods to predict the unobserved ratings. Inspired by the classical Taylor's expansion for differentiable functions, we provide a prediction algorithm that is consistent for all Lipschitz functions. In fact, the analysis through our framework naturally leads to a variant of collaborative filtering, shedding insight into the widespread success of collaborative filtering in practice. Assuming each entry is sampled independently with probability at least $\max(m^{-1+\delta},n^{-1/2+\delta})$ with $\delta > 0$, we prove that the expected fraction of our estimates with error greater than $\epsilon$ is less than $\gamma^2 / \epsilon^2$ plus a polynomially decaying term, where $\gamma^2$ is the variance of the additive entry-wise noise term. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides principled improvements over basic collaborative filtering and is competitive with matrix factorization methods.

Neural Information Processing Systems http://nips.cc/

However, its main drawback is that it neglects possible interactions between predictor variables.

Neural Information Processing Systems http://nips.cc/

In a multivariate nonparametric regression setting we construct explicit asymptotic uniform confidence bands for centered purely random forests. Since the most popular example in this class of random forests, namely the uniformly centered purely random forests, is well known to suffer from suboptimal rates, we propose a new type of purely random forests, called the Ehrenfest centered purely random forests, which achieve minimax optimal rates. Our main confidence band theorem applies to both random forests. The proof is based on an interpretation of random forests as generalized U-Statistics together with a Gaussian approximation of the supremum of empirical processes. Our theoretical findings are illustrated in simulation examples.