In this paper, we prove directional convergence of network parameters of fixed width leaky ReLU two-layer neural networks optimized by gradient descent with exponential loss, which was previously only known for gradient flow. By a careful analysis of the convergent direction, we establish sufficient conditions of benign overfitting and discover a new phase transition in the test error bound. All of these results hold beyond the nearly orthogonal data setting which was studied in prior works. As an application, we demonstrate that benign overfitting occurs with high probability in sub-Gaussian mixture models.

This paper examines the performance of ridge regression in reproducing kernel Hilbert spaces in the presence of noise that exhibits a finite number of higher moments. We establish excess risk bounds consisting of subgaussian and polynomial terms based on the well known integral operator framework. The dominant subgaussian component allows to achieve convergence rates that have previously only been derived under subexponential noise - a prevalent assumption in related work from the last two decades. These rates are optimal under standard eigenvalue decay conditions, demonstrating the asymptotic robustness of regularized least squares against heavy-tailed noise. Our derivations are based on a Fuk-Nagaev inequality for Hilbert-space valued random variables.

Probabilistic principal component analysis (PCA) and its Bayesian variant (BPCA) are widely used for dimension reduction in machine learning and statistics. The main advantage of probabilistic PCA over the traditional formulation is allowing uncertainty quantification. The parameters of BPCA are typically learned using mean-field variational inference, and in particular, the coordinate ascent variational inference (CAVI) algorithm. So far, the convergence speed of CAVI for BPCA has not been characterized. In our paper, we fill this gap in the literature. Firstly, we prove a precise exponential convergence result in the case where the model uses a single principal component (PC). Interestingly, this result is established through a connection with the classical $\textit{power iteration algorithm}$ and it indicates that traditional PCA is retrieved as points estimates of the BPCA parameters. Secondly, we leverage recent tools to prove exponential convergence of CAVI for the model with any number of PCs, thus leading to a more general result, but one that is of a slightly different flavor. To prove the latter result, we additionally needed to introduce a novel lower bound for the symmetric Kullback--Leibler divergence between two multivariate normal distributions, which, we believe, is of independent interest in information theory.

Counterfactual explanations enhance interpretability by identifying alternative inputs that produce different outputs, offering localized insights into model decisions. However, traditional methods often neglect causal relationships, leading to unrealistic examples. While newer approaches integrate causality, they are computationally expensive. To address these challenges, we propose an efficient method called BRACE based on backtracking counterfactuals that incorporates causal reasoning to generate actionable explanations. We first examine the limitations of existing methods and then introduce our novel approach and its features. We also explore the relationship between our method and previous techniques, demonstrating that it generalizes them in specific scenarios. Finally, experiments show that our method provides deeper insights into model outputs.

LLM routing aims to select the most appropriate model for each query, balancing competing performance metrics such as accuracy and cost across a pool of language models. Prior approaches typically adopt a decoupled strategy, where the metrics are first predicted and the model is then selected based on these estimates. This setup is prone to compounding errors and often relies on full-feedback data, where each query is evaluated by all candidate models, which is costly to obtain and maintain in practice. In contrast, we learn from observational data, which records only the outcome of the model actually deployed. We propose a causal end-to-end framework that learns routing policies by minimizing decision-making regret from observational data. To enable efficient optimization, we introduce two theoretically grounded surrogate objectives: a classification-based upper bound, and a softmax-weighted regret approximation shown to recover the optimal policy at convergence. We further extend our framework to handle heterogeneous cost preferences via an interval-conditioned architecture. Experiments on public benchmarks show that our method outperforms existing baselines, achieving state-of-the-art performance across different embedding models.

Differentially private (DP) linear regression has received significant attention in the recent theoretical literature, with several works aimed at obtaining improved error rates. A common approach is to set the clipping constant much larger than the expected norm of the per-sample gradients. While simplifying the analysis, this is however in sharp contrast with what empirical evidence suggests to optimize performance. Our work bridges this gap between theory and practice: we provide sharper rates for DP stochastic gradient descent (DP-SGD) by crucially operating in a regime where clipping happens frequently. Specifically, we consider the setting where the data is multivariate Gaussian, the number of training samples $n$ is proportional to the input dimension $d$, and the algorithm guarantees constant-order zero concentrated DP. Our method relies on establishing a deterministic equivalent for the trajectory of DP-SGD in terms of a family of ordinary differential equations (ODEs). As a consequence, the risk of DP-SGD is bounded between two ODEs, with upper and lower bounds matching for isotropic data. By studying these ODEs when $n / d$ is large enough, we demonstrate the optimality of aggressive clipping, and we uncover the benefits of decaying learning rate and private noise scheduling.

Scientists often want to make predictions beyond the observed time horizon of "snapshot" data following latent stochastic dynamics. For example, in time course single-cell mRNA profiling, scientists have access to cellular transcriptional state measurements (snapshots) from different biological replicates at different time points, but they cannot access the trajectory of any one cell because measurement destroys the cell. Researchers want to forecast (e.g.) differentiation outcomes from early state measurements of stem cells. Recent Schrödinger-bridge (SB) methods are natural for interpolating between snapshots. But past SB papers have not addressed forecasting -- likely since existing methods either (1) reduce to following pre-set reference dynamics (chosen before seeing data) or (2) require the user to choose a fixed, state-independent volatility since they minimize a Kullback-Leibler divergence. Either case can lead to poor forecasting quality. In the present work, we propose a new framework, SnapMMD, that learns dynamics by directly fitting the joint distribution of both state measurements and observation time with a maximum mean discrepancy (MMD) loss. Unlike past work, our method allows us to infer unknown and state-dependent volatilities from the observed data. We show in a variety of real and synthetic experiments that our method delivers accurate forecasts. Moreover, our approach allows us to learn in the presence of incomplete state measurements and yields an $R^2$-style statistic that diagnoses fit. We also find that our method's performance at interpolation (and general velocity-field reconstruction) is at least as good as (and often better than) state-of-the-art in almost all of our experiments.

The power prior is a class of informative priors designed to incorporate historical data alongside current data in a Bayesian framework. It includes a power parameter that controls the influence of historical data, providing flexibility and adaptability. A key property of the power prior is that the resulting posterior minimizes a linear combination of KL divergences between two pseudo-posterior distributions: one ignoring historical data and the other fully incorporating it. We extend this framework by identifying the posterior distribution as the minimizer of a linear combination of Amari's $α$-divergence, a generalization of KL divergence. We show that this generalization can lead to improved performance by allowing for the data to adapt to appropriate choices of the $α$ parameter. Theoretical properties of this generalized power posterior are established, including behavior as a generalized geodesic on the Riemannian manifold of probability distributions, offering novel insights into its geometric interpretation.

The success of unsupervised learning raises the question of whether also supervised models can be trained without using the information in the output $y$. In this paper, we demonstrate that this is indeed possible. The key step is to formulate the model as a smoother, i.e. on the form $\hat{f}=Sy$, and to construct the smoother matrix $S$ independently of $y$, e.g. by training on random labels. We present a simple model selection criterion based on the distribution of the out-of-sample predictions and show that, in contrast to cross-validation, this criterion can be used also without access to $y$. We demonstrate on real and synthetic data that $y$-free trained versions of linear and kernel ridge regression, smoothing splines, and neural networks perform similarly to their standard, $y$-based, versions and, most importantly, significantly better than random guessing.

Modern biomedical technologies have transformed how we diagnose, treat, and prevent diseases in the past decade. In particular, we see surging needs in combining gene expression data with traditional clinical measurements. However, since gene expression data is usually high-dimensional, while the total number of available clinical measurements is rather limited, naive approaches such as pooling all features together will not work well. Particularly, the microarray features would dominate in inference and variable selection, resulting in negligence of valuable information from clinical data or failure of type I error control. To address the issue caused by imbalance in feature dimensions, Tibshirani & Efron (2002) proposed the pre-validation procedure to make a fairer comparison between the two sets of predictors. In the setting of gene expression data and clinical measurements, the procedure includes two steps: first we use high-dimensional gene expression data to train the leave-one-out fits of the response, then we use the fitted values and clinical data to build a final prediction model. It turns out that the pre-validation procedure not only enables us to test whether the gene expression data have predictive power with type-I error control, but also gives a good estimate of the prediction error. Figure 1 illustrates the two-stage pre-validation procedure.