Multivariate Conformal Prediction via Conformalized Gaussian Scoring

While achieving exact conditional coverage in conformal prediction is unattainable without making strong, untestable regularity assumptions, the promise of conformal prediction hinges on finding approximations to conditional guarantees that are realizable in practice. A promising direction for obtaining conditional dependence for conformal sets--in particular capturing heteroskedasticity--is through estimating the conditional density $\mathbb{P}_{Y|X}$ and conformalizing its level sets. Previous work in this vein has focused on nonconformity scores based on the empirical cumulative distribution function (CDF). Such scores are, however, computationally costly, typically requiring expensive sampling methods. To avoid the need for sampling, we observe that the CDF-based score reduces to a Mahalanobis distance in the case of Gaussian scores, yielding a closed-form expression that can be directly conformalized. Moreover, the use of a Gaussian-based score opens the door to a number of extensions of the basic conformal method; in particular, we show how to construct conformal sets with missing output values, refine conformal sets as partial information about $Y$ becomes available, and construct conformal sets on transformations of the output space. Finally, empirical results indicate that our approach produces conformal sets that more closely approximate conditional coverage in multivariate settings compared to alternative methods.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the $k$th most frequent word $\pi_k$ is proportional to $1/k$, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law $\pi_k \propto 1/k^\alpha$ parameterized by the exponent $\alpha > 0$. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the exponent $\alpha$. Existing theoretical investigations in scaling laws assume that the eigenvalues of the data decay as a power law with exponent $\alpha > 1$. This assumption effectively makes the problem ``finite dimensional'' as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case $\alpha = 1$ as found in text data is ``worst-case'' for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement for large vocabularies.

Backward Conformal Prediction

We introduce $\textit{Backward Conformal Prediction}$, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tilde{\alpha}}(X_{\rm test})) \ge 1 - \mathbb{E}[\tilde{\alpha}]$ up to a first-order Taylor approximation for any data-dependent miscoverage $\tilde{\alpha}$, and (ii) a novel leave-one-out estimator $\hat{\alpha}^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tilde{\alpha}]$ based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.

StablePCA: Learning Shared Representations across Multiple Sources via Minimax Optimization

When synthesizing multisource high-dimensional data, a key objective is to extract low-dimensional feature representations that effectively approximate the original features across different sources. Such general feature extraction facilitates the discovery of transferable knowledge, mitigates systematic biases such as batch effects, and promotes fairness. In this paper, we propose Stable Principal Component Analysis (StablePCA), a novel method for group distributionally robust learning of latent representations from high-dimensional multi-source data. A primary challenge in generalizing PCA to the multi-source regime lies in the nonconvexity of the fixed rank constraint, rendering the minimax optimization nonconvex. To address this challenge, we employ the Fantope relaxation, reformulating the problem as a convex minimax optimization, with the objective defined as the maximum loss across sources. To solve the relaxed formulation, we devise an optimistic-gradient Mirror Prox algorithm with explicit closed-form updates. Theoretically, we establish the global convergence of the Mirror Prox algorithm, with the convergence rate provided from the optimization perspective. Furthermore, we offer practical criteria to assess how closely the solution approximates the original nonconvex formulation. Through extensive numerical experiments, we demonstrate StablePCA's high accuracy and efficiency in extracting robust low-dimensional representations across various finite-sample scenarios.

Minimum Volume Conformal Sets for Multivariate Regression

Conformal prediction provides a principled framework for constructing predictive sets with finite-sample validity. While much of the focus has been on univariate response variables, existing multivariate methods either impose rigid geometric assumptions or rely on flexible but computationally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-driven framework based on a novel loss function that directly learns minimum-volume covering sets while ensuring valid coverage. This formulation naturally induces a new nonconformity score for conformal prediction, which adapts to the residual distribution and covariates. Our approach optimizes over prediction sets defined by arbitrary norm balls, including single and multi-norm formulations. Additionally, by jointly optimizing both the predictive model and predictive uncertainty, we obtain prediction sets that are tight, informative, and computationally efficient, as demonstrated in our experiments on real-world datasets.

E-Values Expand the Scope of Conformal Prediction

Conformal prediction is a powerful framework for distribution-free uncertainty quantification. The standard approach to conformal prediction relies on comparing the ranks of prediction scores: under exchangeability, the rank of a future test point cannot be too extreme relative to a calibration set. This rank-based method can be reformulated in terms of p-values. In this paper, we explore an alternative approach based on e-values, known as conformal e-prediction. E-values offer key advantages that cannot be achieved with p-values, enabling new theoretical and practical capabilities. In particular, we present three applications that leverage the unique strengths of e-values: batch anytime-valid conformal prediction, fixed-size conformal sets with data-dependent coverage, and conformal prediction under ambiguous ground truth. Overall, these examples demonstrate that e-value-based constructions provide a flexible expansion of the toolbox of conformal prediction.

Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity

Score-based generative models achieve state-of-the-art sampling performance by denoising a distribution perturbed by Gaussian noise. In this paper, we focus on a single deterministic denoising step, and compare the optimal denoiser for the quadratic loss, we name ''full-denoising'', to the alternative ''half-denoising'' introduced by Hyv{\"a}rinen (2024). We show that looking at the performances in term of distance between distribution tells a more nuanced story, with different assumptions on the data leading to very different conclusions. We prove that half-denoising is better than full-denoising for regular enough densities, while full-denoising is better for singular densities such as mixtures of Dirac measures or densities supported on a low-dimensional subspace. In the latter case, we prove that full-denoising can alleviate the curse of dimensionality under a linear manifold hypothesis.

Convergence of Shallow ReLU Networks on Weakly Interacting Data

We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on $n$ data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order $\log(n)$ neurons suffices for global convergence with high probability. Our analysis uses a Polyak-{\L}ojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of $\frac{1}{n}$. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders $\frac{1}{n}$ and $\frac{1}{\sqrt{n}}$, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order $\frac{1}{\log(n)}$.

Spectral structure learning for clinical time series

We develop and evaluate a structure learning algorithm for clinical time series. Clinical time series are multivariate time series observed in multiple patients and irregularly sampled, challenging existing structure learning algorithms. We assume that our times series are realizations of StructGP, a k-dimensional multi-output or multi-task stationary Gaussian process (GP), with independent patients sharing the same covariance function. StructGP encodes ordered conditional relations between time series, represented in a directed acyclic graph. We implement an adapted NOTEARS algorithm, which based on a differentiable definition of acyclicity, recovers the graph by solving a series of continuous optimization problems. Simulation results show that up to mean degree 3 and 20 tasks, we reach a median recall of 0.93% [IQR, 0.86, 0.97] while keeping a median precision of 0.71% [0.57-0.84], for recovering directed edges. We further show that the regularization path is key to identifying the graph. With StructGP, we proposed a model of time series dependencies, that flexibly adapt to different time series regularity, while enabling us to learn these dependencies from observations.

An Uncertainty Principle for Linear Recurrent Neural Networks

We consider linear recurrent neural networks, which have become a key building block of sequence modeling due to their ability for stable and effective long-range modeling. In this paper, we aim at characterizing this ability on a simple but core copy task, whose goal is to build a linear filter of order $S$ that approximates the filter that looks $K$ time steps in the past (which we refer to as the shift-$K$ filter), where $K$ is larger than $S$. Using classical signal models and quadratic cost, we fully characterize the problem by providing lower bounds of approximation, as well as explicit filters that achieve this lower bound up to constants. The optimal performance highlights an uncertainty principle: the optimal filter has to average values around the $K$-th time step in the past with a range~(width) that is proportional to $K/S$.