Xinwei Shen

Xinwei Shen, Nicolai Meinshausen

Distributional regression aims to estimate the full conditional distribution of a target variable, given covariates. Popular methods include linear and tree-ensemble based quantile regression. We propose a neural network-based distributional regression methodology called `engression'. An engression model is generative in the sense that we can sample from the fitted conditional distribution and is also suitable for high-dimensional outcomes. Furthermore, we find that modelling the conditional distribution on training data can constrain the fitted function outside of the training support, which offers a new perspective to the challenging extrapolation problem in nonlinear regression. In particular, for `pre-additive noise' models, where noise is added to the covariates before applying a nonlinear transformation, we show that engression can successfully perform extrapolation under some assumptions such as monotonicity, whereas traditional regression approaches such as least-squares or quantile regression fall short under the same assumptions. Our empirical results, from both simulated and real data, validate the effectiveness of the engression method and indicate that the pre-additive noise model is typically suitable for many real-world scenarios. The software implementations of engression are available in both R and Python.

Xinwei Shen, Peter Bühlmann, Armeen Taeb

Since distribution shifts are common in real-world applications, there is a pressing need to develop prediction models that are robust against such shifts. Existing frameworks, such as empirical risk minimization or distributionally robust optimization, either lack generalizability for unseen distributions or rely on postulated distance measures. Alternatively, causality offers a data-driven and structural perspective to robust predictions. However, the assumptions necessary for causal inference can be overly stringent, and the robustness offered by such causal models often lacks flexibility. In this paper, we focus on causality-oriented robustness and propose Distributional Robustness via Invariant Gradients (DRIG), a method that exploits general noise interventions in training data for robust predictions against unseen interventions, and naturally interpolates between in-distribution prediction and causality. In a linear setting, we prove that DRIG yields predictions that are robust among a data-dependent class of distribution shifts. Furthermore, we show that our framework includes anchor regression as a special case, and that it yields prediction models that protect against more diverse perturbations. We establish finite-sample results and extend our approach to semi-supervised domain adaptation to further improve prediction performance. Finally, we empirically validate our methods on synthetic simulations and on single-cell and intensive health care datasets.

Mars Liyao Gao, Yuxuan Bao, Amy S. Rude et al.

Reconstructing high-dimensional spatiotemporal fields from sparse sensor measurements is critical in a wide range of scientific applications. The SHallow REcurrent Decoder (SHRED) architecture is a recent state-of-the-art architecture that reconstructs high-quality spatial domain from hyper-sparse sensor measurement streams. An important limitation of SHRED is that in complex, data-scarce, high-frequency, or stochastic systems, portions of the spatiotemporal field must be modeled with valid uncertainty estimation. We introduce UQ-SHRED, a distributional learning framework for sparse sensing problems that provides uncertainty quantification through a neural network-based distributional regression called engression. UQ-SHRED models the uncertainty by learning the predictive distribution of the spatial state conditioned on the sensor history. By injecting stochastic noise into sensor inputs and training with an energy score loss, UQ-SHRED produces predictive distributions with minimal computational overhead, requiring only noise injection at the input and resampling through a single architecture without retraining or additional network structures. On complicated synthetic and real-life datasets including turbulent flow, atmospheric dynamics, neuroscience and astrophysics, UQ-SHRED provides a distributional approximation with well-calibrated confidence intervals. We further conduct ablation studies to understand how each model setting affects the quality of the UQ-SHRED performance, and its validity on uncertainty quantification over a set of different experimental setups.

Xinwei Shen, Kani Chen, Tong Zhang

Generative Adversarial Networks (GANs) have achieved great success in data generation. However, its statistical properties are not fully understood. In this paper, we consider the statistical behavior of the general $f$-divergence formulation of GAN, which includes the Kullback--Leibler divergence that is closely related to the maximum likelihood principle. We show that for parametric generative models that are correctly specified, all $f$-divergence GANs with the same discriminator classes are asymptotically equivalent under suitable regularity conditions. Moreover, with an appropriately chosen local discriminator, they become equivalent to the maximum likelihood estimate asymptotically. For generative models that are misspecified, GANs with different $f$-divergences {converge to different estimators}, and thus cannot be directly compared. However, it is shown that for some commonly used $f$-divergences, the original $f$-GAN is not optimal in that one can achieve a smaller asymptotic variance when the discriminator training in the original $f$-GAN formulation is replaced by logistic regression. The resulting estimation method is referred to as Adversarial Gradient Estimation (AGE). Empirical studies are provided to support the theory and to demonstrate the advantage of AGE over the original $f$-GANs under model misspecification.

Zetai Cen, Jin Zhu, Xinwei Shen et al.

We introduce a simple yet powerful framework for training large language models. In contrast to the standard autoregressive next-token prediction based on an exact prefix, we propose a perturbation-based procedure that first transforms the prefix into a semantic neighbor and then conditions on this perturbed variant for next-token prediction. This yields a hierarchical model with a pre-post-additive noise structure. Within this framework, we develop a rigorous theory of extrapolability, namely, the capacity of a model class to make reliable predictions for token sequences that lie outside the empirical support of the training corpus. We evaluate the finite-sample performance of the proposed procedure using both synthetic and real-world language data. Results show that the proposed method consistently improves out-of-support prediction while maintaining competitive in-support performance, demonstrating that perturbation offers a practical route to language modeling.

Yifan Hao, Yanxin Lu, Hanning Zhang et al.

As overparameterized models become increasingly prevalent, training loss alone offers limited insight into generalization performance. While smoothness has been linked to improved generalization across various settings, directly enforcing smoothness in neural networks remains challenging. To address this, we introduce Distributional Input Projection Networks (DIPNet), a novel framework that projects inputs into learnable distributions at each layer. This distributional representation induces a smoother loss landscape with respect to the input, promoting better generalization. We provide theoretical analysis showing that DIPNet reduces both local smoothness measures and the Lipschitz constant of the network, contributing to improved generalization performance. Empirically, we validate DIPNet across a wide range of architectures and tasks, including Vision Transformers (ViTs), Large Language Models (LLMs), ResNet and MLPs. Our method consistently enhances test performance under standard settings, adversarial attacks, out-of-distribution inputs, and reasoning benchmarks. We demonstrate that the proposed input projection strategy can be seamlessly integrated into existing models, providing a general and effective approach for boosting generalization performance in modern deep learning.

Mengyue Yang, Furui Liu, Zhitang Chen et al.

Learning disentanglement aims at finding a low dimensional representation which consists of multiple explanatory and generative factors of the observational data. The framework of variational autoencoder (VAE) is commonly used to disentangle independent factors from observations. However, in real scenarios, factors with semantics are not necessarily independent. Instead, there might be an underlying causal structure which renders these factors dependent. We thus propose a new VAE based framework named CausalVAE, which includes a Causal Layer to transform independent exogenous factors into causal endogenous ones that correspond to causally related concepts in data. We further analyze the model identifiabitily, showing that the proposed model learned from observations recovers the true one up to a certain degree by providing supervision signals (e.g. feature labels). Experiments are conducted on various datasets, including synthetic and real word benchmark CelebA. Results show that the causal representations learned by CausalVAE are semantically interpretable, and their causal relationship as a Directed Acyclic Graph (DAG) is identified with good accuracy. Furthermore, we demonstrate that the proposed CausalVAE model is able to generate counterfactual data through "do-operation" to the causal factors.

Xinwei Shen, Tong Zhang, Kani Chen

This paper considers the general $f$-divergence formulation of bidirectional generative modeling, which includes VAE and BiGAN as special cases. We present a new optimization method for this formulation, where the gradient is computed using an adversarially learned discriminator. In our framework, we show that different divergences induce similar algorithms in terms of gradient evaluation, except with different scaling. Therefore this paper gives a general recipe for a class of principled $f$-divergence based generative modeling methods. Theoretical justifications and extensive empirical studies are provided to demonstrate the advantage of our approach over existing methods.