Fast Rates for Nonstationary Weighted Risk Minimization

Weighted empirical risk minimization is a common approach to prediction under distribution drift. This article studies its out-of-sample prediction error under nonstationarity. We provide a general decomposition of the excess risk into a learning term and an error term associated with distribution drift, and prove oracle inequalities for the learning error under mixing conditions. The learning bound holds uniformly over arbitrary weight classes and accounts for the effective sample size induced by the weight vector, the complexity of the weight and hypothesis classes, and potential data dependence. We illustrate the applicability and sharpness of our results in (auto-) regression problems with linear models, basis approximations, and neural networks, recovering minimax-optimal rates (up to logarithmic factors) when specialized to unweighted and stationary settings.

Hinge Regression Tree: A Newton Method for Oblique Regression Tree Splitting

Oblique decision trees combine the transparency of trees with the power of multivariate decision boundaries, but learning high-quality oblique splits is NP-hard, and practical methods still rely on slow search or theory-free heuristics. We present the Hinge Regression Tree (HRT), which reframes each split as a non-linear least-squares problem over two linear predictors whose max/min envelope induces ReLU-like expressive power. The resulting alternating fitting procedure is exactly equivalent to a damped Newton (Gauss-Newton) method within fixed partitions. We analyze this node-level optimization and, for a backtracking line-search variant, prove that the local objective decreases monotonically and converges; in practice, both fixed and adaptive damping yield fast, stable convergence and can be combined with optional ridge regularization. We further prove that HRT's model class is a universal approximator with an explicit $O(δ^2)$ approximation rate, and show on synthetic and real-world benchmarks that it matches or outperforms single-tree baselines with more compact structures.

Weighted Sum-of-Trees Model for Clustered Data

Clustered data, which arise when observations are nested within groups, are incredibly common in clinical, education, and social science research. Traditionally, a linear mixed model, which includes random effects to account for within-group correlation, would be used to model the observed data and make new predictions on unseen data. Some work has been done to extend the mixed model approach beyond linear regression into more complex and non-parametric models, such as decision trees and random forests. However, existing methods are limited to using the global fixed effects for prediction on data from out-of-sample groups, effectively assuming that all clusters share a common outcome model. We propose a lightweight sum-of-trees model in which we learn a decision tree for each sample group. We combine the predictions from these trees using weights so that out-of-sample group predictions are more closely aligned with the most similar groups in the training data. This strategy also allows for inference on the similarity across groups in the outcome prediction model, as the unique tree structures and variable importances for each group can be directly compared. We show our model outperforms traditional decision trees and random forests in a variety of simulation settings. Finally, we showcase our method on real-world data from the sarcoma cohort of The Cancer Genome Atlas, where patient samples are grouped by sarcoma subtype.

Improving the Linearized Laplace Approximation via Quadratic Approximations

Deep neural networks (DNNs) often produce overconfident out-of-distribution predictions, motivating Bayesian uncertainty quantification. The Linearized Laplace Approximation (LLA) achieves this by linearizing the DNN and applying Laplace inference to the resulting model. Importantly, the linear model is also used for prediction. We argue this linearization in the posterior may degrade fidelity to the true Laplace approximation. To alleviate this problem, without increasing significantly the computational cost, we propose the Quadratic Laplace Approximation (QLA). QLA approximates each second order factor in the approximate Laplace log-posterior using a rank-one factor obtained via efficient power iterations. QLA is expected to yield a posterior precision closer to that of the full Laplace without forming the full Hessian, which is typically intractable. For prediction, QLA also uses the linearized model. Empirically, QLA yields modest yet consistent uncertainty estimation improvements over LLA on five regression datasets.

Anytime Pretraining: Horizon-Free Learning-Rate Schedules with Weight Averaging

Large language models are increasingly trained in continual or open-ended settings, where the total training horizon is not known in advance. Despite this, most existing pretraining recipes are not anytime: they rely on horizon-dependent learning rate schedules and extensive tuning under a fixed compute budget. In this work, we provide a theoretical analysis demonstrating the existence of anytime learning schedules for overparameterized linear regression, and we highlight the central role of weight averaging - also known as model merging - in achieving the minimax convergence rates of stochastic gradient descent. We show that these anytime schedules polynomially decay with time, with the decay rate determined by the source and capacity conditions of the problem. Empirically, we evaluate 150M and 300M parameter language models trained at 1-32x Chinchilla scale, comparing constant learning rates with weight averaging and $1/\sqrt{t}$ schedules with weight averaging against a well-tuned cosine schedule. Across the full training range, the anytime schedules achieve comparable final loss to cosine decay. Taken together, our results suggest that weight averaging combined with simple, horizon-free step sizes offers a practical and effective anytime alternative to cosine learning rate schedules for large language model pretraining.

Online Vector Quantized Attention

Standard sequence mixing layers used in language models struggle to balance efficiency and performance. Self-attention performs well on long context tasks but has expensive quadratic compute and linear memory costs, while linear attention and SSMs use only linear compute and constant memory but struggle with long context processing. In this paper, we develop a sequence mixing layer that aims to find a better compromise between memory-compute costs and long-context processing, which we call online vector-quantized (OVQ) attention. OVQ-attention requires linear compute costs and constant memory, but, unlike linear attention and SSMs, it uses a sparse memory update that allows it to greatly increase the size of its memory state and, consequently, memory capacity. We develop a theoretical basis for OVQ-attention based on Gaussian mixture regression, and we test it on a variety of synthetic long context tasks and on long context language modeling. OVQ-attention shows significant improvements over linear attention baselines and the original VQ-attention, on which OVQ-attention was inspired. It demonstrates competitive, and sometimes identical, performance to strong self-attention baselines up 64k sequence length, despite using a small fraction of the memory of full self-attention.

Simulation-Based Inference via Regression Projection and Batched Discrepancies

We analyze a lightweight simulation-based inference method that infers simulator parameters using only a regression-based projection of the observed data. After fitting a surrogate linear regression once, the procedure simulates small batches at the proposed parameter values and assigns kernel weights based on the resulting batch-residual discrepancy, producing a self-normalized pseudo-posterior that is simple, parallelizable, and requires access only to the fitted regression coefficients rather than raw observations. We formalize the construction as an importance-sampling approximation to a population target that averages over simulator randomness, prove consistency as the number of parameter draws grows, and establish stability in estimating the surrogate regression from finite samples. We then characterize the asymptotic concentration as the batch size increases and the bandwidth shrinks, showing that the pseudo-posterior concentrates on an identified set determined by the chosen projection, thereby clarifying when the method yields point versus set identification. Experiments on a tractable nonlinear model and on a cosmological calibration task using the DREAMS simulation suite illustrate the computational advantages of regression-based projections and the identifiability limitations arising from low-information summaries.

Mixture of Concept Bottleneck Experts

Concept Bottleneck Models (CBMs) promote interpretability by grounding predictions in human-understandable concepts. However, existing CBMs typically fix their task predictor to a single linear or Boolean expression, limiting both predictive accuracy and adaptability to diverse user needs. We propose Mixture of Concept Bottleneck Experts (M-CBEs), a framework that generalizes existing CBMs along two dimensions: the number of experts and the functional form of each expert, exposing an underexplored region of the design space. We investigate this region by instantiating two novel models: Linear M-CBE, which learns a finite set of linear expressions, and Symbolic M-CBE, which leverages symbolic regression to discover expert functions from data under user-specified operator vocabularies. Empirical evaluation demonstrates that varying the mixture size and functional form provides a robust framework for navigating the accuracy-interpretability trade-off, adapting to different user and task needs.

Handling Covariate Mismatch in Federated Linear Prediction

Federated learning enables institutions to train predictive models collaboratively without sharing raw data, addressing privacy and regulatory constraints. In the standard horizontal setting, clients hold disjoint cohorts of individuals and collaborate to learn a shared predictor. Most existing methods, however, assume that all clients measure the same features. We study the more realistic setting of covariate mismatch, where each client observes a different subset of features, which typically arises in multicenter collaborations with no prior agreement on data collection. We formalize learning a linear prediction under client-wise MCAR patterns and develop two modular approaches tailored to the dimensional regime and communication budget. In the low-dimensional setting, we propose a plug-in estimator that approximates the oracle linear predictor by aggregating sufficient statistics to estimate the covariance and cross-moment terms. In higher dimensions, we study an impute-then-regress strategy: (i) impute missing covariates using any exchangeability-preserving imputation procedure, and (ii) fit a ridge-regularized linear model on the completed data. We provide asymptotic and finite-sample learning rates for our predictors, explicitly characterizing their behaviour with the global dimension, the client-specific feature partition, and the distribution of samples across sites.

Data Distribution as a Lever for Guiding Optimizers Toward Superior Generalization in LLMs

Can modifying the training data distribution guide optimizers toward solutions with improved generalization when training large language models (LLMs)? In this work, we theoretically analyze an in-context linear regression model with multi-head linear self-attention, and compare the training dynamics of two gradient based optimizers, namely gradient descent (GD) and sharpness-aware minimization (SAM), the latter exhibiting superior generalization properties but is prohibitively expensive for training even medium-sized LLMs. We show, for the first time, that SAM induces a lower simplicity bias (SB)-the tendency of an optimizer to preferentially learn simpler features earlier in training-and identify this reduction as a key factor underlying its improved generalization performance. Motivated by this insight, we demonstrate that altering the training data distribution by upsampling or augmenting examples learned later in training similarly reduces SB and leads to improved generalization. Our extensive experiments show that our strategy improves the performance of multiple LLMs-including Phi2-2.7B , Llama3.2-1B, Gemma3-1B-PT, and Qwen3-0.6B-Base-achieving relative accuracy gains up to 18% when fine-tuned with AdamW and Muon on mathematical reasoning tasks.