Despite recent progress, continual learning still does not match the performance of batch training. To avoid catastrophic forgetting, we need to build compact memory of essential past knowledge, but no clear solution has yet emerged, even for shallow neural networks with just one or two layers. In this paper, we present a new method to build compact memory for logistic regression. Our method is based on a result by Khan and Swaroop [2021] who show the existence of optimal memory for such models. We formulate the search for the optimal memory as Hessian-matching and propose a probabilistic PCA method to estimate them. Our approach can drastically improve accuracy compared to Experience Replay. For instance, on Split-ImageNet, we get 60% accuracy compared to 30% obtained by replay with memory-size equivalent to 0.3% of the data size. Increasing the memory size to 2% further boosts the accuracy to 74%, closing the gap to the batch accuracy of 77.6% on this task. Our work opens a new direction for building compact memory that can also be useful in the future for continual deep learning.

Contextual predictability shapes both the form and choice of words in online language production. The effects of the predictability of a word given its previous context are generally well-understood in both production and comprehension, but studies of naturalistic production have also revealed a poorly-understood backward predictability effect of a word given its future context, which may be related to future planning. Here, in two studies of naturalistic speech corpora, we investigate backward predictability effects using improved measures and more powerful language models, introducing a new principled and conceptually motivated information-theoretic predictability measure that integrates predictability from both the future and the past context. Our first study revisits classic predictability effects on word duration. Our second study investigates substitution errors within a generative framework that independently models the effects of lexical, contextual, and communicative factors on word choice, while predicting the actual words that surface as speech errors. We find that our proposed conceptually-motivated alternative to backward predictability yields qualitatively similar effects across both studies. Through a fine-grained analysis of substitution errors, we further show that different kinds of errors are suggestive of how speakers prioritize form, meaning, and context-based information during lexical planning. Together, these findings illuminate the functional roles of past and future context in how speakers encode and choose words, offering a bridge between contextual predictability effects and the mechanisms of sentence planning.

Theoretically exploring the advantages of neural networks might be one of the most challenging problems in the AI era. An adaptive feature program has recently been proposed to analyze the feature learning characteristic property of neural networks in a more abstract way. Motivated by the celebrated Le Cam equivalence, we advocate the over-parametrized sequence models to further simplify the analysis of the training dynamics of adaptive feature program and present several supporting evidences for the adaptive feature program. More precisely, after having introduced the feature error measure (FEM) to characterize the quality of the learned feature, we show that the FEM is decreasing during the training process of several concrete adaptive feature models including linear regression, single/multiple index models, etc. We believe that this hints at the potential successes of the adaptive feature program.

Collecting high-quality labeled data remains a challenge in data-driven research, especially when each label is costly and time-consuming to obtain. In response, many fields have embraced machine learning as a practical solution for predicting unobserved labels, such as annotating satellite imagery in remote sensing [46] and predicting protein structures in proteomics [24]. Prediction-powered inference [1] is a methodological framework showing how to perform valid statistical inference despite the inherent biases in such predicted labels. Active statistical inference [51] was recently introduced to further enhance inference by actively selecting which data points to label. The basic idea is to compute the model's uncertainty scores for all data points and prioritize collecting those labels for which the predictive model is most uncertain. When the uncertainty scores appropriately reflect the model's errors, Zrnic and Cand` es [51] show that active inference can significantly outperform prediction-powered inference (which can essentially be thought of as active inference with naive uniform sampling), meaning it results in more accurate estimates and narrower confidence intervals. However, when uncertainty scores are of poor quality, active inference can result in overly noisy estimates and large confidence intervals.

We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $λ$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

We study an online linear regression setting in which the observed feature vectors are corrupted by noise and the learner can pay to reduce the noise level. In practice, this may happen for several reasons: for example, because features can be measured more accurately using more expensive equipment, or because data providers can be incentivized to release less private features. Assuming feature vectors are drawn i.i.d. from a fixed but unknown distribution, we measure the learner's regret against the linear predictor minimizing a notion of loss that combines the prediction error and payment. When the mapping between payments and noise covariance is known, we prove that the rate $\sqrt{T}$ is optimal for regret if logarithmic factors are ignored. When the noise covariance is unknown, we show that the optimal regret rate becomes of order $T^{2/3}$ (ignoring log factors). Our analysis leverages matrix martingale concentration, showing that the empirical loss uniformly converges to the expected one for all payments and linear predictors.

The high accuracy of large-scale weather forecasting models like Aurora is often accompanied by a lack of transparency, as their internal representations remain largely opaque. This "black box" nature hinders their adoption in high-stakes operational settings. In this work, we probe the physical consistency of Aurora's encoder by investigating whether its latent representations align with known physical and meteorological concepts. Using a large-scale dataset of embeddings, we train linear classifiers to identify three distinct concepts: the fundamental land-sea boundary, high-impact extreme temperature events, and atmospheric instability. Our findings provide quantitative evidence that Aurora learns physically consistent features, while also highlighting its limitations in capturing the rarest events. This work underscores the critical need for interpretability methods to validate and build trust in the next generation of Al-driven weather models.

Quantification has been proven to be a particularly difficult linguistic phenomenon for (Multimodal) Large Language Models (MLLMs). However, given that quantification interfaces with the logic, pragmatic, and numerical domains, the exact reasons for the poor performance are still unclear. This papers looks at three key features of human quantification shared cross-linguistically that have remained so far unexplored in the (M)LLM literature: the ordering of quantifiers into scales, the ranges of use and prototypicality, and the biases inherent in the human approximate number system. The aim is to determine how these features are encoded in the models' architecture, how they may differ from humans, and whether the results are affected by the type of model and language under investigation. We find that there are clear differences between humans and MLLMs with respect to these features across various tasks that tap into the representation of quantification in vivo vs. in silico. This work, thus, paves the way for addressing the nature of MLLMs as semantic and pragmatic agents, while the cross-linguistic lens can elucidate whether their abilities are robust and stable across different languages.

Accurate, detailed, and regularly updated bathymetry, coupled with complex semantic content, is essential for under-mapped shallow-water environments facing increasing climatological and anthropogenic pressures. However, existing approaches that derive either depth or seabed classes from remote sensing imagery treat these tasks in isolation, forfeiting the mutual benefits of their interaction and hindering the broader adoption of deep learning methods. To address these limitations, we introduce Seabed-Net, a unified multi-task framework that simultaneously predicts bathymetry and pixel-based seabed classification from remote sensing imagery of various resolutions. Seabed-Net employs dual-branch encoders for bathymetry estimation and pixel-based seabed classification, integrates cross-task features via an Attention Feature Fusion module and a windowed Swin-Transformer fusion block, and balances objectives through dynamic task uncertainty weighting. In extensive evaluations at two heterogeneous coastal sites, it consistently outperforms traditional empirical models and traditional machine learning regression methods, achieving up to 75\% lower RMSE. It also reduces bathymetric RMSE by 10-30\% compared to state-of-the-art single-task and multi-task baselines and improves seabed classification accuracy up to 8\%. Qualitative analyses further demonstrate enhanced spatial consistency, sharper habitat boundaries, and corrected depth biases in low-contrast regions. These results confirm that jointly modeling depth with both substrate and seabed habitats yields synergistic gains, offering a robust, open solution for integrated shallow-water mapping. Code and pretrained weights are available at https://github.com/pagraf/Seabed-Net.

This study investigates treatment effect estimation in the semi-supervised setting, where we can use not only the standard triple of covariates, treatment indicator, and outcome, but also unlabeled auxiliary covariates. For this problem, we develop efficiency bounds and efficient estimators whose asymptotic variance aligns with the efficiency bound. In the analysis, we introduce two different data-generating processes: the one-sample setting and the two-sample setting. The one-sample setting considers the case where we can observe treatment indicators and outcomes for a part of the dataset, which is also called the censoring setting. In contrast, the two-sample setting considers two independent datasets with labeled and unlabeled data, which is also called the case-control setting or the stratified setting. In both settings, we find that by incorporating auxiliary covariates, we can lower the efficiency bound and obtain an estimator with an asymptotic variance smaller than that without such auxiliary covariates.