A recent line of research investigates how strategic agents may respond to such scoring tools to receive favorable assessments. While prior work has focused on the short-term strategic interactions between a decision-making institution (modeled as a principal) and individual decision-subjects (modeled as agents), we investigate interactions spanning multiple time-steps . In particular, we consider settings in which the agent's effort investment

We identify mechanisms through which normalizing flows can simultaneously increase likelihood for all structured images.

We then use these approximate subclass labels as a form of noisy supervision in a distributionally robust optimization objective.

This paper introduces a novel kernel-based formulation of the Equalized Odds (EO) criterion, denoted as $EO_k$, for fair representation learning (FRL) in supervised settings. The central goal of FRL is to mitigate discrimination regarding a sensitive attribute $S$ while preserving prediction accuracy for the target variable $Y$. Our proposed criterion enables a rigorous and interpretable quantification of three core fairness objectives: independence (prediction $\hat{Y}$ is independent of $S$), separation (also known as equalized odds; prediction $\hat{Y}$ is independent with $S$ conditioned on target attribute $Y$), and calibration ($Y$ is independent of $S$ conditioned on the prediction $\hat{Y}$). Under both unbiased ($Y$ is independent of $S$) and biased ($Y$ depends on $S$) conditions, we show that $EO_k$ satisfies both independence and separation in the former, and uniquely preserves predictive accuracy while lower bounding independence and calibration in the latter, thereby offering a unified analytical characterization of the tradeoffs among these fairness criteria. We further define the empirical counterpart, $\hat{EO}_k$, a kernel-based statistic that can be computed in quadratic time, with linear-time approximations also available. A concentration inequality for $\hat{EO}_k$ is derived, providing performance guarantees and error bounds, which serve as practical certificates of fairness compliance. While our focus is on theoretical development, the results lay essential groundwork for principled and provably fair algorithmic design in future empirical studies.

Standard gradient descent methods yield point estimates with no measure of confidence. This limitation is acute in overparameterized and low-data regimes, where models have many parameters relative to available data and can easily overfit. Bootstrapping is a classical statistical framework for uncertainty estimation based on resampling, but naively applying it to deep learning is impractical: it requires training many replicas, produces post-hoc estimates that cannot guide learning, and implicitly assumes comparable optima across runs - an assumption that fails in non-convex landscapes. We introduce Twin-Bootstrap Gradient Descent (Twin-Boot), a resampling-based training procedure that integrates uncertainty estimation into optimization. Two identical models are trained in parallel on independent bootstrap samples, and a periodic mean-reset keeps both trajectories in the same basin so that their divergence reflects local (within-basin) uncertainty. During training, we use this estimate to sample weights in an adaptive, data-driven way, providing regularization that favors flatter solutions. In deep neural networks and complex high-dimensional inverse problems, the approach improves calibration and generalization and yields interpretable uncertainty maps.

We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\%). We verify that these counterexamples remain valid over $\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\mathbb{R}$. Remarkably, our analysis reveals that matrix density (number of ones) is highly predictive of expressibility, achieving 95.7\% classification accuracy. Using modern machine learning techniques, we discover that expressible matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space, despite the naive parameter count of 24 (12 parameters each for two $4 \times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.

With the advent of wearable recorders, scientists are increasingly turning to automated methods of analysis of audio and video data in order to measure children's experience, behavior, and outcomes, with a sizable literature employing long-form audio-recordings to study language acquisition. While numerous articles report on the accuracy and reliability of the most popular automated classifiers, less has been written on the downstream effects of classification errors on measurements and statistical inferences (e.g., the estimate of correlations and effect sizes in regressions). This paper proposes a Bayesian approach to study the effects of algorithmic errors on key scientific questions, including the effect of siblings on children's language experience and the association between children's production and their input. In both the most commonly used \gls{lena}, and an open-source alternative (the Voice Type Classifier from the ACLEW system), we find that classification errors can significantly distort estimates. For instance, automated annotations underestimated the negative effect of siblings on adult input by 20--80\%, potentially placing it below statistical significance thresholds. We further show that a Bayesian calibration approach for recovering unbiased estimates of effect sizes can be effective and insightful, but does not provide a fool-proof solution. Both the issue reported and our solution may apply to any classifier involving event detection and classification with non-zero error rates.

Contrast-enhanced spectral mammography (CESM) is an imaging modality that provides two types of images, commonly known as low-energy (LE) and dual-energy subtracted (DES) images. In many domains, particularly in medicine, the emergence of image-to-image translation techniques has enabled the artificial generation of images using other images as input. Within CESM, applying such techniques to generate DES images from LE images could be highly beneficial, potentially reducing patient exposure to radiation associated with high-energy image acquisition. In this study, we investigated three models for the artificial generation of DES images (virtual DES): a pre-trained U-Net model, a U-Net trained end-to-end model, and a CycleGAN model. We also performed a series of experiments to assess the impact of using virtual DES images on the classification of CESM examinations into malignant and non-malignant categories. To our knowledge, this is the first study to evaluate the impact of virtual DES images on CESM lesion classification. The results demonstrate that the best performance was achieved with the pre-trained U-Net model, yielding an F1 score of 85.59% when using the virtual DES images, compared to 90.35% with the real DES images. This discrepancy likely results from the additional diagnostic information in real DES images, which contributes to a higher classification accuracy. Nevertheless, the potential for virtual DES image generation is considerable and future advancements may narrow this performance gap to a level where exclusive reliance on virtual DES images becomes clinically viable.

Continual learning is motivated by the need to adapt to real-world dynamics in tasks and data distribution while mitigating catastrophic forgetting. Despite significant advances in continual learning techniques, the theoretical understanding of their generalization performance lags behind. This paper examines the theoretical properties of continual ridge regression in high-dimensional linear models, where the dimension is proportional to the sample size in each task. Using random matrix theory, we derive exact expressions of the asymptotic prediction risk, thereby enabling the characterization of three evaluation metrics of generalization performance in continual learning: average risk, backward transfer, and forward transfer. Furthermore, we present the theoretical risk curves to illustrate the trends in these evaluation metrics throughout the continual learning process. Our analysis reveals several intriguing phenomena in the risk curves, demonstrating how model specifications influence the generalization performance. Simulation studies are conducted to validate our theoretical findings.