Making calibrated online predictions is a central challenge in modern AI systems. Much of the existing literature focuses on fully adversarial environments where outcomes may be arbitrary, leading to conservative algorithms that can perform suboptimally in more benign settings, such as when outcomes are nearly stationary. This gap raises a natural question: can we design online prediction algorithms whose calibration error automatically adapts to the degree of non-stationarity in the environment, smoothly interpolating between i.i.d. and adversarial regimes? We answer this question in the affirmative and develop a suite of algorithms that achieve adaptive calibration guarantees under multiple calibration measures. Specifically, with $T$ being the number of rounds, $K$ being the unknown number of i.i.d. segments of the environment, and $C\in[0,T]$ being another unknown non-stationary measure defined as the minimal $\ell_1$ deviation of the mean outcomes, our algorithms attain $\widetilde{O}(\min\{\sqrt{T}+(TC)^{\frac{1}{3}}, \sqrt{KT}\})$ for $\ell_1$ calibration error and $\widetilde{O}(\min\{(1+C)^{\frac{1}{3}}, K\})$ for both $\ell_2$ and pseudo KL calibration error. These bounds match the optimal rates in the stationary case ($C=0$ and $K=1$) and recover known guarantees in the fully adversarial regime ($C, K=Ω(T)$). Our approach builds on and extends prior work [Hu et al., 2026, Luo et al., 2025], introducing an epoch-based scheduling together with a novel non-uniform partition of the prediction space that allocates finer resolution near the underlying ground truth.

We study feature learning in two-layer neural networks within the linear-width regime, where the number of hidden neurons, sample size, and input dimension scale proportionally. While recent work has analyzed feature learning via a single step of gradient descent on the first layer weights in this regime, such one-step update schemes are fundamentally limited: the update to the weights is approximately rank-one, captures only a single direction, and requires the target function to have an information exponent of one. In this paper, we go beyond one-step updates to provide a full characterization of the features learned during the \textit{second step} of gradient descent with step-sizes $η_1\asymp N^{α_1}$ and $η_2 \asymp N^{α_2}$ for $α_1, α_2 \in [0,0.5)$, where $N$ is the number of hidden neurons. We derive a spectral characterization of the updated weights, demonstrating they behave as a spiked random matrix with multiple outliers, each corresponding to a learned direction. We show that the number of the outliers is determined by the parameters $α_1, α_2$ through $\lfloor \frac{α_2}{1/2 - α_1} \rfloor$. Furthermore, by analyzing the alignment between the learned directions and the target function, we identify a gap between training with independent versus reused batches. While independent batches restrict learning to directions with an information exponent of one, batch reuse enables the second update to capture directions even when the information exponent exceeds one, provided that $α_1, α_2$ are chosen properly. This shows that the benefits of batch reuse, previously observed in narrow-width regimes, persist in the linear-width limit as well. By characterizing these early-phase evolutions, our work proposes a tractable framework for studying optimization and feature learning phenomenology in modern overparameterized networks.

We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko--Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, $\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3$. Here, $D_1$ measures the population quantile movement induced by perturbing the projection direction, $D_2$ measures the empirical quantile fluctuation with the projection direction held fixed, and $D_3$ is the Bahadur-type remainder.

In the era of precision medicine, genome-wide epigenetic modifications offer rich data that could inform risk prediction. However, these data are high-dimensional and exhibit complex dependence structures, which makes it difficult to jointly model them with low-dimensional covariates when the goal is to obtain interpretable effect estimates for covariate adjustment. Standard Bayesian additive regression trees (BART) provide strong predictive performance but treat all predictors uniformly within the tree ensemble, obscuring the contributions of significant covariates and complicating variable selection in high-dimensional settings. We propose a semi-parametric BART model (spBART) that addresses this limitation by modeling low-dimensional covariates through a parametric component with interpretable coefficients, while capturing complex nonlinear associations among high-dimensional predictors through the tree ensemble. To perform stable variable selection, we develop a cross-validation-based procedure that aggregates posterior inclusion probabilities across folds and applies Bayesian false discovery rate control. We apply the proposed method to a pooled case--control analysis of high-dimensional genome-wide 5-hydroxymethylcytosine profiles derived from circulating cell-free DNA in two multiple myeloma studies ($N = 869$). The approach identifies a parsimonious set of candidate loci and achieves strong out-of-sample discrimination (AUC $= 0.96$) in a held-out validation set. Overall, spBART provides a unified framework for combining interpretable covariate inference with flexible modeling and variable selection in high-dimensional biomedical studies.

Discrete probability laws underpin statistical modeling, yet the catalog of interpretable distributions has expanded only gradually through centuries of case-by-case mathematical derivations. We introduce symbolic density estimation (SDE), an unsupervised framework that automatically recovers closed-form probability mass functions by composing elementary analytic operations within a structured search space. Our method integrates domain-specific structural priors with evolutionary search and a validity-aware inference stage, and it extends to richer distribution families such as zero inflation and finite mixtures. To support systematic evaluation and future research, we contribute a benchmark dataset spanning a broad collection of commonly used discrete distributions. The proposed algorithm recovers all benchmark families with accurate parameter estimates. A real data application shows that it identifies concise and interpretable mixture models that improve goodness-of-fit over standard models.

Ensembles of neural networks typically outperform individual networks but incur large computational costs, whereas weight aggregation produces less costly, yet also less accurate, aggregate models. We introduce partial fusion of networks, which interpolates between ensembles and weight aggregation and thus allows for a flexible tradeoff between computational cost and performance. A direct way to achieve this is to extend existing weight aggregation methods based on neuron-level similarity between different networks, where partial fusion then only aggregates weights of neurons which are most similar. We showcase one particular method to jointly identify which neurons are most similar and match them via partial optimal transport. Further, we consider the more general perspective of weight aggregation and partial fusion as generalized pruning of ensemble models, where neurons cannot just be deleted, but also linearly combined. Finally, we show that generalized pruning applied to a single network yields similar benefits as partial fusion by allowing for a tradeoff between isolating, deleting, and linearly combining neurons based on similarity. Our code is available at https://github.com/Fabian-Mor/partial_fusion_nn.

Shapley and Banzhaf interactions capture the complex dynamics inherent in modern machine learning applications. However, current estimators for these higher-order interactions trade off between speed and accuracy. To overcome this limitation, we introduce ProxySHAP. ProxySHAP reconciles the high sample efficiency of tree-based proxy models with a principled path to consistency via residual correction. On a theoretical level, we derive a polynomial-time generalization of interventional TreeSHAP to compute exact interaction indices for tree ensembles, successfully bypassing exponential tree-depth dependencies in prior methods. Furthermore, we formally analyze the residual adjustment strategy, characterizing the specific conditions under which Maximum Sample Reuse (MSR) corrects proxy bias without its variance scaling exponentially with interaction size. Extensive benchmarking demonstrates that ProxySHAP sets a new state-of-the-art standard for approximation quality, including in large-scale applications with thousands of features. By achieving the lowest error in both small- and large-budget regimes, ProxySHAP significantly outperforms the prior best estimators ProxySPEX and KernelSHAP-IQ, while also delivering superior performance on downstream explainability tasks.

Machine unlearning is a fundamental mechanism that enforces the right to be forgotten. Existing unlearning studies that rely on label manipulation or task-gradient reversal often deliver limited unlearning effectiveness. Moreover, they can undermine the original learning objective and typically do not guarantee equivalence to standard unlearning by retraining. In this paper, we propose \textbf{ManiF-SMC} (\textbf{Mani}fold \textbf{F}orgetting with \textbf{S}elf \textbf{M}ode \textbf{C}onnectivity), motivated by the observation that a model retrained on the remaining data tends to classify erased samples by their semantic similarity to the retained data. We begin with systematically recasting the approximate unlearning as pushing each erased sample away from its original learned manifold representation centroid toward its nearest semantic neighbors in the retained data. This reformulation aligns unlearning with retraining behavior and operates purely in representation space, reducing reliance on labels and task-specific gradients. To tackle the manifold representation-based unlearning problem, ManiF-SMC encapsulates the unlearning and representation preservation goals in a margin-based triplet loss. Because finding a suitable margin for unlearning is challenging, we propose a self-mode-connectivity module that rapidly reconstructs the local manifold to guide the adaptive margins generation for each unlearning case. Extensive experiments on four representative datasets show that ManiF-SMC achieves unlearning effectiveness comparable to state-of-the-art approximate methods while operating solely within the model's representation space.

Deep learning is increasingly viewed as a dynamical process in parameter space, yet many existing theories still treat training as a closed optimization system. This view is limited for real-world AI, where models operate under uncertainty, resource constraints, distribution shift, downstream decision risks, and human feedback. We propose Human-Centered Learning Mechanics (HCLM), a dynamical and information-theoretic framework for open and controlled learning systems. The central idea is that entropy regularization is useful only when the chosen entropy surrogate generates a non-degenerate information force along the optimization trajectory. Otherwise, entropy terms may produce weak, unstable, or misaligned gradients, causing the dynamics to collapse toward ordinary loss minimization. We introduce the notion of effective entropy and study tractable geometric entropy surrogates, including variance-based and log-determinant covariance proxies. The paper makes three contributions. First, it formalizes entropy regularization through effective information force and characterizes degenerate entropy regimes. Second, it derives convergence, entropy-flow, Wasserstein-gradient-flow, and noisy-representation generalization results under explicit assumptions. Third, it offers a conditional dynamical interpretation of scaling-law-like behavior as a balance between information injection, entropy dissipation, and residual risk, without claiming an unconditional derivation of empirical neural scaling laws. Controlled representation-learning experiments support the hypothesis that geometric entropy surrogates, especially log-determinant covariance entropy, induce stronger and more stable information forces than softmax-normalized entropy.

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.