Bivariate partial information decomposition (PID) has emerged as a promising tool for analyzing interactions in complex systems, particularly in neuroscience. PID achieves this by decomposing the information that two sources (e.g., different

The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.

We study parametric change-point detection, where the goal is to identify distributional changes in time series, under local differential privacy. In the non-private setting, we derive improved finite-sample accuracy guarantees for a change-point detection algorithm based on the generalized log-likelihood ratio test, via martingale methods. In the private setting, we propose two locally differentially private algorithms based on randomized response and binary mechanisms, and analyze their theoretical performance. We derive bounds on detection accuracy and validate our results through empirical evaluation. Our results characterize the statistical cost of local differential privacy in change-point detection and show how privacy degrades performance relative to a non-private benchmark. As part of this analysis, we establish a structural result for strong data processing inequalities (SDPI), proving that SDPI coefficients for Rényi divergences and their symmetric variants (Jeffreys-Rényi divergences) are achieved by binary input distributions. These results on SDPI coefficients are also of independent interest, with applications to statistical estimation, data compression, and Markov chain mixing.

We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter $α\in (0,1)$. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter $γ_t$ and the posterior contraction rate $ε_t$, and identify conditions on the Gaussian process prior under which $ε_t$ can be controlled. As special cases of our general bound, we recover regret of order $\tilde{\mathcal{O}}(T^{\frac{1}{2}})$ for the squared exponential kernel, $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}} )$ for the Matérn-$ν$ kernel, and a bound of order $\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}})$ for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

In the era of large language models (LLMs), fine-tuning pretrained models has become ubiquitous. Yet the theoretical underpinning remains an open question. A central question is why only a few epochs of fine-tuning are typically sufficient to achieve strong performance on many different tasks. In this work, we approach this question by developing a statistical framework, combining rigorous early stopping theory with the attention-based Neural Tangent Kernel (NTK) for LLMs, offering new theoretical insights on fine-tuning practices. Specifically, we formally extend classical NTK theory [Jacot et al., 2018] to non-random (i.e., pretrained) initializations and provide a convergence guarantee for attention-based fine-tuning. One key insight provided by the theory is that the convergence rate with respect to sample size is closely linked to the eigenvalue decay rate of the empirical kernel matrix induced by the NTK. We also demonstrate how the framework can be used to explain task vectors for multiple tasks in LLMs. Finally, experiments with modern language models on real-world datasets provide empirical evidence supporting our theoretical insights.

A key challenge in probabilistic regression is ensuring that predictive distributions accurately reflect true empirical uncertainty. Minimizing overall prediction error often encourages models to prioritize informativeness over calibration, producing narrow but overconfident predictions. However, in safety-critical settings, trustworthy uncertainty estimates are often more valuable than narrow intervals. Realizing the problem, several recent works have focused on post-hoc corrections; however, existing methods either rely on weak notions of calibration (such as PIT uniformity) or impose restrictive parametric assumptions on the nature of the error. To address these limitations, we propose a novel nonparametric re-calibration algorithm based on conditional kernel mean embeddings, capable of correcting calibration error without restrictive modeling assumptions. For efficient inference with real-valued targets, we introduce a novel characteristic kernel over distributions that can be evaluated in $\mathcal{O}(n \log n)$ time for empirical distributions of size $n$. We demonstrate that our method consistently outperforms prior re-calibration approaches across a diverse set of regression benchmarks and model classes.

The hazard ratio from the Cox proportional hazards model is a ubiquitous summary of treatment effect. However, when hazards are non-proportional, the hazard ratio can lose a stable causal interpretation and become study-dependent because it effectively averages time-varying effects with weights determined by follow-up and censoring. We consider the average hazard (AH) as an alternative causal estimand: a population-level person-time event rate that remains well-defined and interpretable without assuming proportional hazards. Although AH can be estimated nonparametrically and regression-style adjustments have been proposed, existing approaches do not provide a general framework for flexible, high-dimensional nuisance estimation with valid sqrt{n} inference. We address this gap by developing a semiparametric, doubly robust framework for covariate-adjusted AH. We establish pathwise differentiability of AH in the nonparametric model, derive its efficient influence function, and construct cross-fitted, debiased estimators that leverage machine learning for nuisance estimation while retaining asymptotically normal, sqrt{n}-consistent inference under mild product-rate conditions. Simulations demonstrate that the proposed estimator achieves small bias and near-nominal confidence-interval coverage across proportional and non-proportional hazards settings, including crossing-hazards regimes where Cox-based summaries can be unstable. We illustrate practical utility in comparative effectiveness research by comparing immunotherapy regimens for advanced melanoma using SEER-Medicare linked data.