We connect stochastic resetting from non-equilibrium statistical physics with ridge regularization in statistical learning. For linear gradient flow, resetting to the origin at rate $r$ produces stationary mean $(X^\top X+rI)^{-1}X^\top y$, exactly the ridge estimator with penalty $λ=r$. This uses the known Laplace-transform relationship between ridge regression and exponential-time averaging of gradient flow, with the exponential time now interpreted as the stationary age associated with Poisson resetting. We then extend this identity to general renewal reset laws: the exponential reset time distribution is the unique renewal law whose stationary mean reproduces scalar ridge in every eigendirection as an exact filter identity for every positive curvature, while non-exponential renewal laws generate alternative spectral filters. At the fluctuation level, we study a separate additive Ornstein-Uhlenbeck extension with constant diffusion, interpreted as a stylized SGD approximation. In this setting, the equality holds only at the level of the mean, since the reset process has a nonzero stationary covariance from accumulated OU noise and reset-timing variance, whereas deterministic ridge is a fixed estimator with the same center. Stylized experiments compare the deterministic renewal-induced filters directly and illustrate when filters induced by non-exponential reset-time laws can differ predictively from ridge. The results for the stationary mean and the induced spectral filters are established for continuous-time gradient flow with isotropic resetting on quadratic objectives; the covariance and risk formulas additionally assume additive noise with state-independent covariance.

Conformal prediction methods enjoy strong theoretical and empirical predictive inference performance, provided the data is exchangeable, and predictors are trained in a memoryless fashion. However, these assumptions and constraints are impractical in many real-data settings, such as time series (where temporal dependence violates exchangeability, and where memoryless predictors will inevitably have poor predictive accuracy). Recent work shows that the split conformal prediction method is robust to these issues of memory-based predictors and deviations from exchangeability that are common features of time-series data. However, since using sample splitting can lead to lower accuracy, this motivates asking whether other predictive inference methods (that do not rely on data splitting) could also be reliably used in the time series setting. In this work, we show that the vanilla leave-one-out jackknife can suffer an arbitrary loss of coverage even in canonical time series models with mild temporal dependence. As a remedy, we propose a careful modification tailored to such settings, which we term the \emph{leave-a-window-out} (LWO) method, and show that it can achieve valid coverage provided that the model-fitting procedure satisfies mild stability properties. Our proofs are based on quantifying the degree to which the data departs from \emph{cyclic exchangeability}, and we introduce new coefficients to measure the extent of this departure. Experiments on time series data demonstrate that our LWO method often enjoys valid coverage when the vanilla jackknife fails to cover, while producing much narrower intervals than split conformal prediction.

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

Offline Reinforcement Learning from Human Feedback (RLHF) pipelines such as Direct Preference Optimization (DPO) train on a pre-collected preference dataset, which makes them vulnerable to preference poisoning attack. We study label flip attacks against log-linear DPO. We first illustrate that flipping one preference label induces a parameter-independent shift in the DPO gradient. Using this key property, we can then convert the targeted poisoning problem into a structured binary sparse approximation problem. To solve this problem, we develop two attack methods: Binary-Aware Lattice Attack (BAL-A) and Binary Matching Pursuit Attack (BMP-A). BAL-A embeds the binary flip selection problem into a binary-aware lattice and applies Lenstra-Lenstra-Lovász reduction and Babai's nearest plane algorithm; we provide sufficient conditions that enforce binary coefficients and recover the minimum-flip objective. BMP-A adapts binary matching pursuit to our non-normalized gradient dictionary and yields coherence-based recovery guarantees and robustness (impossibility) certificates for $K$-flip budgets. Experiments on synthetic dictionaries and the Stanford Human Preferences dataset validate the theory and highlight how dictionary geometry governs attack success.

Stock clustering algorithms play a pivotal role in quantitative finance and the asset management industry, serving as a core mechanism for understanding market complexity and conducting asset preselection. Their intrinsic value lies in enabling investors to identify the true underlying structure of the stock market, thereby categorizing stocks with similar return characteristics or risk profiles into distinct groups. This data-driven market segmentation not only significantly reduces the computational dimensionality involved in portfolio construction but also provides a solid foundation for formulating differentiated investment strategies. A review of existing literature reveals that scholars both domestic and international have achieved fruitful results in stock clustering. Traditional clustering research predominantly employs classic machine learning algorithms: Xiaojun (2019) and Wu et al. (2022) utilized the K-means algorithm for stock partitioning; Huang et al. (2010) and Lu et al. (2020) explored the sectoral structures of the SSE 50 Index and other markets based on Agglomerative Hierarchical Clustering (AHC) and Spectral Clustering; Korzeniewski (2018) further introduced the Partitioning Around Medoids (PAM) algorithm to construct portfolios with enhanced risk resistance. In recent years, with the advancement of deep learning, L ucio and Caiado (2022) and Siregar and Yosia (2024) have attempted to incorporate time-series models (such as TGARCH) or specific market features (e.g., Indonesian stock data) into clustering frameworks. However, despite their respective merits in capturing market trends, these methods share a common limitation: traditional stock clustering approaches predominantly rely exclusively on stock-specific information (e.g., price, volatility, or financial metrics), neglecting the heterogeneity of market participants--namely, the "investors". In reality, investors are typically categorized into three distinct types based on their risk preferences: risk-averse, risk-seeking, and risk-neutral. Divergent risk attitudes inevitably lead to fundamentally different asset selection logic.

Reward modeling is not only a prediction problem: in KL-regularized policy optimization, the learned reward is exponentiated to define the deployed policy, so downstream value depends on errors in reward-tilted regions. We study this feedback in a Gaussian single-index model with $r^*(x) = σ^*(\langle θ^*, x\rangle)$ and $x \sim N(0, I_d)$. We analyze a two-stage neural reward model that first learns the hidden direction $θ^*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression. Exponential reward weighting changes the Hermite signal available to the first layer; for any feature-learning temperature $β_1$ above a dimension-free $O(1)$ threshold, a constant fraction of neurons recover the hidden direction, with weak-recovery complexity governed by the generative exponent. After feature recovery, we derive tilted-policy value-gap bounds for an idealized label-weighted fit with weights $e^{y/β_2}$ and a more practical surrogate-weighted fit with weights $e^{r_{a_0}(x)/β_2}$. Keeping the $β_2$-dependence explicit yields an admissible set of deployment temperatures, balancing the gain from lowering $β_2$ against the learning cost amplified by exponential weighting; in the surrogate-weighted case, proxy-dependent factors shrink this admissible set.

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability and characterize both discrete and continuous sources of residual uncertainty. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

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.

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions using learnable univariate edge functions, typically parameterized by B-spline bases. Although effective, spline-based implementations can be computationally expensive. A modified version of KANs, called FastKAN, improves efficiency by replacing splines with Gaussian radial basis functions (RBFs), but it relies on a fixed kernel and shape parameter. In this work, we extend the RBF-based KAN framework by introducing a broader family of radial basis kernels and by initializing the kernel shape parameter using leave-one-out cross-validation (LOOCV). To the best of our knowledge, this is the first study that integrates LOOCV-based kernel scale estimation with deep KAN training. We also introduce Matérn and Wendland kernels into the KAN framework for the first time, enabling more flexible basis representations beyond the Gaussian kernel used in FastKAN. The LOOCV estimate provides a data-driven initialization of the kernel scale, which is subsequently refined during network training. The proposed adaptive RBF-KAN is evaluated on several two-dimensional benchmark functions. The results highlight the importance of kernel selection and adaptive shape parameters, with different kernels showing advantages for smooth functions, discontinuities, and oscillatory patterns. Overall, combining LOOCV-based initialization with adaptive kernel learning provides a practical strategy for improving RBF-based KAN models.

GPUs have significantly accelerated first-order methods for large-scale optimization, especially in continuous optimization. However, this success has not transferred cleanly to problems with discrete variables, combinatorial structure, and nonlinear objectives, such as certifying optimal solutions for cardinality-constrained generalized linear models. Major challenges include the sequential processing of heterogeneous nodes in branch and bound (BnB) and frequent data movement between the CPU and GPU. We propose a simple, generic, and modular CPU--GPU framework that processes multiple BnB nodes in batches on GPUs. The framework is built around a small set of GPU-efficient routines and uses padding together with lightweight custom kernels to handle irregular node data structures. Experiments show one to two orders of magnitude speedups and zero optimality gap on challenging instances. The framework can also be extended to collect the entire Rashomon set, enabling downstream statistical analysis such as variable-importance analysis and model selection under secondary user-specific measures (e.g., AUC in classification).