The trustworthiness of AI decision-making systems is increasingly important. A key feature of such systems is the ability to provide recommendations for how an individual may reverse a negative decision, a problem known as algorithmic recourse. Existing approaches treat recourse outcomes as counterfactuals of a fixed unit, ignoring that real-world recourse involves repeated decisions on the same individual under possibly different latent conditions. We develop a causal framework that models recourse as a process over pre- and post-intervention outcomes, allowing for partial stability and resampling of latent variables. We introduce post-recourse stability conditions that enable reasoning about recourse from observational data alone, and develop a copula-based algorithm for inferring the effects of recourse under these conditions. For settings where paired observations of the same individual before and after intervention are available (called recourse data), we develop methods for inferring copula parameters and performing goodness-of-fit testing. When the copula model is rejected, we provide a distribution-free algorithm for learning recourse effects directly from recourse data. We demonstrate the value of the proposed methods on real and semi-synthetic datasets.

We present the Spatial Adapter, a parameter-efficient post-hoc layer that equips any frozen first-stage predictor with a structured spatial representation of its residual field and an induced closed-form spatial covariance. The adapter operates as a cascade second stage on residuals, jointly learning a spatially regularized orthonormal basis and per-sample scores via a tractable mini-batch ADMM procedure, without modifying any first-stage parameter. Because the first-stage parameters are frozen, the adapter does not retrain the backbone; its role is to supply a compressed distributional summary of the residual field. Smoothness, sparsity, and orthogonality together turn a generic low-rank factorization into an identifiable spatial representation whose induced residual covariance admits a closed-form low-rank-plus-noise estimator; the effective rank is determined data-adaptively by spectral thresholding, while the nominal rank K is an optimization-side upper bound only. This covariance enables kriging-style spatial prediction at unobserved locations, with plug-in uncertainty quantification as a secondary downstream use. Across synthetic data, Weather2K for spatial-holdout prediction, and GWHD patch grids as a basis-transferability diagnostic, the adapter recovers residual spatial structure when paired with frozen first stages from linear models to deep spatiotemporal and vision backbones; the added representation uses fewer than K(N+T) parameters alongside a compact residual-trend network.

Soft Actor-Critic (SAC) and its variants dominate Multi-Task Reinforcement Learning (MTRL) due to their off-policy sample efficiency, while on-policy methods such as Proximal Policy Optimization (PPO) remain underexplored. We diagnose that PPO in MTRL suffers from a previously overlooked issue: critic-side gradient ill-conditioning, which may cause tail tasks to stall while easy tasks dominate the value function's updates. To address this, we propose TOPPO (Tail-Optimized PPO), a reformulation of PPO via Critic Balancing -- a set of modules that improve gradient conditioning and balance learning dynamics across tasks. Unlike prior approaches that rely on modular architectures or large models, TOPPO targets the optimization bottleneck within PPO itself. Empirically, TOPPO achieves stronger mean and tail-task performance than published SAC-family and ARS-family baselines while using substantially fewer parameters and environment steps on Meta-World+ benchmark. Notably, TOPPO matches or surpasses strong SAC baselines early in training and maintains superior performance at full budget. Ablations confirm the effectiveness of each module in TOPPO and provide insights into their interactions. Our results demonstrate that, with proper optimization, on-policy methods can rival or exceed off-policy approaches in MTRL, challenging the prevailing reliance on SAC and highlighting critic-side gradient conditioning as the central bottleneck.

We study bilevel optimization with a fixed polyhedral lower feasible set. Such problems are challenging for two reasons: active-set changes can make the upper objective nonsmooth, and existing hypergradient methods typically require lower-Hessian inversions or equivalent linear solves, which are computationally expensive. To address these issues, we adopt a logarithmic barrier smoothing of the lower problem to obtain a differentiable approximation of the constrained bilevel objective, and develop a proxy-gradient algorithm for the resulting barrier-smoothed surrogate. The algorithm uses only gradients of the upper and lower objectives; its only second-order object is the explicit logarithmic barrier Hessian determined by the fixed polyhedral constraints. Barrier smoothing restores differentiability, but Euclidean smoothness constants are not uniformly bounded near the boundary. We therefore develop a local Dikin-geometry analysis in which the barrier-metric provides an oracle-free curvature scale near the moving lower centers. This leads to barrier-aware schedules that keep the iterates inside locally well-behaved regions. For the barrier-smoothed objective, we prove stationarity rates of $\widetilde{O}(K^{-2/3})$ in the deterministic setting and $\widetilde{O}(K^{-2/5})$ under upper-level-only bounded stochastic noise after $K$ outer iterations, together with quantitative bias control as the barrier parameter decreases.

Long-context inference is increasingly a memory-traffic problem. The culprit is the key--value (KV) cache: it grows with context length, batch size, layers, and heads, and it is read at every decoding step. Rotation-based scalar codecs meet this systems constraint by storing a norm, applying a shared random rotation, and quantizing one coordinate at a time. They are universal and random-access, but they discard the geometry created by the normalization step. After a Haar rotation, a block of $k$ consecutive coordinates is not a product source; it is a spherical-Beta source on the unit ball. We introduce \textsc{FibQuant}, a universal fixed-rate vector quantizer that keeps the same normalize--rotate--store interface while replacing scalar tables by a shared radial--angular codebook matched to this canonical source. The codebook combines Beta-quantile radii, Fibonacci\,/\,Roberts--Kronecker quasi-uniform directions, and multi-restart Lloyd--Max refinement. We prove that the resulting vector code strictly improves on its scalar product specialization at matched rate, with a high-rate gain that separates into a cell-shaping factor and a density-matching factor. The same construction gives a dense rate axis, including fractional-bit and sub-one-bit operating points, without calibration or variable-length addresses. On GPT-2 small KV caches, \textsc{FibQuant} traces a memory--fidelity frontier from $5\times$ compression at $0.99$ attention cosine similarity to $34\times$ at $0.95$. End-to-end on TinyLlama-1.1B, it is within $0.10$ perplexity of fp16 at $4\times$ compression and has $3.6\times$ lower perplexity than scalar \textsc{TurboQuant} at $b = 2$ ($8\times$ compression), where scalar random-access quantization begins to fail.

The validity of statistical inference depends critically on how data are collected. When data gathered through active data collection (ADC) are reused for a post-hoc inferential task, conventional inference can fail because the sampling is adaptively biased toward regions favored by the collection strategy. This issue is especially pronounced in black-box optimization, where sequential model-based optimization (SMBO) methods such as the tree-structured Parzen estimator (TPE) and Gaussian process upper confidence bound (GP-UCB) preferentially concentrate evaluations in promising regions. We study statistical inference on actively collected data when the inferential target is constructed in a data-dependent manner after data collection. To enable valid inference in this setting, we propose post-ADC inference, a framework that accounts for the biases arising from both the active data collection process and the subsequent data-driven target construction. Our method builds on selective inference and provides valid $p$-values and confidence intervals that correct for both sources of bias. The framework applies to a broad class of ADC processes by imposing only assumptions on the observation noise, without requiring any assumptions on the underlying black-box function or the surrogate model used by the SMBO algorithm. Empirical results also show that post-ADC inference provides valid inference for data collected by GP-UCB and TPE.

Activation functions play a central role in neural networks by shaping internal representations. Recently, learning binary activation representations has attracted significant attention due to their advantages in computational and memory efficiency, as well as interpretability. However, training neural networks with Heaviside activations remains challenging, as their non-differentiability obstructs standard gradient-based optimization. In this paper, we propose Heavy Tailed Activation Function (HTAF), a smooth approximation to the Heaviside function that enables stable training with gradient-based optimization. We construct HTAF as a sigmoid hyperbolic tangent composite function and theoretically show that it maintains a large gradient mass around zero inputs while exhibiting slower gradient decay in the tail regions. We show that Spiking Neural Networks, Binary Neural Networks and Deep Heaviside neural Networks can be trained stably using HTAF with gradient-based optimization. Finally, we introduce Implicit Concept Bottleneck Models (ICBMs), an interpretable image model that leverages HTAF to induce discrete feature representations. Extensive experiments across various architectures and image datasets demonstrate that ICBM enables stable discretization while achieving prediction performance comparable to or better than standard models.

Probabilistic partial least squares (PPLS) is a central likelihood-based model for two-view learning when one needs both interpretable latent factors and calibrated uncertainty. Building on the identifiable parameterization of Bouhaddani et al.\ (2018), existing fitting pipelines still face two practical bottlenecks: noise--signal coupling under joint EM/ECM updates and nontrivial handling of orthogonality constraints. Following the fixed-noise scalar-likelihood line of Hu et al.\ (2025), we develop an end-to-end framework that combines noise pre-estimation, constrained likelihood optimization, and prediction calibration in one pipeline. Relative to Hu et al.\ (2025), we replace full-spectrum noise averaging with noise-subspace estimation and replace interior-point penalty handling with exact Stiefel-manifold optimization. The noise-subspace estimator attains a signal-strength-independent leading finite-sample rate and matches a minimax lower bound, while the full-spectrum estimator is shown to be inconsistent under the same model. We further extend the framework to sub-Gaussian settings via optional Gaussianization and provide closed-form standard errors through a block-structured Fisher analysis. Across synthetic high-noise settings and two multi-omics benchmarks (TCGA-BRCA and PBMC CITE-seq), the method achieves near-nominal coverage without post-hoc recalibration, reaches Ridge-level point accuracy on TCGA-BRCA at rank $r=3$, matches or exceeds PO2PLS on cross-view prediction while providing native calibrated uncertainty, and improves stability of parameter recovery.

$U$-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for $U$-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for $U$-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting $U$-statistic, which is designed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to $U$-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.