This paper studies P-trainable neural networks, where a proportion P of hidden neurons are trainable and the rest stay fixed as random features, bridging random feature methods and fully trainable networks. The authors establish two-layer approximation bounds showing that a small trainable fraction can capture most of the gain when initialization mismatch is large, yielding a phase transition in rates of the form $\mathcal{O}(((P/\kappa_1^*+(1-P)/\kappa_0^*)m/\log m)^{-\alpha})$. They verify these assumptions explicitly for spherical biased-ReLU features and present numerical experiments showing analogous phase transitions in training losses across architectures with up to 15 layers.

AI systems increasingly make consequential decisions, but fully auditing every output is infeasible at scale. This paper studies selective auditing: sequentially deciding when to trust an AI decision versus escalate it to an expert, using only a scalar confidence score. The authors frame this as an online constrained optimization problem and derive algorithms that guarantee the fraction of unaudited errors stays below a user-specified threshold α without any distributional assumptions. The core contribution is an optimally efficient algorithm that provably minimizes audit effort for both adversarial and i.i.d. data streams.

This paper asks whether transformers that generate intermediate reasoning steps can discover new algorithms beyond the teacher they are trained to imitate. It designs a two-stage training protocol for in-context linear regression—first supervised by gradient descent, then by ground-truth weights—and proves that a one-layer linear self-attention transformer learns the Heavy-Ball method. The result matters because it gives a rigorous case where chain-of-thought reasoning internalizes acceleration rather than simply memorizing the supervised update.

Transformer attention is commonly viewed as an interacting particle system, yet the mean-field limit on probability densities over $\mathbb{R}^d$ lacks rigorous geometric foundations on non-compact domains. This paper derives accelerated attention blocks from Nesterov-type damped Hamiltonian dynamics on spaces of probability measures, introducing a local-mobility geometry for softmax attention and a quotient Stein geometry for linear attention. The theory yields sharp PDE well-posedness and finite-time blow-up results, plus quantitative links to token-norm growth across discrete layers. Small-scale language-modeling experiments report minor validation-loss improvements at increased computational cost.

High-dimensional nonparametric density estimation is notoriously plagued by the curse of dimensionality. This paper proposes to factorize the joint distribution autoregressively into a product of one marginal and $d-1$ conditional densities, estimating each factor with parallel gradient boosting (PGB), a fast tree-based multi-quantile regression method. The authors prove that for subadditive divergences the joint convergence rate is governed by the hardest factor, and that when conditional estimators adapt to sparsity the optimal rate relates to the treewidth of an underlying dependence graph. Together, the theory and experiments suggest that flexible autoregressive factorization offers a practical off-the-shelf alternative to kernels, copulas, and random forests for moderate-to-high dimensional continuous data.

Post-hoc calibration usually adjusts confidence scores globally, which ignores that neural networks can be miscalibrated differently in different regions of feature space. This paper proposes Local Adaptive Probability Estimation (LAPE), a residual framework that keeps an existing global calibrator as a stable anchor and adds a non-parametric local correction drawn from neighboring calibration samples in the learned feature representation. It also extends this idea to multi-view inputs through Cascade Feature Augmentation Fusion (CFAF). By exploiting local geometry without retraining, the work targets high-stakes settings where uniform global corrections leave spatially varying errors untouched.

Standard computer-model calibration focuses on parameter posteriors, not the predictive accuracy of the calibrated simulator. The authors propose a two-layer framework: a flow-matching model amortizes posterior inference, then conformal risk control tunes parameter sets so the simulator pushforward hits a target miscoverage on synthetic calibration data. This targets the quantity that actually matters downstream—predictive coverage—while exploiting amortization to avoid expensive per-observation MCMC. Experiments on synthetic and real epidemiological data show roughly 5× sharper bands than KOH–MCMC with near-nominal coverage.

Conformal prediction for non-Euclidean responses is underdeveloped because conventional scores require residuals or densities that assume a vector-space structure. This paper introduces the conditional expected local radius, $R_\beta(\omega|x)=\frac{1}{\beta}\int_0^\beta Q_x(\omega,\tau)\,d\tau$, as a nonconformity measure in general metric spaces, and estimates the conditional distance profile, the CELR, and its conditional distribution from a single metric distributional random forest. The result is a computationally cheaper alternative to the conditional profile score method of Zhou and Müller (2025) with uniform consistency and asymptotic conditional validity guarantees.

SABLE tackles expensive optimisation in the high-dimensional latent spaces of pretrained generative models — a regime where latent dimension $D$ can reach $40{,}500$ while the evaluation budget $N$ is orders of magnitude smaller. The core idea is to use a Bayesian affine surrogate on the characteristic-radius sphere, decoupling acquisition into a data-sized in-span component and a scalar off-span norm that is prescribed by a finite-horizon schedule rather than left to expected improvement. Together with a fast recompression scheme that bounds model-refitting cost to $\mathcal{O}(M^{2}D+M^{3})$, the method achieves tens-of-millisecond decisions and strong sample efficiency on synthetic and scientific benchmarks.

This paper studies whether width and depth limits commute not just at initialization, but during training, for linear residual networks with $1/\sqrt{L}$ branch scaling and $\mu$P width scaling. Its core result is that under clipped gradient descent, the trained output converges in $\mathbb{L}_2$ as $\min\{n,L\}\to\infty$ for any fixed number of steps, proving the order of limits does not matter. Without clipping, commutativity holds only in probability, because the second moment can diverge with depth even after a single gradient step. The finding matters because it rigorously characterizes joint scaling limits in a tractable deep-learning model and isolates gradient clipping as a key enabler of stable infinite-depth training.

Global AUC can hide regions where a classifier discriminates poorly, which is dangerous in high-stakes clinical risk stratification. This paper introduces LOCATOR, a method that computes a localized AUC ($\mathrm{LAUC}$) for each participant by weighting all event–non-event pairs by Gower-distance similarity, then trains an XGBoost regressor on $\logit(\mathrm{LAUC})$ from baseline covariates to flag high- and low-competence regions without subsetting the data. The authors derive asymptotic properties of the $\mathrm{LAUC}$ estimator and apply the framework to the guideline-endorsed PREVENT model in NHANES, with external validation in UK Biobank.

Classical analyses of the Unadjusted Langevin Algorithm (ULA) in Wasserstein-2 distance predict a dimension-dependent mixing time and bias, yet empirically ULA often estimates low-dimensional statistical functionals accurately without scaling the step size with dimension. This paper proposes using the subspace robust Wasserstein (SRW) distance to analyze ULA on ridge-separable Bayesian posteriors, proving that iteration complexity can be dimension-free in both the overparameterized regime ($N \lesssim d$) and the large-sample regime ($N \gtrsim d^2$), while showing such guarantees are impossible near the proportional regime $N \asymp d$. The results explain a bell-shaped bias profile and provide a rotationally invariant alternative to prior $\ell^\infty$-based dimension-free analyses that fail under isotropy.

Next-event prediction in sequential data is usually handled by opaque neural networks, while existing symbolic rule methods lack a principled way to form a unified predictive model. This paper introduces Rules, a neuro-symbolic network that learns Temporal Conjunctive Prediction Rules via differentiable temperature annealing, and Runes, which adds a residual transformer to correct rule-based logit predictions. It matters because it offers an explicit accuracy-interpretability trade-off, producing compact, inspectable symbolic rules for safety-sensitive domains while remaining competitive with black-box transformers.

Random projections (RP) offer a powerful way to reduce the $O(n^3)$ cost of kernel spectral algorithms, but existing theory has largely been confined to well-specified settings or narrow mis-specified regimes under restrictive boundedness assumptions. This paper establishes high-probability minimax optimal learning rates in Sobolev norms for RP-based spectral algorithms under general source conditions and a Bernstein noise condition, without assuming uniform boundedness of the regression function $f_\rho$. By exploiting the RKHS embedding property and a refined operator similarity analysis, the authors enlarge the optimal source range from $s \geq 1 - 1/\beta$ to $s > \alpha_0 - 1/\beta$, showing that projection dimensions proportional to the effective dimension preserve statistical efficiency while yielding substantial computational savings.

xVGAE tackles graph generation for moderate-sized, potentially dense graphs where diffusion and autoregressive methods become prohibitively slow. It builds on the Aldous-Hoover representation theorem to design a hierarchical VAE with a global graph latent $z_0$ and node latents $z_i$ that are conditionally i.i.d., using a learnable neural graphon decoder for one-shot generation. The core pitch is that this structure yields sample quality competitive with state-of-the-art while sampling one to three orders of magnitude faster, and produces interpretable latent representations.

Standard on-policy distillation forces a student to copy a teacher at every token, even when the teacher's preferred actions hurt the student's eventual success. This paper proposes Advantage-Based On-Policy Distillation (AOPD), which weights each teacher token by its counterfactual advantage under the current student policy. The key insight is that imitation should optimize for student task reward, not teacher agreement, and the authors derive an unbiased estimator that needs only standard student rollouts and terminal rewards.

Standard deep Gaussian processes become increasingly ill-conditioned as depth grows because discrete layer composition amplifies parameter perturbations. This paper proposes replacing the discrete stack with a latent continuous-depth SDE whose drift family carries a GP-based prior, yielding two models: CDGP with a direct separable GP prior, and FlowDGP with a flow-evolved base GP. The central claim is that depth in probabilistic regression is better modeled as continuous latent evolution than as a discrete composition, and that retaining a GP prior over the drift improves stability and competitiveness relative to both discrete DGPs and neural continuous-depth alternatives.

AI agents that act as statistical analysts evolve through model upgrades, prompt edits, and new tools, but each change risks inflating Type I error. This paper introduces continual validation: before any update is deployed, it is frozen and tested on newly simulated null data using exact binomial confidence bounds with a decaying per-update risk budget. The result is a lifetime guarantee that no invalid version passes, which matters because classical fixed-procedure guarantees break under adaptive, open-ended evolution.

Goal-conditioned RL in complex mazes suffers from a geometric mismatch: Euclidean embedding distances model straight-line proximity, yet walls can make actual route distances highly tree-like and deceptive. This paper introduces Hyperbolic CRL, a drop-in replacement of the critic's Euclidean distance with a hyperbolic geodesic distance on the hyperboloid. Its central finding is that this geometric alignment allows a depth-64 hyperbolic network to outperform a 1024-layer Euclidean baseline on hard humanoid maze tasks while training more than $12\times$ faster and using roughly $13\times$ fewer parameters.

LLM workflows rely on fast binary decisions like moderation and reranking, but autoregressive generation is too slow for tight latency budgets. This paper studies System One models—lightweight scorers that emit bounded decision scores—and proposes Learn Then Test (LTT) to certify thresholds that maximize the fraction of inputs receiving the utility-maximizing action while controlling the False Omission Rate (FOR) with a finite-sample guarantee. The authors show across six datasets that naive thresholding at the target risk level exceeds the budget in roughly half of settings, whereas LTT violates it in only 7.8% of cases, making a strong case that a probability-like interface is insufficient for safe deployment under asymmetric risk.