Papers

We study the symmetric and antisymmetric parts of bilinear forms in the attention heads of trained large language models. We introduce an orthogonally invariant profile map from real bilinear forms to a three-dimensional simplex and observe that profiles of trained bilinear forms accumulate near profiles of rank-one bilinear forms. We prove that the symmetric part of a bilinear form in an attention head is the sum of a hyperbolic form and a zero form for a Zariski-dense subset of query-key matrices.

We develop an index policy for finite-horizon Bernoulli multi-armed bandits from minimax solutions to single-arm bandit (SAB) problems. Each SAB problem involves choosing between an unknown Bernoulli arm and a known reward. We show that minimizing worst-case regret of SAB problems over all non-anticipative policies admits an exact semi-infinite linear programming formulation. The resulting stopping policies offer a natural way to compare arms: the higher the known reward against which a policy continues sampling, the more promising the unknown arm. We turn this intuition into indices based on cumulative continuation probabilities, with a monotone adjustment and a reward-shortfall cap. By relating index errors to the regret of single-arm stopping policies, we establish a distribution-free regret bound of $4.45\sqrt{KT}+10.75K$ for $K$ arms and horizon $T$. This bound matches the minimax-optimal regret order established in the literature. The guarantee extends to rewards supported on $[0,1]$ through Bernoulli randomization. We also provide a finite-grid implementation with quantified approximation loss. In numerical experiments, the SAB-based index policy achieves lower worst-case regret than every tested benchmark policy across all evaluated numbers of arms and horizons, while closely matching the grid-based MAB minimax policy in the two-arm setting.

Implanted neurostimulators that sense local field potentials now enable chronic electrophysiology based biomarker tracking in patients at home. The Medtronic Percept PC, the only commercially available sensing-enabled deep brain stimulation (DBS) device, stores spectral amplitudes as 16-bit integers at approximately 0.1 $\mu$V per bit (quantum $q \approx 0.11$ $\mu$Vp). At frequencies where the true amplitude spans only a few quantization levels, consecutive bins round to the same stored value. Standard spectral parameterization (FOOOF, fitting oscillations and one over f), which separates periodic peaks from the aperiodic 1/f activity, treats every value as exact and fits oscillatory peaks to these plateaus. Because these spectra feed clinical biomarker pipelines and spectral foundation models for symptom decoding, spurious peaks can corrupt downstream inference. Across 9,438 spectra from 14 hemispheres in 7 subcallosal cingulate DBS patients, 20.6% of peaks detected at [2, 45] Hz have no match in ground truth synthesized by quantizing clean in-clinic BrainSense recordings, while aggregate beta band power and the aperiodic exponent are preserved. We formalize dequantization as interval-censored subspace estimation and compare five classes of correction methods. Quantized probabilistic PCA is the only tested method that reduces the spurious rate (20.6% to 18.3%) while preserving true peak detection and keeping noise floor RMSE below $q/\sqrt{12}$.

Federated learning (FL) has emerged as a leading privacy-preserving framework for collaborative machine learning across decentralized environments. While considerable progress has been made in horizontal federated learning (HFL), where data with common features is distributed across sites, vertical federated learning (VFL), where sites share observations across distinct feature sets, remains less explored. Advancing Bayesian high-dimensional multivariate reduced-rank regression methods for VFL poses unique challenges: (a) stringent privacy regulations preventing local site data sharing, and (b) fitting local regressions overlooks essential modeling aspects like inter-variable correlations. In contrast HFL allows each site to fit a comparable model independently. We present a novel Bayesian VFL framework for multivariate high-dimensional reduced-rank regression, termed BayesVFLReg, which enables precise coefficient estimation while safeguarding both feature and response privacy. Participating sites use a shared random sketching matrix to compress local variables into privacy-preserving sketches. A central server collects these sketches where Bayesian multivariate reduced-rank regression uses Gaussian scale mixture priors. For feature selection, we introduce a single-step post-processing strategy based on mixture-model clustering of the absolute posterior coefficient means to distinguish signal from noise per response variable. BayesVFLReg is computationally scalable for large, high-dimensional datasets and facilitates efficient variable selection. Theoretically, we establish sharp non-asymptotic bounds on the posterior probability that the fitted density falls within a Hellinger ball centered at the true data-generating density. Comparative simulation studies and real-world data analyses show that BayesVFLReg reliably identifies sparse feature effects, even under feature correlation.

Forecasting a long horizon from only the first observations of a sequence is ill-posed: many trajectories are consistent with the same short history. We study this problem in oil and gas production forecasting, where forecasts made after roughly the first fifth of a well's producing life drive development and abandonment decisions, and where a usable forecast must describe a monotone decline. We present Physics-SIMS-TS, a conditional diffusion forecaster that combines negative guidance against synthetic artifacts, decline-curve constraints and an isotonic projection applied during sampling, spatial training augmentation, and an ensembled stochastic sampler yielding a full predictive distribution. Across three jurisdictions and more than 35,000 wells, under a shared-space, validation-frozen protocol, Physics-SIMS-TS is the most accurate diffusion forecaster in the comparison and is competitive with, but not superior to, ensembled transformer forecasters. Its forecasts are monotone by construction at a cost of at most 0.5% in mean squared error, and its trajectory ensemble yields calibrated intervals after one dispersion factor is fitted per jurisdiction. On six standard benchmarks a reversible-instance-normalization variant of the backbone is the leading diffusion baseline. We also quantify four protocol choices on which the measured ranking depends. Code and evaluation artifacts are released.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

We study binary classification with abstention under separate class-conditional error constraints, with the objective of minimizing abstention while keeping both errors below prescribed thresholds. We characterize the distribution-free minimax rate of excess abstention risk, up to logarithmic factors, in terms of the complexity of the hypothesis class and the sample size. To make the framework amenable to computation with models such as neural networks, we introduce surrogate-loss formulations and derive finite-sample guarantees for excess surrogate ambiguity risk. We formulate the resulting learning task as a constrained optimization problem and characterize its computational complexity in the convex setting. Finally, we evaluate our approach on various datasets and compare its performance with a competing method for this problem.

Reinforcement learning with verifiable rewards (RLVR) often improves pass@1 while falling behind its base model at larger sampling budgets $k$, a crossover read as evidence that RLVR only sharpens existing capability. We identify two limits to this reading. First, a visible crossing need not be statistically established: comparing models on the same prompts, we build confidence bands across sampling budgets $k$ that require evidence of both an early gain and a later loss. Across five public RLVR pairs no crossing is statistically established in the initial evaluations, while a 32k-token evaluation on fresh prompts locates a reversal with first loss between 11 and 61 samples; power analysis shows why failure to detect a crossing need not mean no crossing, and why more prompts can help more than more answers per prompt. Second, base success alone does not determine what RLVR does to a prompt: prompts with the same base success rate have different post-RL success rates, and these differences repeat across independent generation halves. The relationship is a conditional distribution—a Markov kernel—rather than a single curve, and fitting it predicts crossings in independent generations for the same prompts and corrects the simple model's power estimates. Theory further shows how losses on a minority of the hardest prompts can overturn an early lead even when training improves other prompts, separating evidence that a crossover exists from claims about what it means for capability.

Instrumental variable (IV) methods address treatment endogeneity, but with non-compliance and heterogeneous treatment effects a binary instrument generally identifies the local average treatment effect (LATE) among compliers rather than the population average treatment effect (ATE). When treatment effects and compliance probabilities are heterogeneous and dependent through latent factors, the ATE need not be identified by IV variation alone. We develop a proximal instrumental variable framework that uses proxies for these factors to adjust for the compliance weighting in LATE and recover the ATE. We show that the ATE is identified through either an outcome bridge capturing latent treatment effect heterogeneity or a compliance bridge representing inverse compliance probabilities, and combining the two yields a doubly robust representation. We derive the efficient influence function under known and unknown instrument propensities. For estimation, we propose regularized kernel minimax bridge estimators, with orthogonalized conditional moments for estimated propensity, and a three-way sequential cross-fitting scheme. We establish projected bridge-risk bounds, and give conditions for asymptotic normality and semiparametric efficiency. The framework provides a proxy-based route from local IV effects to population causal effects under latent treatment effect and compliance heterogeneity.

Most work in computational ethics treats annotator disagreement on moral content as noise to be voted away, collapsed into majority vote or the more permissive any-annotator rule the moment a single annotator flags an item. We argue this uncertainty should instead be modeled and learned from. We introduce Moral Entropy, a Bayesian framework that keeps a full posterior over the true label and decomposes its entropy into aleatoric uncertainty (irreducible disagreement about the moral content) and epistemic uncertainty (from insufficient or noisy annotation) – and lets any heuristic consensus rule be audited against a calibrated ground truth via entropy methods such as cross-entropy/KL, Brier score, and expected calibration error. Across three corpora and fifteen discourse domains, auditing the standard aggregation rules against this posterior reveals bias that no current pipeline reports: the any-annotator rule disagrees with the calibrated posterior on roughly 30% of items – pooled, almost entirely false positives, though the errors invert at the foundation level (19.9%/38.9% mean FPR/FNR on MFTC) – while the stricter majority and two-vote rules miss 63-83% of true positives.

Masked diffusions are popular generative models for discrete distributions. Unlike standard autoregressive sampling, they reveal several coordinates in parallel, approximating each block's joint conditional law by a product of one-coordinate conditionals. The resulting procedure, usually called tau-leaping, reduces computational cost but introduces a factorization error ($\varepsilon_\text{fact}$), even with perfectly learned predictors. We study the resulting tradeoff between generative accuracy and computational cost, focusing on how to choose a denoising schedule to minimize $\varepsilon_\text{fact}$ for a fixed sampling budget. To do so, we establish an exact integral representation of $\varepsilon_\text{fact}$ separating the schedule from the target's dependence structure, summarized by a dependence density $\rho$. This representation yields recursive stationarity equations for optimal schedules and allows us to quantify how estimation errors in $\rho$ affect schedule selection. As the dimension $N$ and sampling budget grow, we characterize the optimal schedule and quantify the cost of random block sizes relative to a deterministic planner. We highlight a fundamental dichotomy: if $\rho$ converges uniformly to a strictly positive continuous profile as $N\to\infty$, schedule optimization can only improve the leading constant of $\varepsilon_\text{fact}$, while if $\rho$ degenerates, schedule optimization can improve the asymptotic order. Examples based on stationary processes and exchangeable mixtures illustrate these regimes.

We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order $h$ along the base manifold and of order one along the fibre. Its local covariance geometry leads to a Gumbel limit with an explicit intrinsic constant. Combining this limit with Gaussian approximation and cross-fitted covariance estimation yields a feasible simultaneous confidence tube for the regression field. We further discuss improved finite-sample inference with bandwidth selection and high-order bias corrections. Simulations on various manifolds support the proposed inference procedure. A randomized reconstruction of global wind data illustrates how the tube's cross-sections describe spatially varying uncertainty.

As an alternative to the additive and extremal biases of average and max pooling, we introduce Geometric Mean Pooling (GMP), a signed pooling operator that combines the product of feature signs with the geometric mean of feature magnitudes. Motivated by local-to-global composition in quantum many-body physics, GMP retains both joint sign information and a characteristic multiplicative scale without introducing learnable pooling parameters. We show that non-overlapping hierarchical GMP preserves the corresponding global multiplicative statistic and evaluate it on synthetic sequence tasks, iterative coarse-graining, image classification, and molecular lipophilicity regression. On the synthetic tasks, GMP recovers product-based signals more accurately than average and max pooling and maintains predictive performance under the tested levels of multiplicative input noise. On image and molecular data, however, its effectiveness depends on the representation, target parameterization, and placement of local and global pooling. These results position GMP as a complementary, regime-dependent inductive bias for tasks in which equal-weight multiplicative composition is plausible, rather than as a universal replacement for standard pooling operators.

Frontier language models now produce professional deliverables that expert graders judge to match human work on a substantial share of economically valuable tasks, yet most enterprise GenAI initiatives fail to show a measurable business effect and a large fraction of agentic projects are expected to be cancelled. We argue that this is substantially a measurement problem: public benchmarks answer "what can the model do?", whereas a deployment decision requires "is this workflow fit, reliable, safe and worth scaling - here, on our data, under our controls?". We present EnterpriseVal, a use-case-level evaluation system that closes this gap. It comprises (i) a formal specification of the use case and of the frozen socio-technical configuration under test, model, prompts, retrieval, tools, guardrails and human oversight, with an autonomy level and consequence tier that jointly set the required evaluation intensity; (ii) a metric catalogue spanning fidelity, utility, efficiency, reliability, assurance and oversight; (iii) a grading protocol that scales blinded expert judgement with calibrated LLM-as-judge scoring through prediction-powered inference; (iv) a two-tier threshold gate, stated as an executable algorithm, that maps metric vectors with confidence bounds to REJECT/CONDITIONAL/SCALE decisions; and (v) a value-and-risk model in which the reviewer catch rate is a measured parameter. We report a pilot across three workflows in a global bank. In credit-memo drafting, human-graded citation precision reached 88% and hallucination rate 1.6% for the best model against gates of 70% and 5%; in procedure transformation, analyst refinement effort fell from an estimated 27.4 to 2.9 hours per document. We separate established results, documented pilot evidence, the proposed system and open hypotheses, and specify the experiments required for full validation

In classification-oriented adaptive sensing, posterior samples characterize uncertainty at the current measurement state and can serve two roles: they may guide the next sensing direction, while their class labels provide votes for the candidate classes and determine whether sensing should continue. We focus on the stopping layer that turns these votes into a declaration, without modifying the posterior sampler or sensing directions. A natural plug-in rule declares when the observed vote share exceeds a threshold. We show that this threshold is not itself a confidence guarantee: when the underlying vote mass equals the threshold, the plug-in rule declares about half the time. As alternatives, we calibrate a fixed-sample rule and a finite-horizon sequential rule to a prescribed false-declaration probability, and study exact curtailment, which stops a fixed-pool rule once its final verdict is forced. We then derive how one-round declaration probabilities determine posterior-sample cost and classification accuracy along a sensing path. On MNIST with DDRM and a fixed PCA-guided probe sequence, curtailment saves up to 62% of posterior samples. Among the evaluated rules at matched operating points, sequential stopping reduces the cost the most. At a high accuracy, that same sequential rule can trade more posterior samples for fewer measurements.

Simulation based inference (SBI) circumvents the challenge of intractable likelihoods by using a simulator that generates data given parameter values. For instance, neural likelihood estimation (NLE) estimates the likelihood function by training a neural network to perform conditional density estimation on simulated data given corresponding parameters. However such density estimation is only feasible for relatively low dimensional data. We extend the scalability of SBI methods to a higher dimensional problem: long sequences with a complex dependency structure. We introduce Neural Composite Likelihood Estimation (NCLE). This divides the sequence into smaller, equal-sized batches. Instead of training NLE to estimate the likelihood for an entire sequence, we estimate the likelihood for each batch separately. The product of these forms an approximate composite likelihood (CL), and we perform frequentist inference using methods from the CL literature: we get a point estimate from maximising the approximate CL and obtain confidence intervals by estimating the Godambe information matrix. We demonstrate the effectiveness of NCLE with experiments on time series models.

We develop single-loop stochastic projected damped extragradient methods for nonconvex–(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex–strongly concave and nonconvex–concave settings, respectively, where $\kappa=L/\mu$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

Clustering is a fundamental task in data analysis, typically addressed through centroid-based methods such as K-means. In this work, we present a general framework for multi-domain clustering via measure quantization: given samples from multiple domains, we learn a shared set of cluster prototypes by minimizing a probability metric, such as the Sinkhorn divergence or the Maximum Mean Discrepancy, between each domain's probability measure and the measure of prototypes. Data points are then assigned to clusters either via nearest centroid, or via optimal transport, a collaborative strategy that couples all samples within a domain. A mini-batch optimization strategy makes both fitting and assignment scalable, reducing memory and computational cost while preserving clustering performance. Experimental results on 5 multi-domain benchmarks spanning image, audio and sensor data show that our Sinkhorn-based method consistently outperforms classical and multi-domain clustering baselines, and that this advantage persists when scaling to hundreds of thousands of samples.

We revisit Breiman's observation that reducing inter-tree correlation without weakening individual trees can improve random forests. Building on this principle, we introduce two variants: Dirichlet-Multinomial Bagging Random Forest (DM) and Dirichlet-Weighted Random Forest (DW). Both modulate sample reweighting via a concentration parameter $\alpha>0$. We provide a simple theoretical criterion that clarifies when these variants behave indistinguishably from standard random forests, and we use it to guide a lightweight tuning strategy. In a controlled evaluation on public classification benchmarks, DM and DW are consistently competitive and often stronger than other random-forest (RF) baselines, with negligible additional runtime.

Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or Lévy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.

Real data often contain unusual observations that can exert disproportionate effects on variable selection, especially in complex predictor settings. We propose a two-stage sparse median outer product of gradients (smOPG) method for variable selection in single index models with outlier contamination. We first estimate sparse local gradients via \(\ell_1\)-penalized local median regression and then recover the active predictor set from a rank-one sparse approximation of the resulting gradient matrix using regularized singular value decomposition. The combination of median regression and local weighting provides robustness to both response outliers and leverage points. Extensive simulations across varying dimensions and contamination mechanisms demonstrate the favorable variable selection performance of smOPG relative to existing methods. Applications to air pollution and genomic data demonstrate practical utility, while theory establishes active-set recovery without requiring selection consistency of individual local regressions.

In the early stages of a pandemic, identification of a small number of infected individuals through large-scale screening is critical for pandemic control, yet remains challenging under limited reagents and testing capacity. Existing group testing methods suffer from either high computational complexity or low identification accuracy. Even worse, no available methods provide theoretically rigorous analysis for sparse identification with hard constraints caused by the sample usage constraint and the dilution effect existing ubiquitously in practical applications. In this article, we propose the Logic Screening method (LoSc), an ultra fast, accurate, and theoretically grounded framework for large-scale screening. LoSc introduces a novel decoding algorithm with a very simple selection strategy, achieving identification of all positives with only O(klogn) pooled tests. The decoding relies only on logical operations, enabling direct hardware implementation and yielding ultra fast computational implementation. Moreover, LoSc explicitly incorporates dilution and sample usage constraints into pooling designs, and establishes theoretical guarantees to guide optimal pooling configurations. Extensive simulations confirm the superior effectiveness, efficiency, and scalability. We believe LoSc offers a fast and reliable solution for automatic large-scale screening.

Modern text and image representations are often matrix-valued, with rows corresponding to tokens, patches, or other local feature vectors. Predictive information is often sparse but sample-specific, making classical sparse regression methods with a common support poorly suited to this heterogeneity. This paper formalizes an individualized sparse regression framework for matrix-valued covariates in which each observation has its own rows of interest, while the associated regression effects are shared across the population. To estimate this model, we introduce a diagonalized attention mechanism that uses query–key scores to localize sample-specific signal rows and a value matrix for downstream regression. The proposed method has a parameter dimension independent of sample size and can identify rows of interest for new observations without their responses. We establish existence theorems showing that, under suitable score-separation and concentration conditions, single-head and multi-head diagonalized attention models recover the latent rows with high probability, yielding prediction risk bounds. Our theory therefore provides a statistical explanation of how attention-based scoring localizes sample-specific signals in heterogeneous matrix-valued data. Simulations demonstrate strong prediction and localization in regression and misspecified classification across varying sample sizes, dimensions, and signal cardinalities. Real sentiment analyses show improved classification accuracy and interpretable token selection.

Accurate rare-event estimation can be computationally expensive. Classical adaptive importance sampling (IS) schemes often rely on restrictive proposal families and can struggle under multiple failure modes. We propose FAMIS, a flow-based multiple importance sampling (MIS) framework that learns a nonuniform mixture of normalizing flow proposals for rare event estimation. The method does not require presampled failure data or prior knowledge of the number, location, or geometry of the failure modes. Instead, it adaptively learns the mixture through sequential evaluations of the limit state function. To guide training toward the failure domain, FAMIS uses a smooth rare-event surrogate and a tempered target sequence. A defensive exploration mixture improves early-stage coverage, a Rao Blackwellized update adapts the mixture weights, and a Jensen-Shannon repulsion term promotes separation and diversity among the base components. The final failure probability is computed with a deterministic-mixture MIS estimator. Numerical experiments demonstrate that FAMIS accurately approximates quasi-optimal IS densities with fewer training samples and model evaluations, providing stable variance reduction across complex reliability problems.

Sampling many responses and keeping one that passes a verifier lets large language models solve problems beyond their single-response ability, but this search must be paid again for every query, while many deployments answer with a single response. Post-training on verified responses can transfer the benefit of search into the model. With a fixed budget, selecting by correctness alone spends slots on problems the model already answers correctly, leaving fewer to correct its failures. To address this imbalance, we propose GapFT, which trains on the gap between Pass@K and Pass@1: problems that the source model fails with one response but solves within K samples. GapFT keeps the objective and training budget fixed and changes only which verified responses enter training; an exact decomposition splits the resulting Pass@1 change into corrected failures and regressions on problems the source model already solved. On LogiQA 2.0 and ReClor with three model families, GapFT is above budget-matched uniform rejection-sampling fine-tuning (RFT) in every setting, with a positive pooled effect, and on Llama-3.1-8B and Mistral-7B it recovers about two thirds to four fifths of the gain of fine-tuning on the entire verified pool with 11-34% of its problems. Further analyses reveal that the gain comes from failures that the first few search samples recover, while failures found only by deeper search displace replay and add no net gain, that filling the same budget with gold-labeled failures search cannot reach lowers accuracy, and that the gain is bounded by how many transferable failures search exposes.

Industrial predictive maintenance increasingly depends on learning from equipment spread across sites whose sensor data cannot easily be pooled. Federated averaging (FedAvg) solves this with a central aggregation server; gossip learning removes the server, but its behaviour for recurrent failure-detection models has not been measured under controlled conditions. We compare synchronous ring gossip with FedAvg, isolated local training and a centralized reference for a stacked LSTM that detects imminent failure on the NASA C-MAPSS turbofan benchmark. All methods share one open implementation, architecture, initialization, optimizer, data split and training budget, and the primary endpoint uses one terminal window per test engine to avoid the statistical dependence of overlapping windows. On FD001 (five seeds), gossip reached a terminal-window F1 of 89.6 +/- 1.3%, compared with 89.9 +/- 1.1% for FedAvg, 83.6 +/- 6.7% for local training and 93.5 +/- 2.1% for centralized training, while transmitting the same payload as FedAvg without a coordinator. Node models agreed closely but not exactly (1.8% pairwise decision disagreement versus 5.6% without communication). Across FD002-FD004, peer communication improved terminal-window F1 over local training by 13-28 points; gossip matched FedAvg on FD003 and FD004 but was 4.3 points lower on the multi-condition FD002 subset. Simulated message loss, node failure and server outage changed neither method appreciably, whereas larger rings degraded gossip faster. Ring gossip is therefore a practical serverless alternative when data heterogeneity is moderate, and faster-mixing topologies become important as heterogeneity grows.

We propose a decoupled, layerwise method for structurally sparsifying the fully connected layers of pretrained neural networks. Rather than penalizing all layers jointly, our approach extracts shallow two-layer subnetworks, normalizes the inner weights, and applies a structured group penalty to the outer weight matrix of each block, processing layers sequentially to prune neurons and reduce the width of each layer. We prove that the constrained decoupled objective is equivalent at optimality to a specific joint penalty on the inner and outer weights, for any positively homogeneous activation, and thus admits a clean projected and proximal formulation. Our central finding is that this decoupled reformulation is more robust than coupled methods. In numerical experiments it provides a wider usable range of the regularization strength and a lower rate of catastrophic over-pruning than the tested joint baseline while maintaining comparable accuracy. We establish these properties in controlled classification and sparse-recovery studies, and examine their scope in a high-dimensional PINN stress test and in the feed-forward layers of OPT-1.3B.

Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances. Yet, their use in large-scale scientific problems is limited by computational cost. Equivariant neural networks are common but typically lack the uncertainty quantification offered by GPs, which is valuable in applications such as molecular research. High-dimensional inputs and large symmetry groups further demand scalability. We establish results pertaining to the interplay of GP equivariance and conditioning and leverage them to obtain equivariant sparse GPs through suitable mean functions and covariance kernels. We instantiate this framework with a flexible class of integration-free equivariant kernels, yielding scalable and data-efficient GP inference. In particular, we introduce triply scalable equivariant Gaussian processes. We employ equivariant sparse variational Gaussian processes for $\mathrm{SO}(2)$-equivariant vector fields and molecular property prediction. Alongside the SVGP, we develop a matrix-free equivariant full-GP implementation that combines an exact Kronecker reduction with preconditioned conjugate-gradient solves, enabling fast and scalable evaluation of the full joint predictive density. We further compare different approaches for selecting inducing points in the equivariant sparse GP models. Our test cases include synthetic $\mathrm{SO}(2)$-equivariant fields as well as the prediction of electric dipole moments of N-methylformamide based on quantum chemistry simulations, achieving accurate, uncertainty-aware predictions at a fraction of the computational cost of classical GP inference.

We study data-driven early stopping for spectral regularisation methods in the classical non-parametric regression setting. Building on the discrepancy principle, we propose a multi-scale stopping rule that applies to general kernel estimators and show that, unlike previous approaches, it achieves full adaptivity over all smoothness levels in the well-specified case. A key contribution of our work is an extension based on random feature approximations, which reduces computational cost on large datasets while preserving minimax-optimal statistical guarantees. Our procedure not only selects an optimal stopping time but also provides a fully data-driven choice of the number of random features needed to achieve optimal rates. Through the established connection between random features and neural networks in the neural tangent kernel regime, our method further yields a principled, data-driven recommendation for the network width. We prove that the resulting simultaneously chosen width and stopping time allow neural networks to attain minimax-optimal learning rates without prior knowledge of smoothness or capacity parameters.

Latent variable generative models are commonly fit using simple priors over latent variables, but draws from these priors often fail to produce realistic data. This failure is due to a mismatch between the prior and the aggregated posterior, the distribution of latent variables induced by the fitted model and the data. This mismatch is often viewed as evidence that the prior is misspecified and should be replaced. Alternatively, in modern generative models, a two-stage strategy is increasingly used where first, the model is fit, and second, the aggregated posterior is estimated (van den Oord et al.,2017; Rombach et al., 2022.). Synthetic data are then obtained by sampling from this aggregated posterior instead of the prior. To check such procedures, we introduce the aggregated posterior predictive check (APPC). Theoretically, we establish sufficient conditions under which the APPC is asymptotically calibrated. For probabilistic principal component analysis, we show that the APPC can remain calibrated under a misspecified latent prior when pervasive factors permit recovery of the signal space. Experiments with variational autoencoders show that aggregated posterior sampling improves generation for heavy-tailed and clustered data relative to Gaussian prior sampling while performing comparably to models with more flexible latent priors.