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arXiv2026

Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network

Generative Flow Networks (GFlowNets), a class of generative models over discrete and structured sample spaces, have been previously applied to the problem of inferring the marginal posterior distribution over the directed acyclic graph (DAG) of a Bayesian Network, given a dataset of observations. Based on recent advances extending this framework to non-discrete sample spaces, we propose in this paper to approximate the joint posterior over not only the structure of a Bayesian Network, but also the parameters of its conditional probability distributions. We use a single GFlowNet whose sampling policy follows a two-phase process: the DAG is first generated sequentially one edge at a time, and then the corresponding parameters are picked once the full structure is known. Since the parameters are included in the posterior distribution, this leaves more flexibility for the local probability models of the Bayesian Network, making our approach applicable even to non-linear models parametrized by neural networks. We show that our method, called JSP-GFN, offers an accurate approximation of the joint posterior, while comparing favorably against existing methods on both simulated and real data.

cs.LG↗

arXiv2026

Improved off-policy training of diffusion samplers

We study the problem of training diffusion models to sample from a distribution with a given unnormalized density or energy function. We benchmark several diffusion-structured inference methods, including simulation-based variational approaches and off-policy methods (continuous generative flow networks). Our results shed light on the relative advantages of existing algorithms while bringing into question some claims from past work. We also propose a novel exploration strategy for off-policy methods, based on local search in the target space with the use of a replay buffer, and show that it improves the quality of samples on a variety of target distributions. Our code for the sampling methods and benchmarks studied is made public at https://github.com/GFNOrg/gfn-diffusion as a base for future work on diffusion models for amortized inference.

cs.LG↗

arXiv2026

Towards a mathematical theory of superposition

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector $x$ of active features is encoded through an overcomplete dictionary $W$, and feature recovery is performed by applying $\operatorname{ReLU}(W^\top W x+b)$ with an appropriate bias vector $b$. We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order $d/\log n$. In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with $n>d+1$, we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.

stat.ML↗

arXiv2026

Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification

Nearest neighbor classification relies fundamentally on how locality is defined, yet conventional $k$-NN imposes the same neighborhood cardinality throughout the feature space. This assumption can be inadequate for data whose local geometry varies substantially across the underlying manifold. We introduce Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification (CARSANN), a geometry-driven framework that adapts the spatial support of each neighborhood according to local geometric complexity. CARSANN first estimates intrinsic dimensionality using TwoNN and constructs an intrinsic representation through principal component analysis. Local mean curvature is then estimated using a shape-operator-based formulation and controls neighborhood scale: highly curved regions receive stronger radius shrinkage, whereas approximately flat regions retain broader spatial support. Unlike methods that modify only the number of neighbors or the local metric, CARSANN explicitly adapts the spatial extent of local evidence. Experiments on more than 70 real-world OpenML datasets show that CARSANN consistently improves upon standard $k$-NN and is competitive with adaptive nearest-neighbor methods. In a controlled comparison using the same base neighborhood size, CARSANN achieves higher balanced accuracy on 40 of 45 datasets, increasing mean balanced accuracy from 0.6506 to 0.7528. The advantage also persists against $k$-NN with fixed $k=5$. Friedman and Nemenyi tests confirm that the improvements are statistically significant. These results indicate that local manifold curvature can serve as an effective geometric control variable for adapting neighborhood support, providing a complementary paradigm to cardinality-based nearest-neighbor adaptation.

cs.LG↗

arXiv2026

On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.

math.OC↗

arXiv2026

Beyond Search-Imitation: Prior-Directed Exploration for Searchless Chess

Searchless chess networks reach human master strength from a single forward pass by imitating a stronger teacher: the strongest, Leela Chess Zero's (Lc0) released Chessformer, distills the visit counts of an AlphaZero-style Monte Carlo Tree Search (MCTS). Imitating a search is a poor proxy for playing without one, so we fine-tune for single-pass strength with self-play reinforcement learning (RL). Its exploration is usually supplied by an entropy bonus, the reverse Kullback-Leibler (KL) divergence to uniform. We replace it with a forward, mass-covering KL toward the network's own MCTS prior (prior-directed exploration), so exploration covers the moves the prior judges promising, and pair it with an entropy-adaptive sampling temperature, set by the value head's outcome uncertainty, that sharpens once a position is decided. In about two thousand steps it raises puzzle accuracy from 93.9% to 94.9% on a 100,000-puzzle suite and mate-in-four accuracy from 77% to 81% while holding searchless strength at or slightly above the base. Measuring tactical accuracy and playing strength together across a matched-compute sweep, we find the two dissociate: accuracy gains fall in a one-point band while ratings straddle the base, and a control fine-tuned on puzzles alone posts the study's largest tactical gains while shedding roughly 260 Elo; a better puzzle-solver is not thereby a stronger player. Distribution-level measurements show what anchoring buys: without a regularizer self-play collapses onto a single line of play, and the puzzles newly solved are the near misses whose winning move the prior kept alive. The forward-KL prior tops the rating ladder, statistically tied with a reverse-KL anchor that concentrates twice as hard and drops the hardest solutions the mass-covering prior keeps in support.

cs.LG↗

arXiv2026

Fast Weight Attention for Continual Learning

Recurrent fast-weight memories and selective state-space models compress an expanding context into a fixed-size recurrent state, making the state transition an online learning rule. We study this rule under read-after-write autoregressive semantics. For the prefix-prediction objective considered here, the local fast-memory example revealed at step $t$ is the prefix-aligned pair $(\mathbf{x}_t,\mathbf{y}_t)=(ϕ(\mathbf{k}_{t-1}),\mathbf{v}_t)$. The common same-step association $(ϕ(\mathbf{k}_t),\mathbf{v}_t)$ remains causal, but optimizes a different internal objective. We derive normalized first-order updates for squared-error regression and negative inner-product objectives. The regression family comprises Falcon-1 (a scalar NLMS update), Falcon-2 (its per-column extension), and Falcon-3 (a sliding-window mini-batch update); Falcon-1A/Falcon-2A/Falcon-3A are the corresponding inner-product variants. We provide recurrent, masked-parallel, and chunk-parallel forms, together with numerically stable positive-decay renormalization. Representative variants remain competitive in language modeling and improve length extrapolation on variable-digit addition. This framework separates temporal alignment, plasticity, forgetting, and bounded rehearsal in recurrent sequence models.

cs.LG↗

arXiv2026

Towards Scaling Reinforcement Learning to Massive Populations: Learning Mean-Field Representations

Modern multi-agent systems are increasingly deployed at scale over large populations of agents in settings such as ad-auctions, traffic routing, and recommendation systems. The dominant approach in such settings is to optimize each agent's policy independently, treating the other agents as part of a fixed single-agent environment rather than modeling the population dynamics. In many large-population systems, the dynamics depend on an aggregate summary of the population rather than the identity of any individual. Mean-field RL exploits such structure, providing a principled framework that models each agent's environment as an explicit function of the population distribution. However, in large state-action spaces or high-dimensional control problems, modeling the population distribution is itself intractable. How can we design a scalable framework for high-dimensional control problems with large populations? This work explores this question from the perspective of representation learning. We introduce a mean-field RL framework in which the rewards and transition dynamics depend on the population only through an unknown low-dimensional aggregate statistic. We then study this framework in the offline setting and design a provable approach that learns a near-optimal policy by learning a low-dimensional representation. Motivated by real-life supply-chain optimization problems, we design a one-step routing game to test the hypothesis that learning a low-dimensional population representation improves reward prediction and Nash gap estimation relative to baselines that don't exploit this structure. We show that under a fixed neural-network parameter count and optimization budget, learning a low-dimensional population representation improves reward prediction and the equilibrium quality of the resulting policies.

cs.MA↗

arXiv2026

Effective Learning Rate Governs Loss Dynamics in Language Model Pretraining

We uncover ELR collapse in language model pretraining: learning rate (LR) and parameter norm govern loss dynamics primarily through their ratio, the effective learning rate (ELR). When ELR is matched across runs, their loss trajectories collapse throughout training despite substantially different LRs and parameter norms. Across optimizers, architectures, datasets, and model scales, mean collapse errors are typically a few x 10^-3, below the seed-to-seed variation measured in a representative configuration. Systematic ablations identify normalization design and the timescale of LR-norm variation as key determinants of collapse precision. Controlled interventions further show that weight decay and Hyperball shape loss dynamics primarily through the ELR schedules they induce. Replacing LR with ELR enables a fitted functional scaling law (FSL) to transfer across norm-control methods. The resulting ELR-based FSL also explains delayed acceleration, a recurring effect of norm control. Together, these results establish ELR as a common coordinate linking LR scheduling, norm control, and loss dynamics.

cs.LG↗

arXiv2026

Asymptotically-informed neural networks for Black-Scholes implied volatility computation

The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.

q-fin.CP↗

arXiv2026

The Role of Pseudo-labels in Self-training Linear Classifiers on High-dimensional Gaussian Mixture Data

Self-training (ST) is a simple yet effective semi-supervised learning method. However, why and how ST improves generalization performance by using potentially erroneous pseudo-labels is still not well understood. To deepen the understanding of ST, we derive and analyze a sharp characterization of the behavior of iterative ST when training a linear classifier by minimizing the ridge-regularized convex loss on binary Gaussian mixtures, in the asymptotic limit where input dimension and data size diverge proportionally. The results show that ST improves generalization in different ways depending on the number of iterations. When the number of iterations is small, ST improves generalization performance by fitting the model to relatively reliable pseudo-labels and updating the model parameters by a large amount at each iteration. This suggests that ST works intuitively. On the other hand, with many iterations, ST can gradually improve the direction of the classification plane by updating the model parameters incrementally, using soft labels and small regularization. It is argued that this is because the small update of ST can extract information from the data in an almost noiseless way. However, in the presence of label imbalance, the generalization performance of ST underperforms supervised learning with true labels. To overcome this, two heuristics are proposed to enable ST to achieve nearly compatible performance with supervised learning even with significant label imbalance.

stat.ML↗

arXiv2026

Neural means and kernel corrections for operator learning

We combine neural network means with exact Matérn kernel regressions of their residuals and of their learned features, and evaluate the pairing on two public emulation problems with published baselines: the structural-mechanics benchmark of de Hoop et al. and the OCO-2 radiative-transfer emulator of Lamminpää et al. On structural mechanics the combination reaches 4.55% test error, matching the best published architecture, and 5.38% against a published 6.49% in the low-data regime. On OCO-2 it improves on the published Gaussian-process emulator on that problem's own test points, outright on two of the three spectral bands; the same kernel that trails the network tenfold on the raw state overtakes it on the network's features, and we measure why (the target's squared native-space norm drops about fortyfold at fixed effective dimension) and prove the mechanism. Where the two families tie instead, the residuals of every architecture we train correlate above 0.86 and their shared component is flat in diversity and sample size, which reads the published plateau as a property of the data. Supporting results include a second-moment identity that predicts stacking outcomes from measured correlations, an optimal-recovery certificate, and a distribution-free coverage band, the only uncertainty signal that survives our tests.

cs.LG↗

arXiv2026

Skew-adaptive conformal prediction

We develop a skew-adaptive extension of split conformal prediction for regression. The method starts from an asymmetric interval family centered at a point prediction and uses the gauge approach to deduce the conformity score induced by this family. The inverse hyperbolic sine transform of signed scaled residuals provides the training target for an additional predictive model, whose role is to learn how predictive uncertainty should tilt across the feature space. The resulting procedure preserves the finite-sample marginal validity of split conformal prediction under exchangeability, while producing intervals that adapt to both local scale and local skewness. We also develop a calibration-sample-based estimator for comparing the expected relative future width of the skew-adaptive and classical scaled-score intervals. Experiments on a variety of datasets indicate gains in prediction interval efficiency over the scaled-score construction and conformalized quantile regression, and show that the proposed estimator closely matches the corresponding average width ratio observed on the test sample.

stat.ML↗

arXiv2026

Debiased Machine Learning for Conformal Prediction of Counterfactual Outcomes Under Runtime Confounding

Data-driven decision making frequently relies on predicting counterfactual outcomes. In practice, researchers commonly train counterfactual prediction models on a source dataset to inform decisions on a possibly separate target population. Conformal prediction has arisen as a popular method for producing assumption-lean prediction intervals for counterfactual outcomes that would arise under different treatment decisions in the target population of interest. However, existing methods require that every confounding factor of the treatment-outcome relationship used for training on the source data is additionally measured in the target population, risking miscoverage if important confounders are unmeasured in the target population. In this paper, we introduce a computationally efficient debiased machine learning framework that allows for valid prediction intervals when only a subset of confounders is measured in the target population, a common challenge referred to as runtime confounding. Grounded in semiparametric efficiency theory, we show the resulting prediction intervals achieve desired coverage rates with faster convergence compared to standard methods. Through numerous synthetic and semi-synthetic experiments, we demonstrate the utility of our proposed method.

stat.ML↗

arXiv2025

Behavior of prediction performance metrics with rare events

Objective: Area under the receiving operator characteristic curve (AUC) is commonly reported alongside prediction models for binary outcomes. Recent articles have raised concerns that AUC might be a misleading measure of prediction performance in the rare event setting. This setting is common since many events of clinical importance are rare. We aimed to determine whether the bias and variance of AUC are driven by the number of events or the event rate. We also investigated the behavior of other commonly used measures of prediction performance, including positive predictive value, accuracy, sensitivity, and specificity. Study Design and Setting: We conducted a simulation study to determine when or whether AUC is unstable in the rare event setting by varying the size of datasets used to train and evaluate prediction models. This plasmode simulation study was based on data from the Mental Health Research Network; the data contained 149 predictors and the outcome of interest, suicide attempt, which had event rate 0.92\% in the original dataset. Results: Our results indicate that poor AUC behavior -- as measured by empirical bias, variability of cross-validated AUC estimates, and empirical coverage of confidence intervals -- is driven by the number of events in a rare-event setting, not event rate. Performance of sensitivity is driven by the number of events, while that of specificity is driven by the number of non-events. Other measures, including positive predictive value and accuracy, depend on the event rate even in large samples. Conclusion: AUC is reliable in the rare event setting provided that the total number of events is moderately large; in our simulations, we observed near zero bias with 1000 events.

stat.ML↗

arXiv2025

DPGIIL: Dirichlet Process-Deep Generative Model-Integrated Incremental Learning for Clustering in Transmissibility-based Online Structural Anomaly Detection

Clustering based on vibration responses, such as transmissibility functions (TFs), is promising in structural anomaly detection. However, most existing methods struggle to determine the optimal cluster number, handle high-dimensional streaming data, and rely heavily on manually engineered features due to their shallow structures. To address these issues, this work proposes a novel clustering framework, referred to as Dirichlet process-deep generative model-integrated incremental learning (DPGIIL), for online structural anomaly detection, which combines the advantages of deep generative models (DGMs) in representation learning and the Dirichlet process mixture model (DPMM) in identifying distinct patterns in observed data. Within the context of variational Bayesian inference, a lower bound on the log marginal likelihood of DPGIIL, tighter than the evidence lower bound, is derived analytically, which enables the joint optimization of DGM and DPMM parameters, thereby allowing the DPMM to regularize the DGM's feature extraction process. Additionally, a greedy split-merge scheme-based coordinate ascent variational inference method is devised to accelerate the optimization. The summary statistics of the DPMM, along with the network parameters, are used to retain information about previous data for incremental learning. For online structural anomaly detection, DPGIIL can not only detect anomalies by dynamically assigning incoming data to new clusters but also indicate different structural states using distinct clusters, thereby providing additional information about the operating conditions of the monitored structure compared to traditional anomaly detectors. Three case studies demonstrate the dynamic adaptability of the proposed method and show that it outperforms some state-of-the-art approaches in both structural anomaly detection and clustering.

cs.LG↗

arXiv2025

Stacked conformal prediction

We consider a method for conformalizing a stacked ensemble of predictive models, showing that the potentially simple form of the meta-learner at the top of the stack enables a procedure with manageable computational cost that achieves approximate marginal validity without requiring the use of a separate calibration sample. Empirical results indicate that the method compares favorably to a standard inductive alternative.

stat.ML↗

arXiv2025

Symmetry-driven embedding of networks in hyperbolic space

Hyperbolic models are known to produce networks with properties observed empirically in most network datasets, including heavy-tailed degree distribution, high clustering, and hierarchical structures. As a result, several embeddings algorithms have been proposed to invert these models and assign hyperbolic coordinates to network data. Current algorithms for finding these coordinates, however, do not quantify uncertainty in the inferred coordinates. We present BIGUE, a Markov chain Monte Carlo (MCMC) algorithm that samples the posterior distribution of a Bayesian hyperbolic random graph model. We show that the samples are consistent with current algorithms while providing added credible intervals for the coordinates and all network properties. We also show that some networks admit two or more plausible embeddings, a feature that an optimization algorithm can easily overlook.

stat.CO↗