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Machine Learning

Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding

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cs.IT 2026-06-26

Multi-distribution functionals reduce to integrals of coincidence divergences

by Akshay Balsubramani

All you need is log

Monotonicity under data processing and additivity on independent products force every such functional to an integral over four strata

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Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the R\'enyi divergences of order $\alpha\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the R\'enyi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_{\alpha}(\pi_1,\dots,\pi_W) := -\log\int \pi_1^{\alpha_1}\cdots\pi_W^{\alpha_W}$ (with $\sum_k \alpha_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of R\'enyi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with R\'enyi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution R\'enyi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard R\'enyi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.
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Top Pith
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stat.ML 2026-05-20 2 theorems

Contradiction graph decides VC dimension threshold for any m

by Jesse Campbell, Daniel Ibaibarriaga +1 more

Contradiction Graphs Determine VC Dimension

Vertices are realizable label sequences of length m; edges mark label disagreements on shared points, fixing whether dimension meets or tops

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We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024).
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Top Pith
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cs.LG 2026-05-19 2 theorems

Wrapper gives pathwise risk control for updating LLMs

by Hamed Khosravi, Xiaoming Huo

Conformal Selective Acting: Anytime-Valid Risk Control for RLVR-Trained LLMs

CSA maintains per-round selective risk bounds under predictable updates without pooling across deployments.

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A local specialist LLM, fine-tuned with reinforcement learning from verifiable rewards (RLVR) on operator-local data, is installed in a regulated organization with per-deployment error budget $\alpha$. The operator needs a safety certificate for this deployment's stream at every round: no pooling across deployments, no waiting for a long-run average. Existing wrappers cannot deliver this on adaptive, online-updated streams: offline conformal-risk methods require exchangeability; online-conformal methods bound only long-run averages; non-exchangeable extensions are marginally valid; and the closest anytime wrapper, A-RCPS, controls marginal rather than selective risk. Using a (test statistic, validity guarantee, deployment rule) framework, we identify one empty cell forced by deployment requirements: e-process per threshold, selective risk, anytime-pathwise validity, max-certified-threshold rule. Conformal Selective Acting (CSA) fills it as a per-round wrapper maintaining a Ville-type e-process per threshold on a Bonferroni grid, evaluated against the RLVR filtration. Under predictable updates and isotonic-calibrated monotone risk we prove (i) an anytime-pathwise selective-risk bound $R_T^{\mathrm{act}}\le\alpha+O(N_T^{-1/2})$, (ii) rate-optimal certification matching $\Theta(\bar\eta^{-2}\log(1/\delta))$, and (iii) a horizon-independent release-rate gap. Across eight specialist benchmarks ($480$ streams), sixteen adversarial distribution-shift cells ($160$ streams), and five live Expert-Iteration RLVR cells with online LoRA over four base models in three architecture families ($10{,}300$ rounds), CSA is the only method among ten compared that satisfies pathwise validity and non-refusing deployment on every cell. We do not propose a new LLM, training algorithm, or policy class; CSA is the deployment-side complement, orthogonal to the model, for operators who cannot use a frontier API.
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cs.AI 2026-07-03

Simple threshold monitor matches advanced LLM safety checks

by Mona Schirmer, Metod Jazbec +4 more

Online Safety Monitoring for LLMs

Risk-calibrated thresholding on external verifier signals performs competitively on reasoning and red teaming tasks.

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Despite alignment training, LLMs remain prone to generating unsafe outputs at deployment time. Monitoring outputs online and raising an alarm when safety can no longer be assumed is therefore critical. We study a simple real-time monitor that turns a verifier signal from an external model into an alarm decision by thresholding, with the threshold calibrated via risk control. In experiments on mathematical reasoning and red teaming datasets, we show that this simple design is competitive with more advanced monitors based on sequential hypothesis testing.
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stat.ML 2026-07-03

LLM personas split into frame-stable aggregates and frame-sensitive geometry

by Yuan Yuan

The Dual Nature of LLM Persona: Aggregated Tendencies and Frame-Dependent Geometry

Aggregate trait scores resist frame changes while correlation structure drops 42% on mismatch and recovers with alignment.

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Evaluations of LLM personas via psychometric questionnaires typically rely on aggregate scores, discarding within-instance correlation structure. We test whether this geometric structure is intrinsic or frame-dependent. Constructing within-instance correlation matrices from IPIP-50 responses, we analyze geometry on SPD manifolds under manipulated question orderings in GPT-4o simulating American and Chinese-American personas. We find that persona expression comprises two dissociable components: aggregated features (Big Five scores) degrade under randomization (21% drop) but are frame-robust; geometric features (SPD manifold) collapse under frame misalignment (42% drop) but recover substantially (to 84%) under shared frames, surpassing aggregated features (76%). This collapse-recovery pattern reveals that persona geometry is not intrinsic but a frame-dependent coordination pattern encoding information invisible to aggregation. Our findings establish a dual-nature framework for LLM personas, frame-dependent geometry versus frame-robust aggregates, necessitating frame-aware evaluation and challenging static trait conceptions.
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stat.ME 2026-07-03

CAP matches empirical risk at first order and removes second-order bias

by Yijian Huang

Cross-Audit Projection for Model Risk Prediction

Resampling audit plus asymptotic projection corrects over-optimism in binary classification risk estimates without sacrificing leading accur

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For training-data-based model risk prediction, $K$-fold cross-validation~(CV) is widely used to mitigate the well-known over-optimism of the empirical risk and is often regarded as reliable. However, for binary classification via empirical risk minimization, our numerical studies reveal a surprising phenomenon: $K$-fold CV may perform poorly in estimating class-specific risks, even worse than the empirical estimator. We perform a higher-order asymptotic analysis showing that $K$-fold CV may converge at a slower rate, whereas the empirical estimator exhibits a second-order asymptotic bias that explains its over-optimism. These findings motivate a novel two-step procedure for model risk prediction, termed cross-audit projection (CAP). The cross-audit step adopts the same resampling scheme as $K$-fold CV to estimate over-optimism in subsamples, while the asymptotic-theory-informed projection step adjusts for the reduced sample size in bias correction of the empirical risk. The resulting CAP estimator is first-order asymptotically equivalent to the empirical risk while achieving second-order asymptotic unbiasedness. An accompanying inference procedure is also developed. Simulation studies support theoretical advantages of CAP and demonstrate favorable finite-sample performance. An application to breast cancer detection further illustrates the proposed method.
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math.ST 2026-07-03

AEW achieves T log(M)/(n+1) excess risk in expectation

by Mikael M{o}ller H{o}gsgaard, Patrick Rebeschini +1 more

Aggregation with Exponential Weights is Optimal in Expectation

The bound holds for large constant temperatures on bounded Lipschitz strongly convex losses without Bernstein assumptions

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The aggregation with exponential weights (AEW) estimator is not fully understood in the basic setting of model selection aggregation with squared loss. In particular, whether it is minimax-rate optimal in expectation for large enough fixed temperatures and under random design has been an open problem since its introduction, which was explicitly posed by Lecu\'{e} and Mendelson (2013). In this paper, we settle this problem by showing that \emph{without} requiring a Bernstein-type assumption, the AEW indeed achieves the excess risk $T \log (M) / (n+1)$ in expectation, whenever the temperature $T$ satisfies $(L^2/T)\exp(B/T)\leq \mu /2$. Here, the number of dictionary elements is $M$, the estimator has observed $n$ i.i.d. samples from any distribution, and the loss is assumed to be bounded by $B$, $L$-Lipschitz continuous and $\mu$-strongly convex. For squared loss, we show that $T\geq 4 b^2$ suffices when the predictions and labels are $[0,b]$-valued. Because AEW is known to be suboptimal in expectation for temperatures below some constant, this shows that AEW has a sharp phase transition when the temperature is large enough but constant, as conjectured by Lecu\'{e} and Mendelson.
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stat.ML 2026-07-03

Additive MLP-GNN separates chemical and structural solubility drivers

by Sampreeti Bhattacharya, Arkaprava Roy

An Additive MLP-GNN Framework for Characterizing Chemical and Structural Contributions to Aqueous Solubility

MLP and GNN branches stay separate until the final step, enabling direct inspection of each contribution after pretraining on larger data.

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Aqueous solubility is a key property in early-stage drug discovery, but most predictive models merge physicochemical descriptors and molecular graph information into a single representation, obscuring whether a prediction is driven by global chemistry, molecular structure, or both. We present an additive deep-learning framework that keeps these two sources of information separate throughout training: physicochemical descriptors are encoded by a multilayer perceptron (the chemical branch) and molecular graph topology by a graph neural network (the structural branch), with the two outputs combined only at the prediction stage through an additive model with an optional multiplicative interaction. This design provides a direct decomposition of chemical and structural components that can be examined separately after training. Furthermore, pretraining on the larger AqSolDB dataset and fine-tuning on the smaller BigSolDB2 dataset substantially improve accuracy and reduce run-to-run variations, indicating generalizability of the learned features from the data-rich settings. We further interpret the fitted model using best linear projections of the branch outputs, molecule-level embedding summaries across solubility classes, and atom-level GNNExplainer masks aggregated over functional groups. These analyses show that the chemical branch aligns with familiar physicochemical descriptors, while the structural branch captures graph-topological and functional-group patterns associated with solubility. Across both datasets, the framework attains competitive predictive performance while making the distinct roles of chemical and structural information more transparent.
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stat.ML 2026-07-03

Policy-coupled coverage optimizes counterfactual prediction sets

by Yurui Zheng, Ying Jin

Prediction Sets for Counterfactual Decisions: Coverage, Optimality, and Conformal Prediction

Equivalence to risk-averse optimization produces explicit optimal sets and a conformal method with finite-sample coverage guarantees.

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Predictions are increasingly used to guide high-stakes decisions, from treatment selection to policy making. To ensure reliability with imperfect predictions, uncertainty quantification methods such as conformal prediction build prediction sets with coverage guarantees. However, statistical validity alone does not immediately determine the decisions to take, nor the optimality thereof. This gap is especially delicate in counterfactual settings where the outcome that materializes depends on the action taken, so uncertainty cannot be specified independently of the decision rule. We develop a decision-theoretic framework for uncertainty-informed counterfactual decisions. We identify a novel notion of \emph{policy-coupled coverage} -- namely, coverage of the realized outcome under the action induced by the prediction sets themselves -- as the optimal and lossless interface between uncertainty and action. It plays three roles. First, it justifies acting via a natural max-min rule as minimax-optimal under distributional ambiguity. Second, optimizing prediction sets under policy-coupled coverage is equivalent both to a stronger universal-coverage formulation and to the direct risk-averse optimization over policies and utility certificates; this equivalence yields the explicit form of the population-optimal prediction sets. Third, it admits a two-stage procedure, Policy-Coupled Risk-Averse Conformal Prediction (PC-RACP), that approximates these optimal sets with rigorous finite-sample coverage. Simulations and a real email-marketing experiment confirm that PC-RACP delivers higher utility than existing approaches while maintaining valid coverage, and that ignoring the counterfactual structure of the decision problem is suboptimal for both validity and utility.
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stat.ME 2026-07-03

Weighted tilt restores coverage for censored label shift

by Seungjin Choi

Conformal Bayes for Two-Sided Censored Gaussian Regression under Label Shift

Mixed atom-density calibration weights yield smaller valid sets than source-score methods in two-sided censored Gaussian regression.

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Prediction under label shift becomes nonstandard when responses are censored. In a two-sided censored Gaussian model, latent values below $L$ and above $U$ are recorded at the boundary values, so the observed predictive distribution is mixed, with atoms at $L$ and $U$ and a continuous density on $(L,U)$. In this paper we develop conformal Bayes for this mixed-space setting by combining posterior predictive tilting with weighted conformal calibration. Under a two-sided Tobit Gaussian Bayesian prediction head with a Laplace posterior approximation, the tilted predictive distribution has left-atom, interior, and right-atom components, with a three-term closed-form normalizer. The resulting prediction set is a mixed highest density region that can combine boundary atoms with an interior interval and can reduce to atom-only sets under strong censoring. The main technical issue is that latent label shift does not directly give an ordinary density ratio on the observed censored scale. A latent exponential tilt induces tail-averaged atom weights at the censored boundaries, while the interior ratio remains density based. This yields a mixed observed-space calibration weight with two atom ratios and one interior density ratio. The weight corrects the calibration measure, while predictive tilting gives target-adapted mixed-HDR geometry. Synthetic experiments show that weighted tilted conformal Bayes restores marginal coverage with smaller sets than weighted source-score calibration, while revealing a trade-off between marginal coverage and component-wise behavior across atoms and interior observations.
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stat.ME 2026-07-03

Expert portfolio detects PDE model errors missed by residuals

by Ieva Kazlauskaite

Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

Sequential e-process rejects bad fits early using spatial residual patterns with anytime-valid error control.

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Computational models in science and engineering are often assessed by checking whether the residual norm is consistent with the assumed noise level. This can be misleading in smoothing inverse problems: structured model errors may be attenuated in observation space, leaving residual magnitudes below practitioner discrepancy thresholds while coherent residual patterns remain. As a result, residual-norm diagnostics can accept fitted models that still give biased parameters, predictions, or quantities of interest. We propose a structure-sensitive sequential diagnostic based on e-processes. The method uses a portfolio of spatial residual-pattern experts, updates their likelihood-ratio wealth as observations are processed, and rejects the fitted model when the aggregate wealth crosses a prescribed threshold, giving anytime-valid type-I error control for a fixed fitted model. We compare the method with Morozov discrepancy checks, fixed-sample residual tests, and batch projection tests. Across three inverse problems (elliptic diffusion, two-dimensional Stokes flow, and a glaciological ice-stream inversion implemented in the community finite-element model icepack) we demonstrate how standard discrepancy checks accept misspecified fits that produce materially wrong quantities of interest. Structure-sensitive batch tests detect these failures using the full dataset, while the e-process detects them earlier from a fraction of the observations. After rejection, the expert wealth attributes the evidence to residual patterns in the chosen dictionary and provides a basis for exploratory model correction.
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stat.ML 2026-07-03

Shallow network optimum recovered by one linear solve

by Matej Benko, Pierre Bousquet +2 more

Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

A continuum variational problem on parameter densities turns training convex and yields the minimizer directly from a linear system with exp

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Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $\lambda$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost $C^3$ regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of $1/\alpha$ relative to the regularization parameter, and prove that finite-width networks of size $N$ achieve the continuum optimum at an $O(1/N)$ rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.
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stat.ME 2026-07-03

Moment method selects random effects consistently in mixed models

by Yifan Chen, Yuedong Wang +1 more

Moment-Based Selection of Multiresponse Linear Mixed-Effects Models

It reduces the problem to convex optimization using cross-moment identities and establishes finite-sample guarantees under sub-Weibull error

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We propose MOMENT (\textbf{MO}ment-Based \textbf{M}ixed-\textbf{E}ffects Selectio\textbf{N} and Es\textbf{T}imation), a stage-wise moment-based framework that exploits second-order cross-moment identities to select and estimate the random-effects covariance matrix and fixed-effects coefficients. By inducing sparsity through its diagonal under a positive semidefinite constraint, the random-effects selection problem reduces to a smooth constrained convex optimization problem that can be solved efficiently by projected gradient descent. We further establish finite-sample theoretical guarantees for the proposed procedure, including random-effects selection consistency and fixed-effects selection consistency under joint sub-Weibull errors. Simulation studies show that MOMENT performs competitively overall and can substantially outperform separate univariate analyses when responses are correlated. An application to the hemodialysis dataset demonstrates that the proposed method yields an interpretable and flexible approach for multivariate longitudinal data.
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stat.ML 2026-07-03

Autorelevance function recovers lag structure in time series forecasts

by Julian Cardenas, Jamie Arjona +1 more

Autorelevance function and other feature relevance measures for univariate time series

Shapley-based measures with one-step forecast replacement for missing lags identify expected patterns across ARMA and neural models.

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We propose a model agnostic methodology to measure lag relevance in machine learning forecasting models applied to univariate time series. Particularly, we are working in the context of time series using the frameworks of Ghost variables and Shapley values, together with additive importance measures, to introduce the auto-relevance and partial auto-relevance functions as the lag importance values. Additionally, we propose a novel method to replace absent features in coalition based methods with a one step forecast from the same model. We evaluate these proposals under different simulations and real data cases. This combined framework perspective is particularly suitable for time series. In addition, to show our discoveries we use a pull of models from the seasonal ARMA family and recurrent neural networks. We found that the calculated relevance measures successfully demonstrate the expected lag structure in almost all cases.
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stat.ML 2026-07-03

K-means centers from MCAR data converge at √n rate

by Xin Guan

Statistical Properties of k-means Clustering for Data Missing Completely at Random

Recovery of the true centers holds under a missing-probability and separation condition, provided centers differ in every dimension.

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The classical $k$-means clustering cannot be directly used to incomplete data, and existing $k$-means-based clustering for missing data primarily focus on improving the practical accuracy of clustering, whereas most of them lack theoretical guarantees in the asymptotic sense. In this paper, we investigate the statistical properties of $k$-means clustering in the presence of missing data. We first establish the $\sqrt{n}$-excess risk bound and prove the consistency of the estimated cluster centers under general missing mechanisms. For the Missing Completely at Random (MCAR) mechanism, we further derive the $\sqrt{n}$-convergence rate and asymptotic normality of the estimated cluster centers. Moreover, we study in what cases the cluster centers estimated by incomplete data converge to the true cluster centers of original fully observed data, and give a sufficient condition about the missing probability and the separation among true clusters. These results provide a theoretical guarantee for missing-data-$k$-means. Notably, our analysis reveal that under MCAR mechanism, both achieving the $\sqrt{n}$-rate and converging to the true cluster centers require $k$ true centers to be distinct in every dimension, highlighting the significant challenges of application in high-dimensional regimes. Finally, we conduct numerical simulations on synthetic incomplete datasets to support our theoretical analysis results.
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cs.LG 2026-07-03

Variational estimator beats spectral for density ratios with abundant data

by Francis Bach (SIERRA)

Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Model

In the Gaussian location model, the risk ordering reverses with the observation-to-dimension ratio under ridge regularization.

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We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($\Delta$, I), with linear features, where $\Delta$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.
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cs.LG 2026-07-03

Role projections lift directional accuracy on semantic benchmarks

by He Huang, Lu Shen +2 more

Role-Aware Neural Convex Divergence Heads for Asymmetric Representation Learning

Source and target projections before convex neural divergences model asymmetric relations like entailment while keeping scores nonnegative.

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Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but provide limited geometric structure. This paper proposes a role-aware neural convex divergence head for asymmetric representation learning. The head applies source- and target-role projections before evaluating an input-convex neural Bregman divergence, yielding a nonnegative structured score in the role-projected space. We characterize its projected-space identity, source-role convexity, directional-gap decomposition, and Hessian-based local curvature. Experiments on lexical, sentence, ontology, and directed graph benchmarks compare symmetric distances, unstructured asymmetric scorers, order/hyperbolic baselines, plain ICNN-Bregman heads, and the proposed role-aware variant. Across ten random seeds on the main semantic and ontology benchmarks, role-aware projections consistently improve directional accuracy over plain ICNN-Bregman heads while preserving zero observed negative divergence rate. The results also identify a boundary case: on large fixed-feature citation prediction, specialized symmetric or hyperbolic baselines remain stronger in ranking accuracy. Overall, the proposed head is best understood as a structured and interpretable plug-in distance module for tasks where directional relations matter.
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q-bio.QM 2026-07-03

Point source restores identifiability of spatial dynamics from snapshots

by Rujie Gu, Ray Zirui Zhang +1 more

Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots

Distributed sources cannot be uniquely recovered from static patterns, but a transcription site allows physics-informed methods to infer the

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Despite increasing scale and resolution, many biological measurements remain destructive, revealing only spatial information rather than the dynamics it encodes. By combining flexible representations with mechanistic constraints, physics-informed machine learning offers a promising route to inferring these dynamics from static snapshots. Motivated by subcellular imaging of gene expression, we ask when a static spatial pattern of molecules can identify spatially varying diffusivity, creation, destruction, and boundary exchange, and how different inference schemes perform on the task. A structural identifiability analysis shows that distributed sources are non-identifiable, whereas a point source such as a transcription site can restore identifiability. These limits are further shaped by seemingly innocuous modeling choices: the boundary conditions, the spatial regularity of the underlying dynamics, and even the stochastic calculus convention. We then adapt several physics-informed schemes, differing in how they represent the solution and enforce the governing equations, and demonstrate effective inference from a single snapshot. Physics-informed approaches can thus recover spatial heterogeneities of biological dynamics from static data, but their use should be accompanied and guided by careful identifiability analysis for meaningful interpretation of the results.
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stat.ML 2026-07-03

Method brings full Bayesian inference to RL without likelihoods

by Stefano Masini, Cecilia Viscardi +1 more

Full Bayesian Reinforcement Learning via LF-IBIS

LF-IBIS approximates posteriors over parameters and policies from simulation data alone to support uncertainty-aware decisions.

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Reinforcement Learning (RL) is a sequential decision-making framework in which an agent learns optimal policies through interaction with an environment by maximizing cumulative rewards. Among RL methods, Bayesian Reinforcement Learning (BRL) addresses common practical challenges related to data scarcity by leveraging prior knowledge about the environment and sequential belief updates. However, most BRL approaches require an explicit likelihood function, which is frequently inaccessible or intractable in real-world settings. We propose Likelihood-Free Iterated Batch Importance Sampling (LF-IBIS), a novel algorithm for BRL that updates the agent's beliefs online as new interactions become available. By combining Approximate Bayesian Computation with Iterated Batch Importance Sampling, LF-IBIS enables full Bayesian inference in settings where the environment dynamics are not described by an explicit or tractable likelihood. The method yields approximate posterior distributions over both environment parameters and optimal policies, providing a quantification of policy uncertainty useful for a Bayesian treatment of the exploration-exploitation trade-off. We test the method on a simulation study in response-adaptive randomization in clinical trials, where closed-form posteriors enable validation. Additional experiments address settings where the posterior has no closed form and illustrate online policy updating based on the posterior distribution of the optimal policy.
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physics.comp-ph 2026-07-02

Soliton dynamics recovered from scattering data without equations

by Seth Minor, Vanja Dukic +1 more

Learning Effective Soliton Dynamics from Scattering Data

Weak-form identification inside the inverse scattering framework yields low-dimensional models that hold in perturbed regimes.

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The inverse scattering transform (IST) provides the standard theoretical framework for deriving soliton dynamics. Traditionally, such derivations have been of an analytical, rather than data-driven, nature. In this paper, we combine the conceptual framework of the IST with weak-form system identification methods to discover effective soliton dynamics directly from observed scattering data, without assuming prior knowledge of the scattering equations. Our method avoids parameterizing solitary waves via ad hoc curve-fitting by working in the scattering domain, yielding interpretable low-dimensional models that remain valid in perturbed and near-integrable regimes. We demonstrate the performance of the proposed approach on synthetic and experimental data governed by shallow-water equations of Korteweg--de Vries-type and recover models that are consistent with canonical IST theory.
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cs.LG 2026-07-02

Unveiling the Non-Monotonic Effect of Privacy on Generalization under Byzantine Robustness

by Thomas Boudou, Batiste Le Bars +2 more

The benefit reverses when privacy is weaker and the usual tension with robustness returns.

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Recent work has established a fundamental trilemma between Byzantine robustness, local differential privacy (LDP), and optimization error in distributed learning. We show that this trilemma does not universally extend to generalization error, but instead depends critically on the privacy regime. Specifically, in the high-noise regime (strong privacy), we prove that increasing privacy reduces the generalization error, i.e., there is no tension between robustness and privacy. In the low-noise regime (weaker privacy), however, the tension between robustness and privacy reappears and increasing privacy indeed degrades generalization. Our theory explains this surprising non-monotonic behavior of the generalization error via matching lower and upper bounds on the algorithmic stability of Byzantine-robust distributed learning under LDP constraints. We corroborate and further analyze these theoretical findings with empirical evaluations.
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cs.LG 2026-07-02

Three-term law recovers optimal batch size from suboptimal runs

by Fabian Schaipp

How to Allocate Your Tokens? Scaling Laws with Training Steps and Batch Size

Splitting training data into steps and batch size lets the scaling law fit with fewer experiments while predicting best token use.

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We propose a scaling law that takes into account model size and training data while explicitly splitting the latter into training steps and batch size (called three-term law). Fitting the proposed law on a large set of training runs, we find that it correctly recovers the scaling of the optimal batch size. Moreover, because it makes use of training runs with suboptimal batch size, our proposed law can be robustly fit with a significantly smaller amount of training runs. We further show that the three-term law can be used to derive scaling laws for suboptimal batch sizes, and that it matches previous empirical findings related to the critical batch size.
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cs.LG 2026-07-02

Conditional forests rank 3rd-4th as feature selectors

by Robert Milletich, Justin Downes +2 more

Conditional Inference Trees and Forests for Feature Selection

Benchmarks on 30 datasets show competitive top-k performance with bias-controlled split selection.

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Conditional inference trees (CIT) and conditional inference forests (CIF) reduce split-selection bias by testing features before choosing split thresholds, but repeated permutation tests and threshold searches can make these methods computationally expensive. We study CIT and CIF as top-$k$ feature-ranking methods for downstream prediction using real-data benchmarks, runtime ablations, and synthetic feature-recovery experiments. At a fixed node, if the features and permutation budget do not depend on the node responses, Bonferroni-corrected $+1$ Monte Carlo permutation $p$-values control nodewise rejection under the complete permutation null. CIF ranks 4th among 17 classification methods on 22 datasets and 3rd among 18 regression methods on 8 datasets. With Bonferroni correction held fixed, the CIF runtime ablations indicate that adaptive stopping and the number of thresholds searched have the largest measured effect on runtime: turning off adaptive stopping and using exact threshold search increase fitting time by 4.0--8.4$\times$ and 1.9--10.8$\times$, respectively, while downstream score changes are at most 0.011. Sparse high-$p$ simulations indicate that forest feature sampling can leave informative features out of many split decisions. Overall, the results support CIF as a top-$k$ feature-ranking method in the evaluated downstream prediction benchmarks.
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cs.LG 2026-07-02

One narrative links approximation to emergence in deep learning

by Zhilin Zhao

From Approximation to Emergence: A Theory of Deep Learning

The account organizes results on optimization, scaling, transformers, and alignment by what each explains and what each leaves out.

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Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.
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0
cs.LG 2026-07-02

Decision loss augments energy score for cost-aware forecasts

by Kornelius Raeth, Nicole Ludwig

Decision-Aware Training for Sample-Based Generative Models

Sample-based generative models learn to penalize downstream decision costs while retaining full probabilistic output.

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Sample-based generative models are increasingly used for probabilistic forecasting in high-stakes decision settings, yet their training objectives are blind to the decision maker's cost structure. These models are commonly trained with strictly proper scoring rules, such as the energy score, which allocate their training signal in proportion to data density, with no awareness of where forecast errors are most costly for downstream decisions. We therefore propose decision-aware training for sample-based generative models, augmenting the energy score objective with a differentiable decision loss that directly penalises the cost incurred by acting on the model's forecast. This combined loss is theoretically grounded, as the decision loss is itself a proper scoring rule. We validate our method on one synthetic and two real-world tasks, showing targeted improvements in cost-sensitive regions while retaining full probabilistic forecasts.
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0
stat.ML 2026-07-02

Separable graphs unify independence models in mixed graphs

by Christopher Meek, Kayvan Sadeghi

Characterizing and Identifying Separable Graphical Models

Missing edges always admit separating sets, enabling canonical representations and an identification algorithm for equivalence classes

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We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.
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0
stat.ML 2026-07-02

Refined assumption gives dichotomy counts for low-dimensional data

by Konstantin Häberle, Helmut Bölcskei

Function-Counting Theory for Low-Dimensional Data Structures

Extending Cover's counting theory shows how manifold structure shapes classification capacity and generalization.

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The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.
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0
stat.ML 2026-07-02

Monotone transforms unify multitask learning for mixed outcomes

by Huichao Li, Tong Wang +2 more

Deep Multitask Learning for Mixed-Type Outcomes with Shared Sparsity

Shared first layer and group Lasso yield excess risk bounds plus consistent selection of common predictors.

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Most existing multitask learning approaches are limited by their reliance on task-specific loss functions tailored to the scale and type of each outcome. When outcomes differ across tasks, these losses are generally not directly comparable, which makes it difficult to formulate a unified objective and may limit information sharing across tasks. We propose a multitask transformation framework in which task-specific responses may differ through unknown monotone transformations. Motivated by high-dimensional biological applications in which the predictor dimension may diverge with the sample size while only a common subset of predictors is informative, we consider shared sparsity across tasks. Under this framework, we estimate the target functions and identify important predictors by optimizing a smoothed rank-based criterion with a group-Lasso penalty, implemented through a multitask deep neural network with a shared first layer. We establish the nonasymptotic excess-risk bounds, and variable-selection consistency for the proposed estimator. Simulation studies show that the proposed method achieves competitive prediction and variable-selection performance compared with competing approaches. Analyses of gene-expression studies with continuous, binary, and mixed outcomes further illustrate that the proposed method improves prediction and identifies biologically meaningful shared predictors.
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0
stat.ML 2026-07-02

Surrogate variable enables explicit process noise modeling in Kalman filters

by Shilei Li, Dawei Shi +2 more

Hierarchical Variational Kalman Filtering

Reformulated inference cuts iterations and supports higher-order trackers that become zero-phase over full history.

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Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.
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0
math.NA 2026-07-02

Symmetric CAEs yield more accurate latent trajectories in PDE models

by G. Li Causi, N.Tonicello +2 more

Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry

Extending representation consistency to convolutional layers cuts reconstruction errors and boosts robustness on advection, Burgers and Kura

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Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.
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math.ST 2026-07-02

Approximating region contains full-conformal set in multi-task regression

by Davidson Lova Razafindrakoto (SAMM), Alain Celisse (SAMM) +1 more

Approximate full-conformal multi-task regression with reproducing kernels

The construction yields a computable region guaranteed to contain the exact full-conformal one, with a volume bound when task covariances ar

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Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.
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0
cs.LG 2026-07-02

Active-GRPO raises average SRxSim to 0.1773 by updating references

by Xuefeng Liu, Mingxuan Cao +4 more

Active-GRPO: Adaptive Imitation and Self-Improving Reasoning for Molecular Optimization

The method switches from imitation to self-reinforcement per instance and replaces the reference with better policy candidates to exceed pri

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Scientific reasoning is an increasingly important capability of large language models, yet improving the robustness and efficiency of training such reasoning remains a key open challenge. We study this problem in instruction-based molecular optimization, where answer-only supervised fine-tuning (SFT) collapses multi-step reasoning and reinforcement learning with verifiable rewards (RLVR) suffers from sparse feedback. Reference-guided Policy Optimization mitigates both by anchoring policy updates to dataset-provided references, but its effectiveness is tightly coupled to reference quality: weak or misaligned references impose a performance ceiling. To overcome this ceiling, we propose active reasoning, a paradigm in which the policy actively decides, on a per-instance basis, when to imitate a reference and when to reinforce its own discoveries, while continuously upgrading what it imitates. We instantiate this paradigm as Active Group Relative Policy Optimization (Active-GRPO), realized through two coupled mechanisms: active imitate-reinforce and active referencing. The former performs imitation learning when the reference still outperforms the policy's own candidates, and shifts to self-improvement via reinforcement learning once the policy has generated molecules that surpass the reference. The latter continuously upgrades the reference itself by replacing it with the best policy-generated candidate discovered so far, progressively raising the imitation target and ensuring that reference guidance remains informative-rather than restrictive-throughout training. Across TOMG-Bench MOLOPT, Active-GRPO improves average SRxSim from 0.0959 for GRPO and 0.1665 for RePO to 0.1773 under matched three-seed evaluation, with statistically significant gains on LogP, MR, and QED.
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0
cs.LG 2026-07-02

A staged framework uses SEM then OLS then double machine learning to check which survey…

by Ka Ching Chan, Qiana Liu +2 more

From Structural Equation Modelling to Double Machine Learning: Robustness Analysis for Survey-Based Research

Framework tests stability of relationships when moving from structural models to machine learning adjustments on construct scores

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Structural equation modelling (SEM) is widely used in survey-based business and information systems research to assess latent constructs and theory-driven structural relationships. However, SEM path significance is obtained within a particular model specification and may not show whether findings remain stable under alternative estimation frameworks. This study develops and demonstrates a staged robustness analysis framework that connects SEM, ordinary least squares (OLS) regression, and Double Machine Learning (DML). SEM is first used to refine the measurement structure and estimate the robustness-baseline SEM model, in which the full theory-specified structural path system is retained for downstream robustness analysis before final structural path evaluation. OLS regression is then applied to SEM-derived construct scores as a transparent regression benchmark. Finally, DML-style residualisation is used to examine whether each tested focal relationship remains stable after flexible machine-learning-based adjustment for observed controls. Learner-sensitivity checks compare Random Forest, Gradient Boosting, and Support Vector Machine learners, and selected reverse-direction diagnostics are used to examine directional sensitivity. The framework is demonstrated using a FinTech Digital Customer Intimacy survey model. The findings identify which relationships are stable across SEM, OLS, and DML-style checks, and which require more cautious interpretation. A reproducible Google Colab workbook and generated result files are publicly available, providing a reusable template that researchers and students can adapt to other survey-based latent-construct studies. The paper contributes a practical robustness workflow and interpretation guide for survey-based researchers seeking to complement SEM with conventional and machine-learning-based robustness checks.
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0
cs.LG 2026-07-02

Prototype LMs match dense baselines while attributing data 500x faster

by Dan Ley, Giang Nguyen +2 more

Prototype Language Models

Sparse mixtures of learned prototypes keep accuracy within 2.5 points and localize influence to training neighborhoods.

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Knowing which training examples drive outputs is fundamental to auditing, correcting, and understanding language models, yet for modern LLMs this remains expensive, approximate, and largely post-hoc. Standard language models generate tokens through a dense network pathway, causing training data's influence to be distributed across parameters rather than organized along explicit, traceable components. We introduce a prototype language model architecture, Prototypes for Interpretable Sequence Modeling (PRISM), that forms each prediction via a sparse, non-negative mixture of learned prototypes, trained with clustering objectives that anchor each prototype to coherent neighborhoods of training examples. Across architectures from 130M to 1.6B parameters trained on up to 50B tokens, prototype language models either surpass or remain within 2.5 percentage points on average downstream accuracy of matched dense baselines. We show that sparse prototype structure localizes curvature in the loss landscape, yielding a more tractable Hessian and enabling training data attribution that is ~500x faster than post hoc baselines when consuming equivalent memory. Calibrating linear prototype controllers can improve downstream accuracy by roughly 3 points while tracing those corrections back to training neighborhoods, and targeted prototype suppression can remove model behaviors without finetuning or measurable loss in generation quality.
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cs.LG 2026-07-02

Linear transformers map context distributions to responses at dim-free rates

by Peilin Liu, Ding-Xuan Zhou

Ghost in the Kernel: In-Context Learning with Efficient Transformers via Domain Generalization

Domain generalization analysis shows how linear attention achieves in-context learning and informs activation choices for large-model linear

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Transformer-based large models have demonstrated remarkable generalization abilities across different tasks by leveraging a context-aware attention module for in-context learning. With richer context, transformers adapt more effectively to the current use case without any parameter updates. However, the quadratic computational and memory complexity with respect to context length significantly slows data processing in softmax transformers. Linear transformers were proposed to address this issue by reducing the complexity to linear dependence on context length, but the design and understanding of the feature mapping in linear attention, from a theoretical viewpoint, remain unclear. In this paper, we investigate the approximation and generalization abilities of linear transformers under a two-staged sampling process from domain generalization. We show that linear transformers perform in-context learning as learning a mapping from context distributions to response functions. A dimension-independent convergence rate is obtained for our generalization analysis, which also exhibits the tradeoff between the regularities of data distributions and latent features. Guided by our theoretical framework, we propose a new perspective on activation and loss design for linearizing pretrained softmax large language models.
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0
stat.ML 2026-07-02

Neural network recovers time-varying AR coefficients for forecasts

by Agnieszka Kopeć, Pawe{l} Przyby{l}owicz +1 more

Neural Network-Based Estimation of Time-Dependent Parameters in AR(p) Processes

The method keeps an explicit parametric structure while estimating changing coefficients and producing intervals under two noise distributio

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We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.
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0
stat.ML 2026-07-02

FNOs achieve polynomial sample complexity on dissipative PDE operators

by Nisha Chandramoorthy, Daniel Sanz-Alonso +1 more

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

Bounds hold uniformly over families of equations when operators admit spectral discretizations, with rates set by smoothness and dimension.

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We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.
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cs.LG 2026-07-01

Entropy regularization improves sparse recovery in federated learning

by Krishna Harsha Kovelakuntla Huthasana, Alireza Olama +1 more

Entropy-Regularized Probabilistic Gates for Sparse Model Discovery in Scarce-Data Federated Learning

In scarce high-dimensional data with heterogeneous clients, keeping gate entropy high yields better test accuracy and support identification

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Federated Learning (FL) is a distributed machine learning (ML) paradigm with collaboration among multiple clients without sharing data. FL is challenging under data heterogeneity and partial client participation. Learning sparse models is useful for communication and computational efficiency in FL, but it is especially difficult in the small-sample high-dimensional regime (d >> N) where optimization can yield parameter configurations that fail to generalize to unseen test data. While magnitude-based pruning doesn't account for uncertainty exploration in the parameter space, a formulation with probabilistic gates and an L0 constraint allows sampling from competing sparse configurations during training. In this work, we study entropy regularization of gate distributions as a mechanism to maintain uncertainty in sparse federated optimization by preventing early commitment to sparse support. We examine its impact under data heterogeneity, client participation heterogeneity, and sparsity. Experiments on synthetic and real-world benchmarks show consistent improvements over federated iterative hard thresholding (Fed-IHT) and pruning after dense federated averaging (FedAvg) training, both in statistical performance on test data and in sparsity recovery accuracy.
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stat.ML 2026-07-01

X-VAE replaces standard normal prior with data-derived Gaussian

by Qijun Chen, Shaofan Li

eXact-Prior Variational Autoencoder (X-VAE): Learning Data-Adaptive Gaussian Mixture Priors for Latent Distributions

Mean and variance taken from pretrained autoencoder latents produce representations that match empirical statistics more closely.

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Variational Autoencoders (VAEs) commonly assume a standard isotropic Gaussian prior over the latent space, an assumption that often fails to capture the true distribution of latent representations for complex datasets. This mismatch can limit reconstruction accuracy, reduce sample quality, and constrain the expressive power of the learned latent space. We propose the eXact-Prior Variational Autoencoder (X-VAE), a framework that replaces the conventional standard normal prior with a Gaussian prior derived from the latent representations of a pretrained autoencoder (AE). Specifically, the empirical mean and standard deviation of the AE latent codes are used to parameterize a data-adaptive prior that more closely reflects the underlying structure of the training data. During generation, X-VAE introduces a latent scaling factor that enables explicit control over the variance of the sampled latent vectors, providing a simple mechanism for balancing sample diversity and fidelity. This flexibility makes the proposed approach particularly well suited for applications such as industrial and engineering design, where generated solutions must satisfy strict structural or functional constraints while still permitting meaningful design exploration. We present the mathematical formulation of well-suited X-VAE, derive the corresponding KL divergence objective for the proposed prior, and evaluate the method on standard benchmark datasets. Experimental results demonstrate that X-VAE preserves reconstruction quality while producing latent representations that better align with the empirical data distribution, leading to improved controllability and more realistic generated samples.
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cs.LG 2026-07-01

Distributionally Robust Linear Regression With Block Lewis Weights

by Naren Sarayu Manoj, Kumar Kshitij Patel

The reduction yields a (1+ε) solution with Õ(min(rank(A),m)^{1/3} ε^{-2/3}) linear solves and matches ℓ_∞ regression bounds.

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We present an algorithm for the group distributionally robust (GDR) least squares problem. Given $m$ groups, a parameter vector in $\mathbb{R}^d$, and stacked design matrices and responses $\mathbf{A}$ and $\mathbf{b}$, our algorithm obtains a $(1+\varepsilon)$-multiplicative optimal solution using $\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})$ linear-system-solves of matrices of the form $\mathbf{A}^{\top}\mathbf{B}\mathbf{A}$ for block-diagonal $\mathbf{B}$. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of $\ell_{\infty}$ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.
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math.ST 2026-07-01

Full token observation lowers sample needs for watermark proportion estimates

by Shuwen Chai, Qiaosen Wang

Sample Complexities of Estimating Gumbel--Max Watermark Proportions with and without Reduction to Pivotal Statistics

Pivotal reduction to a scalar raises the number of tokens required, with matching bounds in each regime.

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Watermarking promises statistical traceability of large language model (LLM) uses, but real documents rarely arrive as purely human-written or purely LLM-generated. This motivates a quantitative question beyond detection: what proportion of a document is generated from a pre-specified watermarked LLM? We study this watermark proportion estimation problem under the Gumbel--max watermarking mechanism, treating the next-token prediction distributions as unknown and arbitrary nuisance parameters subject to a non-degeneracy condition. We compare two observation regimes: in the full observation regime, the estimator observes the pseudorandom vector and the selected token at each position; in the more prevalent setting of pivotal reduction, it observes only a scalar pivot, which follows a one-dimensional Uniform--Beta mixture distribution. Under pivotal reduction, we develop a Laguerre-polynomial estimator and establish a matching information-theoretic lower bound for the sample complexity. For full observation, we introduce an event-counting estimator and show a matching lower bound, yielding a substantially smaller sample complexity. As our results imply, although reducing to pivotal statistics is an elegant and prevalent choice, it is not always sample-efficient for estimating the proportion of watermarks.
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math.OC 2026-07-01

No constant learning rate guarantees monotone descent in adversarial least squares

by Fabrizzio Sabelli

Homogenization of ell₂-Adversarial Training in High-Dimensions: Exact Dynamics under Stochastic Gradient Descent

High-dimensional ODE analysis shows adversarial risk requires adaptive stepsizes unlike standard least squares.

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We develop a framework for analyzing the learning dynamics of $\ell_2$-adversarial training of single-index models on Gaussian mixtures in the high-dimensional limit under streaming stochastic gradient descent (SGD). We derive deterministic equivalents for a broad class of statistics of the SGD iterates, including the adversarial risk and distance to adversarial optimality, in terms of the solution to a system of ODEs. We use them to study two idealized learning rate schedules: the Polyak stepsize and exact line search. In the case of $\ell_2$-adversarial least squares with a single class, we show that, unlike noiseless standard least squares, no constant learning rate guarantees monotone descent of SGD towards a minimizer of the adversarial risk. We identify anisotropic covariance and a mismatch in ridge parameters as the main sources of suboptimality of exact line search relative to the Polyak stepsize. We also introduce a stochastic differential equation (SDE), called adversarial homogenized SGD, that captures the evolution of statistics of the iterates of SGD. For $\ell_2$-adversarial least squares, using this SDE, we show the evolution of the risk is equivalent, up to dimension-free constants, to that of SGD on standard least squares with an adaptive learning rate and adaptive $\ell_2$-regularization. When the dynamics converge, the limiting adversarial risk and SGD iterate are determined by a fixed-point equation, with the limiting iterate being equivalent to the solution of a ridge regression problem whose regularization parameter is the limiting effective regularization of SGD.
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cs.LG 2026-07-01

Three RL methods for LLMs reduce to one standard-deviation dial

by Yong Yi Bay, Kathleen A. Yearick

GRPO, Dr. GRPO, and DAPO Are Three Operations on One Number: The Group-Standard-Deviation Identity

Binary rewards make disagreement the exact size of each training update

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Three of the most popular methods for training language models to reason look like three different tricks. They are not. All three adjust a single number: standard deviation, reflecting how much a prompt's sampled answers disagree. When such a model is trained, it answers each problem many times, and an automatic checker marks every answer right or wrong. The standard deviation of those marks measures the disagreement: largest when the answers split evenly between right and wrong, and zero when they all agree. Group Relative Policy Optimization (GRPO) divides by this number, GRPO Done Right (Dr. GRPO) drops the division, and Decoupled Clip and Dynamic Sampling Policy Optimization (DAPO) discards the groups where it is zero. Each is presented as its own fix, yet this paper proves they are three settings of one dial. That dial is not cosmetic: for right-or-wrong rewards, the disagreement is exactly the size of the training update, the group-standard-deviation identity. A split group teaches the most, while a unanimous group teaches nothing and falls silent. The same result says which problems deserve the most weight and how many tries each one needs. This paper confirms the intuition on a large real difficulty dataset (Big-Math) and in a controlled training run. What looks like a harmless normalization step is the dial that decides where learning happens and how strongly.
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math.PR 2026-07-01

SVGD particles stay close to mean-field limit for all time

by Krishnakumar Balasubramanian, Sayan Banerjee +1 more

Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent

Cutoff and moment-closure arguments give log or N^{-1/2} rates that hold uniformly rather than only at short times.

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We study uniform-in-time propagation-of-chaos for continuous-time Stein Variational Gradient Descent (SVGD). Classical finite-time propagation-of-chaos estimates for mean-field systems typically deteriorate rapidly with time and therefore do not directly explain the long-time relation between the finite-particle system and its mean-field limit. We obtain two complementary classes of uniform-in-time propagation-of-chaos results. For broad distributional metrics, we introduce a cutoff strategy which combines finite-time propagation-of-chaos estimates up to an $N$-dependent horizon with independent quantitative long-time convergence estimates for the finite-particle and mean-field SVGD flows. This yields uniform-in-averaging-time propagation-of-chaos bounds in Langevin kernel Stein discrepancy, Wasserstein-1 distance, and Wasserstein-2 distance, with logarithmic or iterated-logarithmic rates depending on the metric, target and kernel class. We also develop a finite-dimensional theory for matrix-valued finite-rank kernels. For Gaussian targets with bilinear kernels, the SVGD dynamics close exactly on first and second moments, yielding genuine uniform-in-physical-time parametric propagation-of-chaos rates in finite-dimensional Stein-feature metrics. We then prove a conjugacy principle showing that these feature-level estimates transfer to conjugate target-kernel pairs under orientation-preserving diffeomorphisms, thereby extending the theory to broad classes of nonlinear, including multimodal, targets. Together, these results highlight the contrast between generic distributional metrics, for which our general approach yields logarithmic rates, and closed finite-dimensional Stein observables, for which parametric $N^{-1/2}$ propagation-of-chaos rates persist uniformly in time.
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math.OC 2026-07-01

Random reshuffling beats SGD for any reasonable stepsize

by Zijian Liu

Random Reshuffling Dominates Stochastic Gradient Descent

New proof removes the 1/n stepsize limit and epoch threshold that previously made theory favor standard SGD over RR.

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Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$). A particularly popular strategy in $\textsf{Shuffling SGD}$ is Random Reshuffling ($\textsf{RR}$), which has achieved great empirical success across numerous experiments. Despite its strong performance, $\textsf{RR}$ has long been considered a heuristic due to a lack of theoretical support. Over the last decade, people have finally established provable convergence rates for $\textsf{RR}$, thus justifying its observed superiority. However, for smooth convex optimization, two clouds over the convergence theory of $\textsf{RR}$ remain to this day. More precisely, according to the current theory, $\textsf{Shuffling SGD}$ under $\textsf{RR}$ converges only when the stepsize is smaller than a threshold proportional to $1/n$, where $n$ is the number of summands in the objective (or the number of data points). Consequently, the optimally tuned theoretical rate of $\textsf{Shuffling SGD}$ under $\textsf{RR}$ is strictly worse than that of $\textsf{SGD}$ when the number of epochs is smaller than another threshold proportional to $n$. These two restrictions heavily limit the applicability of existing theories and leave a critical mismatch with practice. In this work, for the first time, we prove that $\textsf{RR}$ dominates $\textsf{SGD}$ in smooth convex optimization under any reasonable stepsize after any finite number of epochs, thereby addressing a longstanding open question.
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cs.LG 2026-07-01

Signed gauges recover 91% of RMSNorm coordinates across fine-tunes

by John Sweeney

Signed-Permutation Coordinate Transport for RMSNorm Transformers

Composing local B_d gauges along trajectories preserves SAE features and steering effects that permutation matching destroys.

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Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge, which we show is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$ (up to a global sign flip), while RMSNorm charts with generic per-channel gain have signed-permutation gauge $B_d = S_d \ltimes \{\pm 1\}^d$. Permutation-only alignment is therefore symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp failure mode: with decorrelated coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling at the positive-sign fraction of the true gauge, which sign-marginalization removes. We then make coordinate-preserving transport, not function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE 0.004 under $B_d$ versus 1.08 under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$; coordinate-preserving merges behave the same way. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state follows a different one from a functionally identical checkpoint. Finally, gauge-sweep audits show index-level interpretability claims are reproducible only relative to an explicit gauge.
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0
stat.ML 2026-07-01

ALO cuts conformal prediction runtime while matching coverage

by Jiachen Cong, Jingbo Liu

Accelerating Conformal Prediction via Approximate Leave-One-Out

Approximate leave-one-out estimators deliver asymptotic coverage and efficiency at far lower cost than exact refits.

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While conformal prediction provides a general framework for uncertainty quantification in predictive inference, its application is often limited by computational cost. Recent methods, including Jackknife+ and Jackknife-minmax, achieve faster computation by trading a slight loss of efficiency relative to full conformal prediction, but still requires computing leave-one-out refits for all observations. In this paper, we further accelerate conformal prediction by incorporating approximate leave-one-out (ALO) estimators, and establish asymptotic coverage and efficiency. While our proof draws on methods developed for analyzing the consistency of ALO cross-validation risk estimators in high-dimensional statistics, it requires adaptations to handle conformal prediction, where leave-$i$-out residuals are needed for predictions at $x_{n+1}$ rather than just at the training covariate $x_i$. Simulation results validate our theoretical findings, showing that the ALO-based methods achieve coverage and efficiency comparable to the exact methods, while significantly reducing the runtime.
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0
cs.LG 2026-07-01

Tabular foundation models calibrate conformal intervals on energy graphs

by Keivan Faghih Niresi, Alice Cicirello +1 more

Relational and Sequential Conformal Inference for Energy Time Series over Graphs via Foundation Models

STOIC reshapes STGNN residuals for zero-shot calibration and beats baselines on electricity and heating networks.

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Accurate energy demand forecasting is essential for the reliable operation and planning of modern sustainable energy systems. Spatial-temporal graph neural networks (STGNNs) have recently achieved strong performance in point forecasting by jointly modeling temporal dynamics and relational dependencies across interconnected energy nodes. However, in real-world energy systems, accurate point forecasts alone are insufficient, as operators also require reliable uncertainty estimates to support risk-aware decision-making, grid stability, and operational planning under uncertainty. Conformal prediction provides a principled and model-agnostic framework for uncertainty quantification with statistical coverage guarantees, making it particularly attractive for safety-critical energy applications. However, existing conformal prediction approaches often fail to fully capture the complex spatial-temporal structure of energy systems. To address these limitations, we propose STOIC (Spatial-Temporal Graph Conformal Prediction with In-Context Learning), a novel framework that integrates graph-based forecasting with the zero-shot calibration capabilities of tabular foundation models. STOIC first generates point forecasts using an STGNN and subsequently reformulates spatial-temporal residuals into a tabular representation suitable for in-context learning. Leveraging a tabular foundation model, STOIC calibrates prediction intervals without task-specific retraining, effectively capturing both sequential and relational dependencies. We evaluate STOIC on five diverse benchmarks, including synthetic simulations as well as real-world electricity and district heating networks. Across all datasets, STOIC consistently outperforms existing conformal prediction baselines, delivering more reliable and robust uncertainty estimates for complex graph-structured energy time series.
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0
cs.LG 2026-07-01

First-order regret bounds now hold for MDPs despite unknown transitions

by Mingyi Li, Taira Tsuchiya +1 more

Policy Optimization Achieves Data-Dependent Regret Bounds in MDPs with Unknown Transitions

The new optimistic FTRL method adds a transition complexity term but matches prior adaptive performance and recovers polylog stochastic regr

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We study policy optimization for online episodic tabular Markov decision processes with unknown transition kernels, aiming for best-of-both-worlds guarantees together with data-dependent regret bounds. Recent work (Dann et al., 2023; Li et al., 2026) has shown that policy optimization can adapt to both adversarial and stochastic losses with first-order, second-order, and path-length bounds, but only under known transitions, leaving open whether such data-dependent guarantees are achievable by policy optimization when the transition kernel is unknown. We resolve this by developing a new algorithm based on optimistic follow-the-regularized-leader that attains these guarantees under unknown transitions. The key ingredient is a new design of optimistic $Q$-function estimators together with a data-dependent transition bonus that controls estimator bias through the loss-prediction error. Our analysis further identifies an unavoidable transition-dependent complexity term that captures the intrinsic cost of estimating the transition kernel. As a result, we obtain first-order, second-order, and path-length bounds with the transition-dependent complexity term while simultaneously achieving gap-dependent $\mathrm{polylog}(T)$ regret in the stochastic regime.
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0
math.ST 2026-07-01

Optimal split minimizes conformal prediction interval length

by Sayan Das, Bahram Yaghooti +2 more

On Optimal Data Splitting for Split Conformal Prediction

Analytical expressions give the training-calibration ratio that shortens intervals while keeping coverage

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Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.
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0
cs.LG 2026-07-01

Reverse KL regularization yields global convergence for SAIL

by Xudong Wu, Pangpang Liu +2 more

On the Convergence of Self-Improving Online LLM Alignment

The modified objective satisfies the PL condition inside a bounded region, delivering near-linear sample complexity for LLM alignment.

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The Self-Improving Alignment (SAIL) algorithm addresses distribution shift by reducing a bilevel formulation of the problem to an efficient, single-level method. Empirically, SAIL has demonstrated strong performance on this task. However, a formal analysis of its convergence properties has been lacking. We identify a key theoretical challenge: the standard SAIL objective function is not guaranteed to be strongly concave due to unfavorable properties of its Hessian. To address this limitation, we propose a regularized objective, SAIL-RevKL, which incorporates a reverse Kullback-Leibler (KL) divergence penalty to improve the optimization landscape. Our central theoretical contribution is to prove that this regularized objective satisfies the Polyak-Lojasiewicz (PL) condition within a bounded parameter space. We establish global convergence guarantees, achieving a near-linear sample complexity. We further validate the effectiveness and stability of SAIL-RevKL through empirical evaluations, demonstrating that it outperforms the vanilla SAIL on both MuJoCo benchmarks and LLM alignment tasks.
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0
cs.LG 2026-07-01

Limited adaptivity yields kappa-free regret for slate GLM bandits

by Tanmay Goyal, Sukruta Prakash Midigeshi +1 more

Contextual Slate GLM Bandits with Limited Adaptivity

Batched and rarely-switching algorithms reach O(Nd^{3/2}√T) and O(Nd√T) bounds under diversity while using only poly(N) time per round.

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We investigate the contextual slate bandit problem with generalized linear rewards under limited adaptivity. At each round, the learner is presented with $N$ sets of items, where each item is represented by a $d$-dimensional feature vector. The learner then constructs a slate by selecting one item per set; the resulting slate yields a scalar reward sampled from a Generalized Linear Model (GLM). We propose algorithms under two limited-adaptivity settings: (a) Batched and (b) Rarely-Switching. For the batched setting, we introduce B-SlateGLinCB, which partitions the time horizon into $\mathcal{O}(\log\log T)$ batches such that each batch's policy relies only on data from previous batches. For the rarely-switching setting, we propose RS-SlateGLinCB, which adaptively performs only $\mathcal{O}(Nd\log T)$ parameter updates. Under a diversity assumption on the item sequences, we prove that B-SlateGLinCB and RS-SlateGLinCB achieve regret bounds of $\mathcal{O}(Nd^{3/2}\sqrt{T})$ and $\mathcal{O}(Nd\sqrt{T})$, respectively. Notably, both bounds are independent of the non-linearity parameter $\kappa$ that is typically found to scale the regret of GLM bandit algorithms. Our algorithms are computationally efficient, requiring only $\text{poly}(N)$ time per round despite $2^{\Omega(N)}$ possible slates. Simulations show our algorithms outperform existing baselines with limited adaptivity and remain competitive with Slate-GLM-OFU, a fully adaptive state-of-the-art algorithm. Notably, a slightly modified B-SlateGLinCB empirically matches this baseline. Finally, we demonstrate strong performance in a practical in-context example selection task for language models.
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0
stat.ML 2026-07-01

Constrained k-means recovers true centers from magnitude-decaying MNAR data

by Xin Guan

MNAR-k-means: A k-means Clustering for Data Missing Not at Random with Magnitude-Decaying Probability

The method bounds imputation size so that estimated clusters converge to those of the complete data.

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The classical $k$-means clustering, based on distances computed from all data features, cannot be directly applied to incomplete data with missing values. A natural extension of $k$-means to missing data is to involve only the observed positions in clustering, which is equivalent to imputing missing values by corresponding cluster means. However, for data missing not at random (MNAR), since missingness is related to data values, such a mean-imputation-based method may lead to the distortion of estimated cluster centers, resulting in a poor clustering result. Since MNAR mechanisms are very common in reality, it is necessary to improve the performance of $k$-means-based clustering methods for such data. In this paper, we focus on a magnitude-decaying MNAR scenario where data is more likely to be missing at positions with smaller absolute values, and we propose a novel $k$-means clustering method based on the constraint of the size of imputation values, which enjoys a good mathematical interpretation. Moreover, we establish the statistical consistency of the estimated cluster centers of the proposed method to the true cluster centers of fully observed data, and solve the optimization of the proposed loss function via an alternative minimization algorithm. Simulation experiments verify the effect of the proposed method in improving clustering results and reducing the bias of estimated cluster centers. Applications to real-world missing data further show the utility of the proposed method.
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0
cs.LG 2026-07-01

Algorithm recovers graph from one Glauber trajectory without mixing

by Eric Shen, Tony Wu +2 more

Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing

Polynomial-time recovery succeeds with trajectory length independent of mixing time under d-sparsity and minimum edge strength.

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We study the task of learning the structure of a $d$-sparse Gaussian graphical model on $n$ variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-$n$ sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.
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0
cs.LG 2026-07-01

Tabular in-context learners match on protein fitness with fixed reps

by Davy Guan, Lu Zhang +8 more

Can Tabular In-Context Learners Generalize to Biomolecular Property Prediction?

They stay competitive on ProteinGym using ESMC but performance on molecules hinges on descriptor choice.

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Predicting biomolecular properties from limited labeled data is a central bottleneck in protein engineering and small-molecule design. As strong pretrained encoders now supply rich fixed-length representations, the difficulty has shifted from representation learning to building a data-efficient predictor for the few-shot regime. Tabular foundation models such as TabPFN3 and TabICL are unlikely candidates for this role: they are in-context learners pretrained on synthetic tables drawn from random causal graphs, a generative prior with no obvious correspondence to the processes that produce protein sequences or molecular graphs. That this tabular, causal inductive bias should transfer to biomolecular data at all is unintuitive, yet we find it does. Treating each method as a predictor-representation pair, we evaluate across two domains. Over a fixed ESMC representation, tabular in-context learning is consistently competitive for protein fitness regression on ProteinGym and a diverse esterase dataset. For small-molecule classification with ECFP/RDKit descriptors, no single pairing dominates across TDC ADMET, MoleculeNet, FS-Mol, and DrugOOD; representation choice becomes a primary determinant, as expected when the predictor's own prior is indifferent to molecular structure. We conclude that tabular foundation models are strong performers on biomolecular prediction tasks, but that their performance depends strongly on the sequence or molecular representation used.
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0
stat.ML 2026-07-01

Hybrid swap cuts cost of dynamic GP inference on disjoint grids

by Rui-Yang Zhang, Lachlan Astfalck +3 more

Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

Vanilla-SPDE Exchange uses formulation equivalence to avoid inflating spatial dimension when observations and predictions do not coincide.

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Gaussian process inference is often limited by cubic computational costs, a challenge that becomes more pronounced in spatio-temporal settings where posterior inference is required over dense grids. While state-space SPDE formulations enable linear complexity in time, exact inference remains cubic in space and deteriorates further when observation locations are disjoint from the prediction locations, which inflates the number of considered spatial points. To address this, we propose the Vanilla-SPDE Exchange, which exploits an equivalence between the standard and SPDE formulations of GP inference to construct a hybrid scheme with improved computational cost. We demonstrate these gains through complexity analysis and numerical experiments.
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cs.LG 2026-07-01

Sparse tree chains match ensemble accuracy for most samples

by Zakk Heile, Hayden McTavish +2 more

Multistage Defer Trees for Hybrid Interpretability: If at First You Can't Succeed, Tree Again

Multistage defer trees route hard cases onward while classifying the rest with one or two simple trees.

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Recent work has shown that well-optimized individual decision trees can match complex black box models in some settings, primarily in noisy domains. For the remaining settings, however, complex ensembled compositions of trees often achieve higher accuracy at the cost of interpretability, leaving practitioners with difficult modeling decisions along an accuracy-interpretability tradeoff. Ideally, we would like to classify as much of the data as possible with one or a small number of trees, achieving interpretability for most samples while maintaining state-of-the-art accuracy. We introduce Multistage Defer Trees: a sequence of sparse decision trees that each make predictions for most samples, while deferring a small proportion to the next tree in the sequence or, ultimately, to a black box. We demonstrate that we can train this model class to match the performance of complex tree-based ensembles while routing most samples through only one or a small number of sparse decision trees. We discuss a range of techniques for training these models while maintaining simplicity. Our method expands the accuracy--interpretability frontier in settings where single-tree methods remain insufficient, demonstrating that even when complex models are necessary, they need not be fully opaque.
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0
stat.ME 2026-06-30

Dual-TV regularizers bound tensor error near minimax rate

by Wenfei Cao, Yang Chen +3 more

Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization

Upper bounds reach O(n3 rt sk^2 log / n) and close the gap to the lower bound by O(sk^2 / n) for exponential-family data.

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With the emergence of various tensor data, tensor completion from partial measurements has attracted widespread attention in data science and signal processing. Total Variation (TV) has been widely used as an effective regularization technique for tensor completion; however, theoretical studies on TV regularization in this context remain limited. In this work, we present a rigorous theoretical analysis of TV regularization for tensor completion. Specifically, we consider tensor completion under exponential-family noise, which generalizes the standard settings such as Gaussian and Poisson tensor completion. To handle exponential-family tensor completion, we propose a family of dual-TV (DTV) regularizers based on the transformed L1 function, which simultaneously capture sparsity and low-rank structures in the gradient tensor. Moreover, we establish the theoretical upper bounds on the recovery error of the proposed estimator. In certain cases, these upper bounds can attain the convergence order of $\mathcal{O}\big( n_3 r_t\big(\max_{k} s_k^2\big) \log\big((n_1+n_2)n_3\big) /n \big)$, and the minimax lower bound analysis is further presented to show that the upper-bounds can approach the lower bound with the gap of order $\mathcal{O}(\max_k s_k^2/max(n_1, n_2))$ up to a logarithmic factor. Finally, multiple groups of experiments on synthetic, image and video tensor data sets are conducted to support our theoretical results and demonstrate the effectiveness of our method.
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0
stat.ML 2026-06-30

SGD self-stabilizes at edge of stability for large steps

by Konstantinos Emmanouilidis, Lachlan MacDonald +2 more

SGD at the Edge of Stability: Stochastic Stabilization with Large Learning Rates

Stochasticity forces return to stable regime in fixed steps, enabling best-iterate convergence on cross-entropy loss.

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Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory. Most prior analyses of the edge of stability phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD self-stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.
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0
cs.LG 2026-06-30

Noisy experts force exponential horizon dependence for offline imitation

by Ved Sriraman, Peihan Liu +2 more

Behavior Cloning is Not All You Need: The Optimality of On-Policy Distillation for Noisy Expert Feedback

Online on-policy distillation recovers only polynomial dependence and explains OPD's advantage over SFT with imperfect teachers.

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Imitation Learning is a natural framework for learning in sequential decision-making systems and has emerged as the dominant paradigm through which we understand language model training. A central puzzle is that, while in theory offline IL can be horizon-free and optimal, in practice online methods such as on-policy distillation often outperform offline methods such as supervised fine-tuning. We propose a noisy expert model to explain this gap, in which the learner only has access to a noisy version of the expert's policy, but wishes to compete against the reward achieved by a clean expert, motivated by the fact that in many applications, e.g. training language models to perform long chains of thought, the expert is often imperfect. In this setting, we show a sharp separation between offline and online IL. Offline learning from noisy trajectories is fundamentally hard: to compete with the clean expert, the sample complexity must grow exponentially, in contradistinction to the clean expert setting where no explicit horizon dependence exists. In contrast, we prove that online interaction with the noisy expert via a novel variant of OPD enables polynomial dependence on the horizon in general. We further show that, under a natural condition on the expert noise distribution, which we show to be necessary for any horizon-free sample complexity, one can obtain such a guarantee, although our proposed algorithm sacrifices statistical efficiency in its dependence on the size of the policy class. Our analysis leads to an alternative loss function that is commonly considered empirically for LM training. We further provide algorithms and lower bounds, and extend our results to the more realistic setting of unknown corruption when the clean expert is deterministic, thereby providing a theoretical foundation for why OPD can outperform SFT when training language models from imperfect teachers.
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0
stat.ML 2026-06-30

Neural network forecasts next alternating event time

by Abigail Loe, Susan Murry +1 more

Dynamic Prediction of Alternating Recurrent Events via Neural Network

Inverse probability weighted pseudo-observations let the model handle censoring and dependence when predicting event-free intervals.

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Alternating recurrent events -- event-times of a specific nature that trigger a secondary refractory period -- occur in a wide-range of fields, including behavioral science, criminal justice, and biostatistics. Analysis of these events requires careful attention to the statistical nuance, including correlated observations and repeated outcomes subject to potential censoring. We develop an online dynamic prediction framework appropriate for predicting subsequent alternating recurrent events, by developing neural network theory for a statistical audiences and applying inverse probability weighted pseudo-observations. The proposed model is applied to dynamically predict alternating recurrent event-free time, showing good performance in simulation, and outstanding capability in application to predicting periods of low mood for first-year medical residents. We close with a discussion.
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0
cs.LG 2026-06-30

Random forest plateau tuning settles at ensemble size scaling as 1/ε²

by Andrey A. Dukhovny, Andrey M. Lange

A Stationary-Distribution Theory for Triplet-Based Plateau Search in Random Forest Ensemble-Size Selection

Stationary distribution of triplet search centers at O(ε^{-2}) with relative spread fixed by scale factor and update rule.

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The number of trees is a central computational parameter in Random Forests: increasing it reduces finite-ensemble variability but increases training and prediction cost. Plateau-based tuning adapts this parameter through local comparisons of out-of-bag scores at a geometric triplet of tree counts. After the remaining hyperparameters have stabilized, however, the central triplet point need not converge to a deterministic value; instead, it fluctuates around a stationary regime. This paper develops a stationary-distribution theory for this process. The central ensemble size $B_t$ is modeled as a birth-death Markov chain on a geometric grid, and its stationary distribution is derived through local balance. Under a leading centered folded-normal approximation, equilibrium equations are obtained for the original update rule and a symmetric modified variant, implying that the stationary center $B_*=O(\varepsilon^{-2})$ as $\varepsilon\downarrow 0$. The stationary spread is also characterized. A local Gaussian approximation and a Fokker-Planck interpretation give grid-level variance constants. After conversion to the ensemble-size scale, $\sigma_{B,*}=O(\varepsilon^{-2})$, while the variance is $O(\varepsilon^{-4})$. The leading relative spread is independent of $\varepsilon$ and controlled by the scale factor and update rule. These results interpret plateau-based Random Forest tuning as a stochastic process rather than a deterministic stopping rule.
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math.PR 2026-06-30

Matrix products converge to free log-normal spectral law

by Mufan Li, Jaume de Dios Pont +2 more

Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices

Sequential mean-field limits turn geometric Dyson motion into a Burgers equation whose solution is the free log-normal or its convolution, m

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We study the squared singular value spectrum of a product of non-square random matrices, a setting that also corresponds to the feature covariance eigenvalues of a deep linear neural network at initialization. We first take a proportional depth-width $d,n$ limit with the number of data points $m$ held fixed, and show that the resulting covariance eigenvalue process satisfies a geometric version of Dyson Brownian motion. We then take a second, sequential mean-field limit corresponding to the scaling $dm/n\to\bar\tau$, and show that the limiting $T$-transform of the spectrum solves a Burgers equation. In the identity-start case this equation yields the free log-normal law, and the general limit is obtained by free multiplicative convolution with the free log-normal. We further obtain the free log-normal support formula, a fixed-point iteration for numerical evaluation, and a formal small-time Marchenko--Pastur approximation. We also use the limiting spectral law to predict a toy random-feature regression risk, finding close agreement with a finite-dimensional simulation.
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0
stat.ML 2026-06-30

Extended counting theory identifies scattering network design factors

by Konstantin Häberle, Helmut Bölcskei

Separation Capacity of Scattering Networks

The number of binary label assignments realizable by a scattering network is controlled by its wavelet, layer, and pooling choices.

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In this paper, we attempt to enhance the theoretical understanding of convolutional neural networks (CNNs) as feature extractors in classification tasks by analyzing them through the lens of Cover's function-counting theory. Specifically, our focus lies on the notion of separation capacity, a combinatorial quantity derived from counting the number of realizable dichotomies (i.e., binary label assignments). Our contributions are threefold. First, we extend Cover's framework by establishing a conceptually insightful and practically useful formulation for the separation capacity. Second, leveraging this formulation, we identify the factors governing the separation capacity of feature extractors that employ a specific CNN architecture, so-called scattering networks, in terms of their network building blocks. Third, we provide practical insights for scattering network design.
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cs.LG 2026-06-30

GRPO training follows a closed-form damped oscillator

by Rajat Ghosh, Datta Nimmaturi +5 more

Predictable GRPO: A Closed-Form Model of Training Dynamics

Mean-field reduction fixes mass, damping, and stiffness from hyperparameters plus one curvature scale, yielding group-size invariance and st

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We develop a first-principles reduced-order model of these dynamics. Under a single mean-field assumption that summarizes the policy by its expected reward, we reduce the GRPO update to a stochastically-forced damped oscillator whose mass, damping, and stiffness are fixed in closed form by the optimizer hyperparameters together with a single measured curvature scale -- momentum supplies the inertia, off-policy lag erodes the damping, and the group size enters, to leading order, as a noise temperature. The reduction has three consequences. First, it subsumes the empirical single-exponential saturation law as its overdamped limit, recasting the fitted plateau, timescale, and size exponent as the fixed point, inverse stiffness, and curvature-scaling exponent of the underlying potential, and adding, through the retained inertial term, the slow-start phase the single exponential cannot represent. Second, it yields predictions tied to independently measurable quantities rather than fitted ones: group-size invariance of the deterministic trajectory with a $1/G$ stationary fluctuation, a sharp stability threshold in the refresh interval, and an overdamped-to-oscillatory transition. Third, it furnishes diagnostics that separate failure modes a reward curve alone conflates -- reward hacking, advantage degeneracy, policy concentration, and dynamical instability. Across three models and two group sizes, the closed-form trajectory fits training reward to $R^2 \geq 0.91$ and the mean trajectory is group-size invariant to leading order -- on both the reward curve and out-of-distribution transfer to eight math benchmarks -- while the within-group reward spread retains a residual $G$-dependence that the leading-order temperature picture does not capture.
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0
cs.LG 2026-06-30

Conservative offline training increases reward hacking in online adaptation

by Subramanyam Sahoo, Aman Chadha +2 more

Pessimism's Paradox: Conservative Offline Training Amplifies Reward Hacking During Online Adaptation in Reasoning Models

Higher beta values enlarge Goodhart gaps during adaptation despite closer data support, pointing to a need for calibrated rather than maxima

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Conservative offline training is widely advocated as a safe foundation for subsequent online adaptation: if a policy stays close to well-supported behaviour, the argument goes, it is less likely to exploit imperfections in a learned reward model. We challenge this intuition empirically and mechanistically. We train a Qwen3-14B policy under Direct Preference Optimisation (DPO) with three levels of conservatism ($\beta \in \{\beta_{\mathrm{lo}}, \beta_{\mathrm{mid}}, \beta_{\mathrm{hi}}\}$ derived from empirical log-ratio percentiles), then adapt each checkpoint online against a learned reward ensemble (3\,$\times$\,Qwen3-1.7B) while measuring true performance on GSM8K exact-answer accuracy. We find that \emph{higher offline conservatism monotonically increases reward-hacking damage}, measured by the Goodhart gap and its area under the curve (AUGC), with Spearman $\rho = 1.0$ across all three conditions. Mechanistic analysis reveals a three-link causal chain: (i) high-$\beta$ DPO compresses policy entropy, (ii) Low-entropy policies generate responses with reduced diversity, concentrating in a narrow region of the reward model's training distribution (lower pairwise cosine distance), and (iii) despite this proximity, ensemble disagreement (epistemic uncertainty) increases with $\beta$ and is exploited faster during online optimisation. We further fit a power-law curve to the $(\beta, \augc)$ data and identify a practical optimal conservatism level $\beta^{\star}$ that balances alignment fidelity against hacking vulnerability. Our results suggest that the field needs \emph{calibrated}, not \emph{maximal}, conservatism.
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stat.ML 2026-06-30

Norms encode semantic specificity in contrastive embeddings

by Ziwei Su, Junyu Ren +1 more

Optimization Dynamics Imprint Semantic Specificity in Contrastive Embedding Norms

Scale-invariant contrastive training imprints concept specificity and frequency into embedding lengths as a natural byproduct.

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Contrastive embedding models trained with scale-invariant losses are typically paired with distance metrics like cosine similarity, effectively ignoring embedding magnitudes. However, surprisingly, empirical studies reveal that despite this, these "discarded" norms seem to correlate with semantic properties such as concept specificity, token frequency, and human uncertainty. In this work, we provide a formal theoretical framework explaining this phenomenon. By analyzing the optimization dynamics, we derive an analytic formula demonstrating that embedding length naturally encodes this information as a byproduct of the training process. We also show how this gives rise to signals that can serve as "free" calibration tools in specific models and retrieval tasks, providing a grounded explanation for a previously heuristic observation.
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cs.LG 2026-06-30

Any valid transport map is as hard to estimate as the OT map

by Sivaraman Balakrishnan

The Fundamental Limits of Valid Transport Map Estimation

Stability assumptions make minimax lower bounds identical for all valid maps, including those produced by diffusion and flow-matching models

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Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.
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cs.LG 2026-06-30

Continual learning reduces to sequential projections in homogeneous nets

by Matan Schliserman, Gon Buzaglo +2 more

Convergence of Continual Learning in Homogeneous Deep Networks

Weak regularization frames updates as projections onto task margins, showing local convergence is possible but global is not.

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We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.
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cs.LG 2026-06-30

Faster updates align covariance matrices under Wasserstein distance

by Woojoo Na, Jennifer Dy

ITSPACE: Monotone Gaussian Optimal Transport Updates

ITSPACE uses closed-form proximal steps on square-root factors to guarantee descent and outperform gradient methods on alignment benchmarks.

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Covariance matrices serve as compact descriptors of feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings. Under a centered Gaussian approximation, the unregularized Wasserstein-2 optimal-transport (OT) discrepancy admits a closed form on covariances given by the Bures-Wasserstein (BW) objective on the symmetric positive definite (SPD) cone. We propose ITSPACE (Iterative Transport for Stable Proximal Alignment of Covariance Embeddings), a proximal majorization-minimization method that directly optimizes this exact BW objective through closed-form updates in a square-root factorization. In exact arithmetic, each iteration satisfies a sufficient-decrease inequality for the BW objective; under inexact polar computations, we provide an explicit certificate-gap bound controlling deviations from exact descent. The resulting iterations preserve PSD structure by construction and naturally support rank-restricted factors, making ITSPACE well-suited as a lightweight inner-loop primitive in settings where adaptation must be performed from unlabeled target batches under strict step and compute budgets. Across real-world covariance-alignment benchmarks, ITSPACE reaches low-BW-gap solutions substantially faster than BW-gradient descent, methods based on other covariance geometries, and entropically regularized sample-OT baselines.
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stat.ML 2026-06-30

DR-ACI gives prediction intervals for causal effects under time dependence

by Andreas Koukorinis, Ricardo Silva

Doubly Robust Adaptive Conformal Inference for Causal Effects Under Temporal Dependence

The method applies adaptive conformal inference to doubly robust pseudo-outcomes to maintain coverage when observations are serially depende

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We propose doubly robust adaptive conformal inference (DR-ACI), which constructs prediction intervals for doubly robust pseudo-outcomes under temporal dependence.
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stat.ML 2026-06-30

Factorizable flows learn each parameter's density effect in isolation

by Davide Valsecchi, Mauro Donegà +1 more

Factorizable Normalizing Flows for parameter-dependent density morphing

Effects are summed at inference to handle any combination without sampling the full joint space, keeping cost linear in the number of parame

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Normalizing Flows excel at modeling a single fixed density, yet many problems across the sciences, such as high energy physics, instead require modeling how that density deforms as a function of continuous parameters: the strength of a physical effect, a calibration constant, or a source of systematic uncertainty. Learning a separate flow for every parameter configuration quickly becomes intractable, since the number of joint settings grows exponentially with the number of parameters. We introduce Factorizable Normalizing Flows (FNFs), which represent the parameter-dependent density as a fixed, high-fidelity flow for a reference configuration composed with a learnable transformation that is polynomial in the parameters and factorized over them. This structure has a practical consequence: each parameter's effect is learned in isolation, from samples in which that parameter alone is varied. The combined response of many parameters is then recovered by summation at inference, without ever sampling their combinatorially large joint space. On a controlled problem with two interpretable deformations applied jointly to the data, the learned transformation reproduces the true deformations and matches the optimal likelihood, while optional interaction terms capture residual correlations when several parameters vary strongly at once. The resulting model is interpretable, scales linearly with the number of parameters, and keeps the likelihood tractable. This provides a general tool for any inference workflow requiring continuous density morphing, and directly enables the next generation of unbinned likelihood fits in high energy physics.
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stat.ML 2026-06-30

Causal drift recovered from equilibrium snapshots

by Richard Schwank, Mathias Drton

Non-parametric recovery of causal diffusion mechanisms from steady-state observations

Proves non-parametric identification of the full drift function when the acyclic graph is known and supplies a consistent kernel estimator.

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We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.
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cs.LG 2026-06-30

Curvature-weighted noise halves SGD error floor on quadratics

by Muhammad Hamza (1), Ayush Goel (1) ((1) Indian Institute of Technology Kharagpur)

Curvature-Weighted Gradient Diversity: A Noise Measure for Geometry-Adaptive SGD Schedules

A geometry-aware measure lets the cosine schedule adapt to directional curvature, cutting final error by up to 50 percent in diagonal quadra

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The standard convergence analysis of mini-batch stochastic gradient descent (SGD) models gradient noise using a single variance term that treats all parameter directions equally, ignoring the fact that noise in high-curvature directions has less impact because learning rates are already constrained there. We introduce Curvature-Weighted Gradient Diversity (CWGD), a geometry-aware measure that weights per-sample gradient diversity by the inverse square root of the Hessian, providing a tighter proxy for the effective optimization noise. For strongly convex quadratic objectives with diagonal Hessians and isotropic noise, we prove that a CWGD-modulated cosine learning-rate schedule can reduce the asymptotic optimization error floor by up to a factor of two compared with standard cosine annealing. We implement this idea as CWGD-Cosine using a Hutchinson-based diagonal Hessian estimator that is exact for quadratic objectives. Across a range of condition numbers, batch sizes, and noise structures, CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine annealing while incurring negligible overhead in the quadratic setting. We also identify and correct a degenerate curvature estimator, analyze the robustness of the proposed estimator, and explicitly discuss the limitations of the method, including Hessian staleness in non-convex optimization. These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.
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stat.ML 2026-06-30

SGD learns spurious shortcut exponentially fast before XOR signal

by Tyler LaBonte, Vidya Muthukumar

SGD Provably Prioritizes a Shortcut Spurious Feature in the XOR Model

Theory for two-layer ReLU networks on Boolean data shows optimization first grows the linear correlation then suppresses the quadratic signa

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Neural networks are known to be susceptible to over-reliance on spurious correlations. However, the precise mechanism by which models exploit shortcut features is not fully understood, and algorithms to mitigate this behavior rely on as yet unjustified assumptions about the learned representations. In this work, we provide the first end-to-end theoretical characterization of spurious feature learning for two-layer ReLU neural networks trained by online minibatch SGD on the logistic loss. We consider data drawn from the high-dimensional Boolean hypercube with a quadratic signal function (namely XOR) and a linear spurious correlation. We show that SGD learns the spurious feature first, and exponentially fast. Moreover, the optimization dynamics couple the spurious and signal features, with a stronger spurious component inhibiting signal feature learning. Our analysis reveals precise phase transitions in the learning dynamics. In the first phase, alignment between the signs of the spurious feature and second-layer weight drives rapid growth of the spurious feature. In the second phase, large majority group margin slows learning and the signal feature remains suppressed. When the spurious correlation is maximally strong, we show theoretically that the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation (i.e., if the spurious feature was absent). In contrast, when the correlation strength is constant, we provide preliminary empirical evidence that the model can eventually learn the XOR signal, although the spurious feature is not forgotten.
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stat.ML 2026-06-30

Shell-core geometry produces grokking scaling laws

by Róisín Luo, Christian Gagné +3 more

A Stochastic--Geometric Theory of Scaling Laws in Grokking

Stopping-time analysis on the manifolds of memorization and generalization solutions predicts how learning rate, batch size and regularizati

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Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.
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stat.ML 2026-06-30

Extrapolation beats single-nugget solves for ill-conditioned systems

by Disha Hegde, Jon Cockayne +1 more

Extrapolating from Regularised Solutions for Solving Ill-Conditioned Linear Systems in Machine Learning

Richardson extrapolation across multiple regularized linear solves improves accuracy and stays compatible with automatic differentiation for

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Rapid prototyping of algorithms is a critical step in modern machine learning. Most algorithms exploit linear algebra, creating a need for lightweight numerical routines which -- while potentially sub-optimal for the task at hand -- can be rapidly implemented. For the numerical solution of ill-conditioned linear systems of equations, the standard solution for prototyping is Tikhonov-regularised inversion using a nugget. However, selection of the size of nugget is often difficult, and the use of data-adaptive procedures precludes automatic differentiation, introducing instabilities into end-to-end training. Further, while data-adaptive procedures perform multiple linear solves to select the size of nugget, only the result of one such solve is returned, which we argue is wasteful. This paper aims to circumvent the above difficulties, presenting autonugget; a Python package for automatic and stable numerical solution of linear systems suitable for rapid prototyping, and fully compatible with automatic differentiation using JAX. autonugget combines multiple linear solves using Richardson extrapolation to determine the solution of the ill-conditioned system, improving in accuracy over approximations based on a single nugget.
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stat.ML 2026-06-30

CDFs replace sorting for sliced Wasserstein distance

by Christophe Vauthier, Quentin Mérigot +1 more

Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions

New estimators compute the distance from projected cumulative distributions, allowing parallelism and federated aggregation without raw data

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The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.
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cs.LG 2026-06-30

Level sets yield exact acceptance certificates for speculative decoding

by Aaryam Sharma

When Is a Draft Accepted? A Theory of Acceptance in Speculative Decoding

Rejection regions of greedy and relaxed rules coincide with lower level sets of the target distribution, producing KL and margin bounds that

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Speculative decoding accelerates language model inference by using a fast drafter to propose candidate tokens that are then verified by a larger target model. Existing theory largely studies the stochastic, distribution-preserving setting, where the goal is to exactly sample from the target distribution. In contrast, many practical systems use greedy decoding, relaxed acceptance rules, or tree-based candidate sets, where success is governed by local ranking and threshold events rather than exact distributional equality. We develop a theory for these regimes. We identify that many common acceptance criteria have rejection regions that can be characterized as lower level sets of the target distribution. For these, we characterize the exact KL divergence required for rejection yielding exact certificates and sharp margin-based bounds for strict greedy decoding, additive and multiplicative relaxed acceptance, top-(m) relaxed criteria, and entropy-thresholded acceptance. We then extend the framework to greedy tree decoding, deriving exact and margin-only certificates for when the target greedy token remains covered by the drafter's top-(m) candidates. Finally, we evaluate the resulting certificates on Qwen3 models, showing that relaxed and tree-based criteria substantially enlarge the region of certified acceptance, especially on decoding steps with low target model distribution margin. These results complement existing distribution-preserving analyses of speculative decoding by characterizing the deterministic local acceptance events common in practical inference systems.
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cs.LG 2026-06-30

TabPFN v2 leads NHANES benchmark for HbA1c and CRP

by Federico Felizzi

Accelerometry-Derived Digital Biomarkers for Cardiometabolic Risk: A Population-Representative Tabular Benchmark with Uncertainty Quantification

Triglycerides stay unpredictable and 90% intervals under-cover some demographic groups.

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Structured tabular data dominates clinical medicine, yet existing benchmarks fail to reflect real-world properties like complex survey sampling, demographic oversampling, and subgroup fairness. We introduce the NHANES Accelerometry Cardiometabolic Benchmark, derived from NHANES 2003-2006, comprising 1,381 adults with hip-worn accelerometry, fasting laboratory biomarkers, dietary intake, and anthropometrics. We evaluate three tabular learning methods -- ridge regression, XGBoost, and the foundation model TabPFN v2 -- to predict glycated haemoglobin (HbA1c), fasting triglycerides, and C-reactive protein (CRP) from activity phenotypes and lifestyle covariates. TabPFN v2 achieves the best overall performance (HbA1c R^2=0.156, CRP R^2=0.383), while triglycerides remain largely unpredictable (R^2 < 0.05), consistent with known genetic dominance. We apply split conformal prediction to generate distribution-free 90% prediction intervals and evaluate demographic coverage equity across sex and race/ethnicity subgroups. Marginal coverage aligns with the 90% target for CRP and HbA1c but falls below for triglycerides. At the subgroup level, we observe localized undercoverage (e.g., HbA1c for Mexican American participants), illustrating the gap between marginal guarantees and the conditional coverage required for clinical fairness. Code and data are at https://github.com/felizzi/nhanes-accel-cardiometabolic-benchmark.
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