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arXiv2026

Robust Streaming PCA

We consider streaming principal component analysis when the stochastic data generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world datasets.

stat.ML↗

arXiv2026

DLM-One: Diffusion Language Models for One-Step Sequence Generation

This paper introduces DLM-One, a score-distillation-based framework for one-step sequence generation with continuous diffusion language models (DLMs). DLM-One eliminates iterative refinement by aligning the scores of a student model's outputs with the score function of a pretrained teacher DLM in the forward-diffused noisy space. We demonstrate that our framework is architecture-agnostic and robust across diverse continuous manifolds, including standard token embedding spaces and logit simplex spaces. Through experiments on multiple representative DLMs, we show that DLM-One achieves up to $\sim$2000$\times$ speedup in sampling steps and $\sim$500$\times$ in wall-clock time, while maintaining competitive performance on benchmark text generation tasks. We further analyze failure modes in language-domain diffusion distillation and propose an adversarially-regularized two-stage training scheme to prevent student degeneration. Our findings position one-step score distillation as a viable path for the efficient deployment of continuous diffusion models operating in continuous space for natural language processing.

cs.CL↗

arXiv2026

Neural Variational Cut Posteriors without Upstream Data

In many applications, one must propagate parameter uncertainty from an earlier (upstream) analysis, available as samples, to subsequent (downstream) analyses without feedback. This problem is called cutting feedback or cut-Bayes, and the cut-posterior, the optimal posterior preserving information-flow constraints, is well characterized. However, sampling from it (e.g., via nested MCMC) is computationally intensive, while existing variational inference methods for cut-Bayes require access to upstream data and model, often unavailable. We propose a modular and provably accurate cut-Bayes approach requiring no access to upstream data or model. We leverage the characterization of the cut-posterior as the minimizer of the expected downstream conditional Kullback-Leibler divergence over the upstream posterior, replacing the expectation with the sample average over upstream draws. Our method, NeVI-Cut (neural variational inference for cut-Bayes), employs conditional normalizing flows as the variational family for downstream parameters. We provide fixed-data convergence rates of NeVI-Cut in terms of the richness of neural architecture and complexity of the cut-posterior. We establish, to our knowledge, first results on uniform Kullback-Leibler approximation rates of conditional distributions by common flow classes, yielding widely applicable fixed-data error rates for variational flows. A stochastic algorithm implements NeVI-Cut efficiently, and we demonstrate its speed and accuracy on multiple applications.

stat.ML↗

arXiv2026

Gradient Prediction with Control Variates in the Cheap-Forward Regime

We study whether otherwise-idle inference resources could reduce the scarce-GPU cost of training. Our analysis uses a simulated compute ledger in which fleet work is billed at a fraction of a scarce-GPU forward; all experiments run on a regular GPU. Our algorithm predicts gradients with a reduced-precision, inference-style reverse-mode program and combines many predictions with a few exact gradients through a control variate, so approximation error becomes variance rather than bias. On a 124M-parameter language model and selected short training windows, the method can lower simulated ledger cost relative to the tested baselines when fleet work is sufficiently cheap. Experiments spanning 10M-774M parameters show both transfers and failures. We do not test inference-only hardware, end-to-end distributed latency, or a full optimizer-by-batch-size baseline sweep.

cs.LG↗

arXiv2026

What Drives Success in Physical Planning with Joint-Embedding Predictive World Models?

A long-standing challenge in AI is to develop agents capable of solving a wide range of physical tasks and generalizing to new, unseen tasks and environments. A popular recent approach involves training a world model from state-action trajectories and subsequently use it with a planning algorithm to solve new tasks. Planning is commonly performed in the input space, but a recent family of methods has introduced planning algorithms that optimize in the learned representation space of the world model, with the promise that abstracting irrelevant details yields more efficient planning. In this work, we characterize models from this family as JEPA-WMs and investigate the technical choices that make algorithms from this class work. We propose a comprehensive study of several key components with the objective of finding the optimal approach within the family. We conducted experiments using both simulated environments and real-world robotic data, and studied how the model architecture, the training objective, and the planning algorithm affect planning success. We combine our findings to propose a model that outperforms two established baselines, DINO-WM and V-JEPA-2-AC, in both navigation and manipulation tasks. Code, data and checkpoints are available at https://github.com/facebookresearch/jepa-wms.

cs.AI↗

arXiv2026

Non-Stationary Functional Bilevel Optimization

Functional bilevel optimization (FBO) provides a powerful framework for hierarchical learning in function spaces, yet current methods are limited to static offline settings and perform suboptimally in online, non-stationary scenarios. We propose SmoothFBO, the first algorithm for non-stationary FBO with both theoretical guarantees and practical scalability. SmoothFBO introduces a time-smoothed stochastic hypergradient estimator that reduces variance through a window parameter, enabling stable outer-loop updates with sublinear regret. Importantly, the classical parametric bilevel case is a special reduction of our framework, making SmoothFBO a natural extension to online, non-stationary settings. Empirically, SmoothFBO consistently outperforms existing FBO methods in non-stationary hyperparameter optimization and model-based reinforcement learning, demonstrating its practical effectiveness. Together, these results establish SmoothFBO as a general, theoretically grounded, and practically viable foundation for bilevel optimization in online, non-stationary scenarios.

stat.ML↗

arXiv2026

Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings

Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor (QMLP) is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We derive non-asymptotic performance guarantees for QMLP in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs. We also consider the related problem of quantum information projection and generalize the quantum Pythagorean theorem to mixture families specified by possibly non-self-adjoint linear constraints. We further show that the Pythagorean inequality continues to hold in the infinite-dimensional setting whenever the convex information-projection problem attains a finite minimum.

cs.IT↗

arXiv2026

Stabilizing Private LASSO under Heterogeneous Covariates via Anisotropic Objective Perturbation

We study high-dimensional LASSO under differential privacy via objective perturbation with heterogeneous covariate scales. In practical scenarios, covariates often exhibit diverse scales; however, standard preprocessing is problematic under privacy constraints, as it consumes additional privacy budget. This heterogeneity induces effective anisotropy in the objective perturbation via the inverse Gram matrix of covariates, which can degrade the stability and accuracy of algorithms. To address this, we propose a Gram-based anisotropic objective perturbation, a ``pre-distortion" strategy that counteracts the distortion from the covariate structure to restore isotropy in the estimation process. Using an Approximate Message Passing (AMP) framework and state evolution analysis, we demonstrate that our proposed perturbation significantly stabilizes convergence and improves both statistical efficiency and privacy performance compared to standard uniform noise injection. Our results provide theoretical insights into designing stable and efficient private estimators without relying on data-dependent preprocessing.

stat.ML↗

arXiv2026

Connections between the Föllmer process and the denoising diffusion probabilistic model

The Föllmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an ``augmented'' time-compressed version of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, the connection between direct discretization of the Föllmer process and the DDPM sampler has not yet been fully explored. This paper clarifies this point while surveying relevant results from the literature. We show that discretized Föllmer processes give natural hyper-parameter settings of the DDPM sampler while accommodating a broader class of variance schedules than discretized reverse SDEs. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds, along with slight improvements.

stat.ML↗

arXiv2026

Theoretical Foundations and Effective Algorithms for Policy-Aware Simulator Learning

Model-based reinforcement learning (MBRL) agents typically learn world models by minimizing predictive loss. However, powerful RL optimizers inevitably exploit minor model inaccuracies, leading to simulator exploitation and a reality gap where policies succeed in simulation but fail in the real world. We propose that the objective for learning simulators should be strategic robustness rather than predictive accuracy, and formulate this as a zero-sum minimax game between a model player and an adversarial policy player. We provide a comprehensive theoretical analysis: (1) an online learning guarantee showing the game is learnable with sublinear regret bounds; (2) a tractable critic-based simplification bounding the global policy-value gap by the local critic's loss; and (3) an Error-MDP duality, proving that finding the worst-case policy is formally dual to a standard RL problem where the reward is the one-step critic error. This duality yields a provably convergent active data selection algorithm. Experiments on continuous control tasks demonstrate that our approach reduces prediction error in strategically important regions by $1.5$-$2.2\times$ and enables policies trained purely in simulation to match near-optimal real-world performance.

cs.LG↗

arXiv2026

SPACR: Single-Pass Adaptive Training of Uncertainty-Aware Conformal Regressors

Conformal Prediction (CP) provides robust uncertainty guarantees for predictive models, but is typically applied post hoc, which misaligns model training with the conformal goal of producing efficient (i.e., narrow) intervals. We propose SPACR (Single-Pass Adaptive Conformal Regressor), a novel method for directly training uncertainty-aware regressors within a differentiable loss. SPACR jointly optimizes accuracy, efficiency, and validity without batch-splitting or a predefined confidence level during training. As a result, a single SPACR model yields valid prediction intervals at multiple confidence levels during inference, avoiding the costly retraining required by methods like Directly Optimized Inductive Conformal Regression (DOICR). Experiments on diverse tabular and image datasets show that SPACR consistently gives tighter intervals and better coverage-efficiency trade-offs compared to standard CP and DOICR, while significantly reducing computational costs relative to retraining-dependent baselines.

cs.LG↗

arXiv2026

Breaking the Likelihood Trap: Variance-Calibrated Modulation for Large Language Model Decoding

In open-ended generation, LLMs frequently fall into the "likelihood trap", characterized by repetitive degeneration and vocabulary dullness, resulting in a discrepancy between machine-generated and human-written text. While post-hoc tail truncation (e.g., Top-p, Min-p) avoids sampling from the unreliable tail, it can misalign generation with human lexical preferences by over-sampling from the uncalibrated head; fixed scalar repetition penalties, in turn, ignore how the scale of the logit distribution varies across inference steps, which can disrupt semantic coherence. To address both shortcomings, we propose Variance-Calibrated Modulation (VCM), a training-free pre-decoding intervention. VCM directly reshapes the probability distribution prior to truncation via two dynamic mechanisms: (1) Contextual Searchlight via PMI, which naturally suppresses global stopwords and elevates context-evoked tokens, and (2) Adaptive Self-Debiasing, which utilizes real-time logit standard deviation to provide scale-invariant penalization. In experiments across open-ended generation, factual QA, and mathematical reasoning, we show that VCM consistently mitigates the likelihood trap. With negligible computational overhead, VCM integrates with existing decoding strategies, improving diversity and coherence and, particularly at higher decoding temperatures, reasoning accuracy. Our code is publicly available on GitHub: https://github.com/AetherDing/VCM

cs.CL↗

arXiv2026

Diagonal Multi-omics Integration of Heterogeneous Datasets

In this paper, we consider methods for the diagonal multi-omics integration of heterogeneous datasets. Several approaches to the nature of biological heterogeneity are analyzed and developed to comprehend more clearly the generated differences. Specifically, the extremal trace problems for the coupled Laplacian on sets homeomorphic to the Stiefel manifold embedded in the complex Euclidean space are investigated. The gradient ascent method for the maximization problem is elaborated in the classical terms of functional analysis, which is of significant interest in itself. On this basis, we introduce a novel characteristic of dataset heterogeneity by employing the norm of the difference between the maximum and minimum points.

stat.ML↗

arXiv2026

Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling

We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d^{-1}$, our general regret bound yields the best known regret bound of $O(d^{3/2}\sqrt{T}\log T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.

stat.ML↗

arXiv2026

HyperMC: Multi-Fidelity Hyperparameter Tuning for Stochastic Gradient MCMC

Stochastic gradient Markov chain Monte Carlo (SGMCMC) methods enable scalable Bayesian inference, but their performance depends strongly on hyperparameters such as the step size, mini-batch size, and number of leapfrog steps. Since most SGMCMC algorithms lack a Metropolis-Hastings acceptance rate, standard acceptance-based tuning methods are not directly applicable. We propose HyperMC, a multi-fidelity tuning framework that combines Hyperband-style resource allocation with kernel Stein discrepancy (KSD) evaluation. By running multiple successive-halving brackets, HyperMC balances broad exploration of a continuous hyperparameter space with increasingly accurate evaluation of promising configurations under a fixed computational budget. We further introduce Robust HyperMC, which uses global grid initialization followed by elite-guided local refinement to reduce sensitivity to random candidate generation and noisy finite-budget evaluations. Under suitable approximation and concentration conditions for the estimated KSD, we establish that the successive-halving component selects a near-optimal configuration among the sampled candidates with high probability and derive a sufficient computational budget for successful selection. Experiments on logistic regression, probabilistic matrix factorization, and Bayesian neural networks show that HyperMC improves posterior approximation or predictive calibration relative to MAMBA, grid search, and heuristic baselines, while Robust HyperMC yields more stable and reproducible tuning results.

stat.ML↗

arXiv2026

Online Non-Monotone DR-Submodular Maximization Matching the Offline $0.401$ Factor

We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $δ\in[0,1/4]$, batching gives $O(T^δ)$ calls per round and $O(T^{4/5-δ/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.

cs.LG↗

arXiv2026

Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation

Generative modeling directly on geometric manifolds can avoid errors introduced by flattening non-Euclidean data, repeated ambient projection, and coordinate inconsistency in Euclidean representations. Schrodinger bridges provide a probabilistic generative framework for entropy-regularized transport between prescribed endpoint distributions. We study Schrodinger bridges for kinetic dynamics on Lie group manifolds with state X_t = (g_t, xi_t) in G x g, allowing endpoint observations to constrain only the variables that are actually measured. In particular, the entropy projection determines the conditional law of the unobserved endpoint velocities. For the same observed endpoint bridge, we develop two computational realizations: Wrapped-Kernel Bridge Calibration (WKBC) uses an explicit periodized kinetic kernel on compact Abelian groups, whereas Reciprocal Conditional-Control Bridge Matching (RCCBM) handles compact non-Abelian groups through two-sided endpoint calibration and mollified conditional-control matching. The canonical teacher-mixture path law is itself a Markov reciprocal law, so forward generation uses a calibrated initial law and one learned Doob controller. Moreover, we establish a modular error bound in the bounded-Lipschitz path metric that provides a clean separation of errors due to endpoints, control regression, initialization, discretization, and related approximations. Experiments on multiple Lie group manifold datasets validate the feasibility and consistency of our proposed method, covering protein and RNA torsions, SO(3), U(n), and the Protein Conformational Transition Pathway Generation task using mdCATH trajectories in a compact reduced representation. The source code is publicly available at https://github.com/cafferyzhang12/Schr-dinger_Bridge_on_LieGroup.

stat.ML↗

arXiv2026

From topology learning to graph generation: A unifying perspective

Learning graph structures from data is a fundamental problem that spans a wide range of signal processing and machine learning tasks. While significant effort has been made to tackle the problem, existing research has largely evolved along two parallel directions. The first seeks to infer the topology of an individual graph from observations supported on it, whereas the second seeks to learn a generative distribution from observed graph instances, enabling the sampling of new graphs. This review presents a unified framework that connects these formulations by viewing them as inverse problems of a common generation process for graph data. We review the major methodologies within this framework, highlight their relationships, strengths, and limitations, and identify opportunities for integrating ideas across paradigms. By bridging graph topology learning and graph generation, this review provides a broader cross-disciplinary perspective on the field and outlines promising directions for future research.

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