Learning independence is fundamental to statistical inference and machine learning, yet learning independence between entire time series by a test remains challenging when only a single observation record is available. Existing guarantees often rely on structural or mixing assumptions, while stationarity and ergodicity alone do not permit consistent testing. In this work, we identify a class-level boundary governed by shared nontrivial Koopman eigenvalues, which represent common persistent oscillatory frequencies of the two processes. When no such eigenvalues are shared, we extend shift-based rank calibration with finite-window dependence witnesses and a data-adaptive threshold to construct a deterministic, weakly consistent test with finite-sample false-positive control, without quantitative mixing assumptions. Conversely, we show that no possibly randomized test can achieve vanishing false-positive probabilities under every independent jointly totally ergodic law while consistently detecting every dependent jointly totally ergodic law with a shared nontrivial eigenvalue. This impossibility already holds when the shared point spectrum is exactly the cyclic group generated by any single prescribed irrational eigenfrequency, identifying a spectral obstruction to universal learning of full-process independence.
Small area estimation (SAE) traditionally borrows strength across domains through a prespecified working model. We introduce small area learning (SAL), a novel learner-agnostic framework that formulates SAE as a multi-target semi-supervised learning problem and treats outcome prediction as a nuisance task. Areas may differ through unrestricted covariate shift and structured conditional shift, with information shared through common and area-specific components. We develop estimators with and without cross-fitting, accommodating low-complexity and flexible learners and unifying synthetic, model-based, and semi-supervised estimators. For area means, we establish a unified asymptotic theory for these estimator families, characterizing conditions for consistency, asymptotic normality and providing variance estimators and confidence intervals without prescribing a particular learning algorithm. We extend the theory to settings with only aggregated population covariates and characterize the resulting aggregation bias. SAL reveals clearly a bias-variance trade-off: partial residual correction in classical estimators can reduce mean squared error, whereas full correction yields estimators robust to the misspecification of working models, potentially at higher variance. Simulations yield RMSE reductions of 34%-49% relative to direct estimation, with the best configuration varying across settings. The numerical studies also illustrate how shrinkage can produce overly optimistic confidence intervals at the expense of strong modeling assumptions.
The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes. We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $\Theta(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove this result, we relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds in analogy to new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.
We propose a principled framework for unsupervised domain adaptation under covariate shift in kernel Generalized Linear Models (GLMs), encompassing kernelized linear, logistic, and Poisson regression with ridge regularization. Our goal is to minimize prediction error in the target domain by leveraging labeled source data and unlabeled target data, despite differences in covariate distributions. We partition the labeled source data into two batches: one for training a family of candidate models, and the other for building an imputation model. This imputation model generates pseudo-labels for the target data, enabling robust model selection. We establish non-asymptotic excess-risk bounds that characterize adaptation performance through an "effective labeled sample size", explicitly accounting for the unknown covariate shift. Experiments on synthetic and real datasets demonstrate consistent performance gains over source-only baselines.
Modern AI models are not static. They go through multiple updates in their lifecycles. We propose to design Sequential Membership Inference (SeMI) attacks leading to tighter privacy audits by exploiting the sequence of models and injecting a target canary at a controlled insertion time. First, for empirical mean computation, we develop SeMI*, an optimal SeMI attack to identify the presence of a target inserted at a specific insertion step. We derive the power of SeMI* to show that accessing the model sequence yields more powerful MI attacks than scrutinising only the final model. SeMI* exhibits an isolation property. Its power depends on the statistics obtained right before and after insertion of the target. Leveraging this insight, we develop practical white-box (accessing model gradients) and black-box (accessing loss) SeMI attacks against models trained with (DP-)SGD. Across datasets and models trained with (DP-)SGD, our experiments show that SeMI attacks achieve higher power than snapshot-independent baselines, and yield tighter privacy audits thanks to (a) control over the insertion time and (b) observations across the model sequence. Code is available at https://github.com/Thomick/pysemi.
Eldan's stochastic localization is a probabilistic construction that has proved instrumental to modern breakthroughs in high-dimensional geometry and the design of sampling algorithms. Motivated by sampling under non-Euclidean geometries and the mirror descent algorithm in optimization, we develop a functional generalization of Eldan's process that replaces Gaussian regularization with regularization by any positive integer multiple of a log-Laplace transform. We further give a mixing time bound on the Markov chain induced by our localization process, which holds if our target distribution satisfies a functional Poincar\'e inequality. Finally, we apply our framework to differentially private convex optimization in $\ell_p$ norms for $p \in [1, 2)$, where we improve state-of-the-art query complexities in a zeroth-order model.
Large reasoning models (LRMs) have achieved remarkable progress in complex problem-solving tasks. Despite this success, LRMs typically suffer from high computational costs during deployment, highlighting a need for efficient inference. A practical direction is to switch the LRM between thinking and non-thinking modes dynamically. However, such approaches often introduce additional reasoning errors and lack statistical guarantees for the performance loss, which are critical for high-stakes applications. In this work, we propose Probably Approximately Correct (PAC) Reasoning that controls the performance loss under the user-specified tolerance. Specifically, we construct an upper confidence bound on the performance loss and determine a threshold for switching to the non-thinking model. Theoretically, using the threshold to switch between the thinking and non-thinking modes ensures bounded performance loss in a distribution-free manner. Our experiments on reasoning benchmarks show that the proposed method can save computational budgets and control the user-specified performance loss.
Stochastic gradient descent (SGD) is a fundamental optimization algorithm widely used in modern machine learning. In this paper, we propose Factor-Augmented SGD (FSGD), a new optimization method that leverages latent factor representations in high-dimensional learning tasks. Unlike standard two-stage dimension reduction approaches that rely on offline representation learning and full data storage, a key novelty of FSGD is that it operates purely on streaming data, making it scalable to large-scale and high-dimensional problems. Furthermore, we establish the first theoretical framework that explicitly incorporates latent factor estimation error into the analysis of SGD, and provide moment convergence in $\ell^s$ norm under decaying step sizes and mini-batch updates. Our results provide a new foundation for employing SGD reliably and scalably in high-dimensional machine learning systems.
We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation for KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain a near-minimax posterior contraction rate. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard multilayer perceptrons, is that the KAN depth can be kept fixed: owing to the expressive power of learnable spline edge functions, the required approximation complexity is controlled through the network width, the range and size of the spline grid, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, these results advance the theoretical understanding of Bayesian neural networks and establish rigorous Bayesian guarantees for KANs.
Unbalanced optimal transport extends classical optimal transport to measures with different total masses, but statistical guarantees for estimation of the transport map and the growth map remain limited. We study unbalanced optimal transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural target of estimation is a transport-growth pair. Consequently, we develop two estimators for the transport-growth pair under two setups: a nearest-neighbor estimator for transporting between general densities, and a kernel-based estimator for transporting between smooth densities. We prove that the kernel-based estimator achieves the minimax optimal rate up to a logarithmic factor by deriving a matching lower bound on the minimax risk. Our main technical contribution is a reduction of estimation to a stability bound on the unbalanced optimal transport value. This bound controls the transport map and growth map risks through the duality gap of the true dual potentials for the estimated measures. These results provide a statistical foundation for estimation of the transport-growth pair in unbalanced optimal transport.
We study the fundamental problem of learning with respect to the squared loss in a convex class. In the subgaussian setting, the sharpest known sample complexity bounds are characterised via fixed points associated with the limiting gaussian process. In more general settings (for example, heavier-tailed scenarios) the state-of-the-art bounds rely on localised Rademacher complexities, which are often difficult to control and can be considerably larger than their limiting gaussian counterparts. We show that if one has access to coarse information on the covariance structure of the underlying class, it is possible to derive an upper bound on the sample complexity that matches the state-of-the-art rates in the gaussian setting---even in heavy-tailed regimes. As a consequence, under rather minimal assumptions, learning problems that share the same $L_2$-structure---even those involving heavy-tailed distributions---admit the same upper bound on the sample complexity as if the problem were fully gaussian. This constitutes the first universality result for general convex learning problems. The proof is based on a novel learning procedure, and its performance is studied by combining optimal mean estimation techniques for real-valued random variables with Talagrand's generic chaining method.
Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.
We study the minimax regret rate for episodic kernel-based reinforcement learning, where the expected reward and transition density belong to a Sobolev reproducing kernel Hilbert space (RKHS), with continuous state and action spaces, and adversarial initial states. For Sobolev smoothness parameter $\nu>0$, state dimension $d_S$, and action dimension $d_A$, we prove that the minimax expected regret over $T$ episodes is $\widetilde\Theta(T^{\max\{e_A,e_B\}})$, where $e_A=\frac{\nu+d_A}{2\nu+d_A}$, $e_B=\frac{\nu+d_A+3d_S/2}{2\nu+d_A+2d_S}$ with the other model parameters fixed. The two exponents reflect the distinct difficulties of action selection and learning across states. We introduce Sobolev Optimal Value Iteration (\texttt{SOVI}), which attains this rate from observed rewards and transitions. For fixed smoothness and reference-kernel lengthscale, its regret bound has a prefactor polynomial in the state-action dimension $d=d_S+d_A$, horizon $H$, reward and transition RKHS norm bounds $B_r,B_p$, and the norm-comparison factor $\Lambda$, up to logarithmic factors; $\Lambda=1$ for normalized Mat\'ern kernels. The algorithm separates initial action selection from continuation-value estimation and combines local polynomial approximation with the shared RKHS structure of the approximation errors. Our matching upper and lower bounds determine the optimal dependence on the number of episodes for Sobolev kernel-based reinforcement learning regret minimization. Thus, we partially address the open problem in Vakili(2024) under Sobolev kernel RL.
Learned barrier certificates provide a principled mechanism for establishing safety of stochastic systems, but their validity may deteriorate during deployment when the underlying dynamics differ from those used for training. We propose an anytime-valid runtime monitor that audits a learned barrier certificate using observed closed-loop trajectories. The monitor is intended to detect loss of statistical support for the deployed barrier condition; it does not by itself guarantee detection before unsafe entry or safety of a subsequent intervention. Specifically, the conditional barrier-drift condition is formulated as an episode-wise sequential hypothesis test, and observed barrier increments are transformed into a nonnegative test supermartingale, yielding finite-sample false-alarm control under arbitrary stopping times. Inspired by finite-horizon safety verification, we further introduce a deterministic temporal shaping of the monitoring boundary, making the monitor more conservative early in a prescribed deployment horizon while preserving anytime-validity. The monitor is designed to detect statistical evidence that the safety certificate is no longer consistent with the observed dynamics, rather than to predict unsafe entry. To assess the monitor empirically, we evaluate empirical alarm behavior under valid and invalid certificates. Code and data are available at https://github.com/FormalCert-AI/runtime-barrier-monitor.
Steinke and Zakynthinou(2020) introduces the Conditional Mutual Information (CMI) framework of analyzing the information complexity of learning algorithms based on algorithm-dependent information-theoretic quantities. We study one of these quantities, the evaluated Conditional Mutual Information (eCMI). It has been an interesting question whether the optimal PAC guarantee for VC classes can be recovered from the algorithm-dependent analyses via CMI. And we show that it is possible to recover this guarantee by constructing a learning algorithm whose eCMI is of order O(d) in the realizable case, where d is the VC-dimension of the concept class. Specially, our algorithm is a randomized Majority-of-5 base learners with optimal in-expectation generalization guarantee.
Knowledge graph learning offers a powerful framework for representing and inferring structured knowledge, with broad practical applications. Two obstacles limit current methods: labeled triples are scarce for each entity and relation, which makes expressive models hard to train, and scoring functions are designed ad hoc, which limits their generalizability and leaves them without theoretical grounding. We address both obstacles with a theoretically grounded framework, with end-to-end guarantees, that extends and subsumes existing methods. The framework proceeds in two stages: multi-view unsupervised pretraining that integrates heterogeneous sources of entity side information into a shared representation, followed by supervised learning across multiple relation types. We establish a unified nonasymptotic risk bound that combines the subspace recovery error of pretraining and the sample complexity of the labeled data in a single guarantee, quantifying formally how large-scale unlabeled data from multiple views benefits downstream knowledge prediction. Synthetic experiments validate each component of the theory, and experiments on large-scale real-world knowledge graph benchmarks confirm the effectiveness of our approach.
We prove the population-level identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: *domain contrast*, which trivializes the mixture symmetry group, a label-free counterpart of the sufficient-variability assumption of nonlinear ICA; *mechanism contrast*, which ensures every decoder branch gets an observational boundary across which it toggles alone; and *interaction contrast*, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from *law identifiability* (LID; latent distribution up to a global affine map) through *map identifiability* (MID; decoder up to the same map) to *posterior* and *pointwise identifiability*. The ICA-form ambiguity emerges under conditions on diagonal component covariances. The interaction contrast is the only one of our contrasts required on the estimator side. To our knowledge this is the first to develop algebraic symmetry-breaking into an identifiability engine for a concrete nonlinear model class, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.
Digital Twins support real-time decision-making through predictive models of physical systems, yet their reliability can deteriorate as operating conditions evolve. Maintaining predictive fidelity under such changes is challenging, particularly when model adaptation relies on limited streaming data, and updated models must be evaluated before deployment. Here, we present an online adaptation framework that integrates gradient-based drift detection, parameter-efficient fine-tuning, and statistical model validation within model predictive control. Changes in prediction-loss gradients trigger batch-wise model updates using Low-Rank Adaptation, while a paired moving-block bootstrap test evaluates predictive improvement before deploying the updated model. The framework is evaluated using a stochastic dynamical system with abrupt changes and a simulated directed energy deposition process with gradual changes in material properties. Results show improvements in predictive accuracy and uncertainty estimation during the evaluated post-adaptation intervals, alongside improved reference tracking and reduced constraint violations in the manufacturing case study compared with the investigated baselines. These findings demonstrate the potential of integrating continual validation with model adaptation while acknowledging the trade-off between responsiveness and predictive reliability in evolving Digital Twins.
Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \citep{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theory and methodology at scale and under DAG misspecification. We then model a latent-collider DAG, a protein signalling network, and a multi-fidelity \mbox{heavy-ion} collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability over the simulator hierarchy.
We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.