ALT Low-rank adaptation (LoRA) is now a standard approach to parameter-efficient fine-tuning of large neural networks. Its low-rank factorization is inherently non-identifiable, making LoRA a singular statistical model whose learning complexity cannot be described by parameter dimension alone. We study this complexity using the real log canonical threshold (RLCT) from singular learning theory and how it depends on both the low-rank parameterization and the pretrained network. Under a local linearization, LoRA induces a low-rank trace regression model whose Kullback–Leibler geometry is controlled by a forward-backward covariance, constructed from activations and backpropagated sensitivities at the adapted layer. This covariance determines which low-rank perturbations are statistically observable. When it is nondegenerate along the relevant low-rank directions, we recover the classical learning complexity of reduced-rank regression; when its kernel meets these directions, additional statisti

ALT There is growing evidence that the performance of a neural network depends largely on a small subset of its weights, which are typically identified by training a dense network and pruning it afterwards. Within a Bayesian framework, pruning can be directly incorporated through a spike-and-slab prior, providing a proper treatment of the uncertainties involved, both with respect to pruning and to parameter uncertainty. In Bayesian neural networks (BNNs), sampling by MCMC is notoriously hard. We demonstrate that modern Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) can be used to sample posteriors of BNNs asymptotically exact whenever numerically constructed rate bounds hold. Spike-and-slab priors can be incorporated through sticky versions of the PDMPs, setting weights exactly to zero during sampling. We examine the predictive performance and convergence of the samplers, and show that sparse BNNs sampled this way can be more robust to input noise than dense

ALT Continual learning (CL) aims to learn from sequential tasks while mitigating catastrophic forgetting when historical data are no longer accessible. We study continual learning for high-dimensional sparse linear regression, where ℓ₁ regularization creates a difficulty absent from continual ridge regression: that recursively reusing an ℓ₁-penalized estimator can propagate earlier shrinkage and accumulate regularization bias across tasks. We propose score-preserving continual Lasso (SP-CL), which separates sparse estimation from state transmission by preserving the historical squared-loss score. Under squared loss, its cumulative estimator exactly recovers a full-history Lasso minimizer. When task coefficients coincide, it inherits the full-history sparse rate. For related task-specific coefficients, SP-CL uses a regularized current-task correction that leaves the transmitted state unchanged and attains the offline source-assisted rate under an oracle related-task condition. We also deriv

ALT Recent work demonstrated that executing Bayesian Optimization (BO) using simply a linear surrogate with a clever transformation is enough to approximate the performance of the standard Gaussian-Process (GP) based version on high-dimensional test functions. When the decision region is a sphere, this strategy admits rapid exact acquisition. However, it is the unit cube we encounter most often in the BO literature, and in this setting a standard approach, namely a multi-start quasi-Newton method, is employed for solving the acquisition subproblem. In this article, we will discuss a series of results, both positive and negative, regarding the complexity of global resolution of various acquisition subproblems over the unit cube. On the basis of this discussion, we will propose a new acquisition function which is globally optimizable using significantly less execution time than the incumbent approach while maintaining or improving performance. The resulting method is competitive with GP-base

ALT Hybrid forecasting systems combining machine learning and numerical weather prediction offer a promising route toward improved forecast skill while preserving physical consistency. In this study, we investigate online learning of state-dependent model-error corrections within the Integrated Forecasting System using 4D-Var. Building on previous work, we represent model error using a neural network embedded within the forecast model and trained from observations through the data assimilation cycle. First, we introduce a targeted variant of neural-network 4D-Var in which only the neural-network parameters are estimated, while the atmospheric state is provided by a precomputed analysis. We then propose a batched extension that processes assimilation windows in random order and averages parameter increments over multiple windows, drawing inspiration from mini-batch stochastic optimization. Experiments are conducted with the Integrated Forecasting System of the European Centre for Medium-Ran

ALT We study squared-error estimation when a nuisance parameter is unidentified on a submodel but affects a target that vanishes there. For models with higher-order profile expansions, we give sufficient conditions under which the largest critical-path minimax risk is the exact asymptotic minimax risk over the full parameter space. An efficient observable feature, negligible calibration error and stable recovery yield one estimator attaining the leading constant uniformly, with the profile coordinates unknown. The estimator uses calibration estimates, a local feature mean and an empirical recovery feature. The constant is expressed through a bounded-normal decision problem. We verify the conditions in two models. In centered Gaussian logistic errors-in-variables regression with one surrogate and two unknown variances, we determine the fifth-order global coefficient risk and its exact constant. A sparse extension separates variance-decomposition uncertainty from direction estimation and spe

ALT Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptio

ALT Whether a federated predictor benefits an individual client depends on both population heterogeneity and estimation uncertainty. We study this question for Nadaraya–Watson regression in a triangular array with potentially different fitting and evaluation distributions. Each client transmits local kernel summaries, allowing the server to construct a covariate-density-weighted federated estimator without centralizing individual observations. A client combines its fitted local predictor with the federated predictor through a prescribed weight, and gain is the paired reduction in squared prediction loss on an independent audit observation. Exact identities characterize both the gain of the realized fitted pair and its repeated-fitting mean. We also establish pointwise inference for the federated estimator. A two-stage analysis of client gains separates fitting variability from audit variability, retains the covariance created when a client's observations enter both its local estimator and

ALT We propose a neural exploratory framework for variational problems over graphons, i.e., symmetric measurable functions [0,1]² → [0,1], that arise as limits of dense graph sequences. Such problems are central to two parts of the mathematical literature: extremal graph theory, which studies graph parameter optimisation under given restrictions, often for those graphs with a large number of vertices, and the large deviation theory of dense random graphs, which characterises the structure of rare events through constrained graphon optimisation. We represent graphons by implicit neural networks and optimise graphon objectives by gradient descent. Our design combines three ingredients: a multi-scale sinusoidal residual architecture biased toward sharp, step-like graphons; an embedded solver that enforces a single empirical density constraint and is differentiated by implicit differentiation; and symmetry-aware Monte Carlo estimators. On generalised Turán problems that previously required sub

ALT Offline maximum entropy reinforcement learning seeks to estimate the optimal soft action-value function Qᶠrom a fixed dataset without further interaction with the environment. Although Qⁱs characterized by the soft optimal Bellman equation, the corresponding Bellman operator cannot be evaluated directly because both the reward distribution and the transition kernel are unknown. Consequently, estimating Qʳequires the joint treatment of environment modeling and value function learning. We propose a framework that addresses these tasks separately. Conditional diffusion models based on probability flow ODEs estimate the conditional reward distribution and transition kernel. The resulting distributional estimators define a data driven approximation of the soft optimal Bellman operator, whose empirical Bellman residual is minimized over a class of deep neural networks to estimate Q. We establish nonasymptotic guarantees for the proposed method. For operator learning, we derive total variatio

ALT We study the PAC-learnability of function classes that map inputs to probability distributions over a finite space. The learner observes i.i.d. input-label pairs, with each label sampled from the conditional distribution of an unknown target hypothesis in the class. We measure learning through excess risk under a bounded proper loss that satisfies two mild assumptions. We introduce the fatᵧNatarajan dimension, which generalizes both fatᵧ and Natarajan dimension, and prove that its finiteness at every positive scale is necessary and sufficient for learnability. We also introduce a fatᵧGraph dimension extending the classical \gra dimension and establish inequalities comparing the two dimensions at different scales, leading to an equivalent characterization of learnability. Finally, we show that a distribution-valued function class is learnable if and only if each of its coordinate classes is learnable in the probabilistic concept model.

ALT We study the early training dynamics of nonlinear residual networks with blocks of zero hidden layers as a model of more general residual networks. We analyze networks in a sequential limit where first width and depth diverge, then the number of training data points diverges, and finally the input dimension diverges. This is a more refined setup compared to the typical mean-field theory analysis in which depth and width diverge but the number of training datapoints and input dimension are fixed. Our main technical result is a diagrammatic computation of both the network output and the hidden representations after the few steps of gradient descent in terms of the statistics of inner products of inputs from the training set. One of our principle conceptual findings is a negative result: in the limit of large input dimension the network output can learn only a linear function of its inputs early in training since higher learned monomials are more and more suppressed in the input dimension

ALT We consider online monitoring of an AI system that produces a sequence of decisions. At each round, the system outputs a proposed decision together with a confidence score estimating the trustworthiness of the decision. The decision and confidence score need not be produced by the same model: for example, one agent may propose a decision, while a separate verifier agent evaluates it and outputs a correctness or safety score. Based on this information, we must decide whether to accept the AI decision automatically or consult an expert, such as a human auditor, who reviews the proposal and either accepts or corrects it. We formulate this task as an online constrained optimization problem in which the goal is to minimize the frequency of expert auditing while guaranteeing that the fraction of AI errors that go unaudited stays below a user-defined threshold. We derive a family of algorithms with provable error-control guarantees with no assumptions, including a method that is optimally eff

ALT Nonparametric estimation of joint probability distributions is a core problem in statistics, which is notoriously challenging with high-dimensional data. Such challenges also arise with conditional distribution estimation and have led to flexible algorithms that aim to tackle the ”curse of dimensionality” by exploiting predictive structures of lower dimension than the ambient space. In this paper, we study the use of such conditional distribution estimators to estimate joint distributions by factorizing the latter in an ”autoregressive” way. We show that for loss functions satisfying a subadditivity property, bounds on the convergence rate of an autoregressive estimator can be derived from bounds on the convergence rate of the conditional distribution estimators. These bounds relate the difficulty of estimating the joint distribution to that of estimating the most challenging factor in the autoregressive factorization. When the conditional estimators adapt to sparsity in their conditio

ALT The Unadjusted Langevin Algorithm (ULA) is widely used for high-dimensional Bayesian inference. Classical analyses in Wasserstein-2 distance predict an asymptotic bias and a mixing time that grow with dimension, but this metric is stronger than necessary for many statistical tasks. We instead analyze the bias in subspace robust Wasserstein distance, which controls the mean and arbitrary low-dimensional marginals and is therefore well suited to observables typically estimated in practice. For ridge-separable potentials formed with i.i.d. data, we prove high-probability bounds showing a bell-shaped mixing profile as the number of data points N varies relative to the dimension d. In the overparameterized regime, the effective dimension is the rank N of the data-dependent perturbation, leading to complexity depending on N rather than d. In the large-sample regime, Gaussian-perturbation estimates yield dimension-free complexity once N is sufficiently large. In contrast, near the proportiona

ALT We introduce the conditional expected local radius (CELR) and develop a new conformal prediction method for non-Euclidean responses in general metric spaces. CELR measures local conditional probability concentration, allowing prediction sets to accommodate heterogeneity and multimodality without requiring densities or a vector space structure. Its conditional distribution yields a nonconformity score that is comparable across covariate values and avoids costly transport comparisons between distance profiles. We further adapt a metric distributional random forest that handles multivariate covariates and nonlinear interactions, while a single forest fit supplies all score components, reducing computation by eliminating repeated local fitting. We establish uniform consistency of the estimators and asymptotic conditional validity under suitable regularity conditions, together with finite-sample marginal coverage under exchangeability. Numerical studies demonstrate effective uncertainty qua

ALT Deep learning (DL) methods have substantially improved predictive performance across a wide range of downstream prediction tasks, enabling new discoveries in multiple domains. However, principled DL methods for non-Euclidean random objects remain relatively scarce, despite the growing importance of modern biomedical outcomes taking the form of probability distributions, random graphs, functions, and other random objects in metric spaces. Motivated by a new generation of digital and clinical biomarkers, we propose a deep learning framework for regression with random objects in metric spaces. Our approach operates directly on the metric structure of the response space, avoiding restrictive manifold or Euclidean embedding assumptions imposed by many existing methods. At the same time, it retains modern statistical guarantees under the stated statistical assumptions. We establish statistical consistency and convergence rates for the proposed estimator, and characterize conditions under whi

ALT Computer-model calibration is usually undertaken in service of prediction, yet neither Bayesian Kennedy-O'Hagan (KOH) calibration nor amortized simulation-based inference is built to control the predictive error of the calibrated simulator. When parameters are weakly identified or the data-model discrepancy is misspecified, the simulator-output bands induced by a nominal posterior level can be far from nominal. We propose a two-layer framework that targets predictive miscoverage directly. First, a conditional flow-matching model learns an amortized posterior for the calibration parameters, trained on simulator outputs perturbed with Gaussian-process discrepancies of varying structure. Second, conformal risk control arranges posterior draws for the target observation into nested candidate sets and selects the smallest one whose fixed-rank pointwise bands attain a prescribed time-averaged miscoverage on a synthetic calibration sample. Under exchangeability of calibration and target obser

ALT Random projections (RP) offer an effective method for reducing computational and storage overhead while preserving the geometric structure of data. However, existing studies have primarily focused on the optimal generalization performance of specific kernel-regularized algorithms with RP in the well-specified setting and under restrictive conditions. In this paper, we provide a comprehensive and improved analysis of the generalization performance of RP-based spectral algorithms under general conditions changewithout sacrificing statistical efficiency. By leveraging the embedding property of the reproducing kernel Hilbert space (RKHS) and a refined analysis of the operator similarity, we establish optimal learning rates in Sobolev norms that match the minimax lower bounds up to logarithmic factors. For two common implementations of RP, randomized sketches and Nyström sub-sampling (uniform or leverage-based), we show that the projection dimension required for optimality is proportional t

ALT Deep Gaussian processes (DGPs) are appealing Bayesian regression models, but their standard discrete-depth parameterization often becomes harder to optimize as depth grows. We argue that depth is better treated as a latent continuous evolution than as a long discrete stack. We therefore propose a continuous-depth formulation in which the latent representation follows a stochastic differential equation with a depth-indexed drift family equipped with a GP-based prior. This reframes the core design choice as the prior placed on that family, rather than adopting a purely neural continuous-depth parameterization as in neural ODE or neural SDE models. We study two instantiations of this idea: CDGP, a direct GP prior over the state–depth drift, and FlowDGP, a flow-evolved prior that captures richer depth dependence at lower training cost. To motivate the redesign, we analyze how the single-sample DGP objective can become increasingly sensitive under repeated layer composition, while residual