Building on the large-sample analysis of infinitesimal gradient boosting (Dombry and Duchamps, 2024b), we study the fluctuations of the process around its deterministic limit and establish a functional central limit theorem: the rescaled deviations converge in distribution to a Gaussian process. The analysis is carried out in a reproducing kernel Hilbert space (RKHS) naturally associated with the softmax gradient tree base learner, in which the boosting process is characterized as the solution of an autonomous ordinary differential equation (ODE). The proof rests on a general stochastic perturbation analysis of ODEs in Banach spaces, which is of independent interest: whenever a sequence of vector fields converges and satisfies a central limit theorem, so does the associated ODE solution. We first illustrate this perturbation approach in the simpler setting of kernel gradient flow, where the Gaussian limit admits an explicit characterization, and then consider the more complicated tree-based gradient boosting setting.

Hyperparameter selection is a critical step in the deployment of modern artificial intelligence systems, given the need to tune degrees of freedom such as inference-time parameters, implementation-level settings, and thresholds driving decision rules. Despite its practical importance, hyperparameter selection is typically performed using best-effort empirical methods such as grid search or Bayesian optimization, which provide no formal statistical guarantees on reliability or safety. This monograph presents a unified statistical framework for reliable hyperparameter selection, centered on the learn-then-test (LTT) paradigm, which formulates the problem as multiple hypothesis testing over a candidate set of hyperparameters. The framework enables the selection of hyperparameters that provably satisfy application-specific reliability requirements -- such as bounds on average risk, quantile risk, or information-theoretic constraints -- with explicit, finite-sample control of error probabilities. The supporting statistical machinery, namely p-values, e-values, and concentration inequalities, is developed from first principles in a dedicated appendix.

Modeling and sampling from the underlying distribution of asynchronous event sequences are crucial in various real-world applications, including social networks, medical diagnosis, and financial transactions. Existing autoregressive methods suffer from error accumulation during multi-step generation, while non-autoregressive diffusion methods are typically limited to fixed-length output sequences. In this paper, we propose Latent Block-Diffusion Temporal Point Processes (LBDTPP), a novel semi-autoregressive TPP framework that introduces a latent block diffusion mechanism for high-quality and variable-length event sequence generation. The core idea is to define an autoregressive probability distribution over event blocks in latent space and perform Gaussian diffusion within each block. By sequentially generating blocks while simultaneously sampling events in each block, LBDTPP preserves the length flexibility of autoregressive TPPs and inherits the parallel high-quality generation capability of diffusion models. Theoretically, we derive Wasserstein error bounds showing that, under suitable local approximation and prefix-stability assumptions, block-wise generation can reduce error accumulation compared with event-wise autoregressive generation. Extensive experiments on six real-world benchmark datasets demonstrate that LBDTPP outperforms state-of-the-art TPP baselines in both unconditional and conditional generation tasks. Further empirical analyses verify the benefits of latent-space diffusion and block-wise generation, and reveal the trade-off between generation quality and block size. Our code is available at https://github.com/Zh-Shuai/LBDTPP.

We study the problem of model selection among probabilistic forecasting models evaluated on datasets of multiple time series. The performance of a model on a single time series is quantified by the average value (score) of a proper scoring rule over a test set, but extending model selection to data sets of time series requires aggregating these scores. Common approaches either rely on scaling scores and averaging them (mean scaled score) or avoid scaling by using alternative statistics such as mean ranks or win rates. However, these approaches can yield conflicting conclusions. We show that such discrepancies arise from the skewness of the distribution of the scores, which is particularly pronounced when test sets are short. The skewness can cause non-mean criteria (e.g., mean rank, median, win rate) to select misspecified models. In contrast, the mean score is immune from this problem. We further show that, as the size of the test sets increases, all aggregation criteria converge to the same model selection decision, mitigating these discrepancies. Our experiments on intermittent demand time series, including data from the M5 competition, highlight the importance of sufficiently large test sets; the mean scaled score appears to be the more reliable approach, also because empirically we found its decision to remain consistent when different scaling factors are adopted.

We prove a single algebraic mixed coincidence identity that unifies a broad swath of information-theoretic variational results. For any family of priors $\{π_i\}$ and real exponents $\{ α_i \}$, the log of the mixed count $E_{x\simν}\!\left[\prod_{i=1}^W π_i^{α_i}(x)\right]$ is simultaneously a Boltzmann coincidence weight, an exponential-family normalizer, a maximum-entropy value, and a KL-barycenter optimum. The identity yields a unified derivation of classical cornerstones of information theory: concentration of empirical distributions (Sanov-type decompositions and Gibbs conditioning), hypothesis-testing error exponents (Chernoff information and its multi-way analogue), change-of-measure inequalities (Donsker-Varadhan and PAC-Bayes), and laws governing rare-pattern coincidences (Erdos-Renyi run-length, iterative guesswork, rate-distortion, and birthday thresholds). Each is recovered as a specialization of the same algebraic equality. It strictly generalizes the classical Renyi entropy and divergence variational formulas (one and two priors respectively) to a $W$-prior simplex, and holds for unnormalized and continuum-indexed priors. Among its consequences are an exact multi-prior PAC-Bayes penalty that subtracts an explicit "coincidence bonus" from the usual single-prior posterior penalty, and the asymptotic MAP error exponent for $W$-ary hypothesis testing as an edge-restricted simplex optimum. We demonstrate the calculus at scale on two large alphabets encoding richly modeled sequential languages: on language-model next-token predictives where we recover contrastive decoding, and on human genomic regulatory sequence where it separates correlated from diverse prior families along a sliding-window trace.

Abstract--As black-box models become foundational to mod-solution that can be applied across a wide range of scientific ern research, ensuring their stability is paramount for the realiza-and industrial domains. The inherent diversity of inputs--ranging from structured Gaussian distributions to Notwithstanding its widespread application, the framework complex data with unknown structures--poses a significantexhibits certain shortcomings when dealing with complex challenge: how to stabilize black-box outputs while effectivelydatasets. First, standard resampling schemes often fail to leveraging available prior information. This paper introduces aaccount for the underlying data structures; as a result, the task-oriented randomization methodology that adaptively tailorsdrawn samples cannot reflect the true data distribution, thereby its strategy to the underlying generative mechanisms of the input data, specifically addressing unstructured complexities. Second, effective sampling requires prior comprehensive suite of stability guarantees is proposed. Beyondknowledge of the distribution, which is often unattainable establishing rigorous theoretical foundations for stability, thein practical environments.

Rank aggregation combines information from ordered lists ranking items by preference. Classical parametric models for such data, including the Mallows and Plackett-Luce models, assume the orders concentrate around one or more complete consensus rankings. Recent work relaxes the total-order assumption by allowing the consensus structure to be a partial order (poset), allowing for incomparabilities in preferences. However, in many applications preference data exhibit group structure. We introduce hierarchical partial order (HPO) models, which extend poset-based models to accommodate grouped data through a hierarchy of latent posets. This framework, which parallels mixture model extensions of the Mallows and Plackett-Luce models, enables principled sharing of information across groups while preserving partial-order structure. We show that the Plackett-Luce model and its hierarchical variants are special cases of HPO-models. We develop a hierarchical clustering extension (HCPO) for unsupervised clustering in settings where group labels are unknown. Bayesian inference for the latent poset hierarchy is performed using Markov chain Monte Carlo methods. Experiments on synthetic and real-world datasets, including pairwise acoustic preference data and LLM agent traces, demonstrate that the proposed HPO and HCPO models outperform existing approaches in both predictive performance and structural interpretability.

Federated learning has emerged as the foremost approach for decentralized model training with privacy preservation. The global class imbalance and cross-client data heterogeneity naturally coexist, and the mismatch between local and global imbalances exacerbates the performance degradation of the aggregated model. The agnosticism of global class distribution poses significant challenges for data-level methods, especially under extreme conditions with severe class absence across clients. In this paper, we propose FedReLa, a novel data-level approach that tackles the coexistence of data heterogeneity and class imbalance in federated learning. By re-labeling samples with a feature-dependent label re-allocator, FedReLa corrects biased global decision boundaries without requiring knowledge of the global class distribution. This modular, model-agnostic approach can be integrated with algorithmic methods to deliver consistent improvements without additional communication overhead. Through extensive experiments, our method significantly improves the accuracy of minority classes and the overall accuracy on stepwise-imbalanced and long-tailed datasets, outperforming the previous state of the art.

Learning effective policies for adaptive data acquisition remains challenging: posterior-based methods rely on surrogate models and posterior approximations that can be misspecified or biased, while direct policy-learning methods map from historical observations and fail to exploit available model representations, making learning harder. We introduce policy learning with belief representations (POLAR), based on the insight that optimal data acquisition depends on the observation history only through a sufficient belief state. Specifically, POLAR decouples representation learning from policy learning by leveraging pretrained predictive foundation models as belief-state encoders, training a policy head on top of their representations. This yields a simple, unified amortised policy learning framework for Bayesian experimental design, Bayesian optimisation, and active learning, differing only in the task-specific utility used to train the policy. Empirically, we find that POLAR outperforms state-of-the-art amortised methods across diverse tasks while requiring far fewer training samples, demonstrating a significant step in the scalability and efficiency of amortised data acquisition.

Synthetic data augmentation is widely used to mitigate class imbalance, but its theoretical effects on score-based classification remain poorly understood. This paper develops a framework for characterizing when synthetic minority augmentation can improve threshold-integrated and threshold-optimized metrics, including AUROC, AUPRC, best-threshold balanced accuracy, and best-threshold \(\F_1\) score. We separate the effect of augmentation into two components: a change in effective class weighting and a discrepancy between the synthetic and true minority distributions. Under well-specified score models, the raw estimator already targets the likelihood-ratio ordering, which is population-optimal for the metrics considered. Consequently, augmentation cannot provide a fundamental population-level improvement beyond possible finite-sample variance reduction, and may introduce additional bias through synthetic distributional error. We further establish minimax lower bounds showing that the raw estimator already achieves the optimal metric-regret rate in the well-specified regime. Under misspecification, however, augmentation can play a qualitatively different role: by changing the effective class balance, it can alter the restricted-class projection and correct ranking errors induced by the raw imbalanced objective. We provide explicit improvement bounds quantifying the roles of approximation error, finite-sample estimation error, and synthetic distributional error. Simulation studies corroborate the theory, demonstrating limited gains under well-specification and nontrivial but nonmonotone improvements under misspecification.