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.

We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models by extending the diffusion-based framework of Mou et al. (2024) to the infinite-dimensional setting. The posterior is represented as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, which allows us to control posterior moments and obtain non-asymptotic concentration rates in Hilbert norms under various likelihood curvature and regularity conditions. We also establish a quantitative Laplace approximation for the posterior. The theory is illustrated in a nonparametric linear Gaussian inverse problem.

Practitioners monitoring deployed probabilistic models face a fundamental trap: any fixed-sample test applied repeatedly over an unbounded stream will eventually raise a false alarm, even when the model remains perfectly stable. Existing methods typically lack formal error guarantees, conflate alarm time with changepoint location, and monitor indirect signals that do not fully characterize calibration. We present PITMonitor, an anytime-valid calibration-specific monitor that detects distributional shifts in probability integral transforms via a mixture e-process, providing Type I error control over an unbounded monitoring horizon as well as Bayesian changepoint estimation. On river's FriedmanDrift benchmark, PITMonitor achieves detection rates competitive with the strongest baselines across all three scenarios, although detection delay is substantially longer under local drift.

We study theoretical properties of a broad class of regularized algorithms with vector-valued output.

Community detection in random graphs or hypergraphs is an interesting fundamental problem in statistics, machine learning and computer vision.

We consider the problem of estimating the mean of a set of vectors, which are stored in a distributed system.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods.

In addition to solving (1), we also wish topreserve privacyfor the people who contributed to the dataset Z.

In particular,SQuARM-SGD[45]can be viewed asCHOCO-SGD with momentum, but its theoretical convergence rate is slower than the originalCHOCO-SGD. We provide some examples of compression operators satisfying Definition 1 that are used in our experiments. Line 6), the penultimate line follows from W1 = 1, and the last line follows from the induction hypothesis at thet-th iteration. Line 3), in the second line we use the propertyofthemixingmatrix 1 W =1,andinthethirdline,weapplyYoung'sinequality(cf.(9)). Bounding Ωt2 in (14b) Similar to the derivation of (14a), by applying the update rule ofGt in BEER(Line 8),thedefinition ofcompression operators (Definition 1),andYoung'sinequality,we have It then boils down to establish (26).

(Theorem 3).