Contextual online decision-making problems with constraints appear in a wide range of real-world applications, such as adaptive experimental design under safety constraints, personalized recommendation with resource limits, and dynamic pricing under fairness requirements. In this paper, we investigate a general formulation of sequential decision-making with stage-wise feasibility constraints, where at each round, the learner must select an action based on observed context while ensuring that a problem-specific feasibility criterion is satisfied. We propose a unified algorithmic framework that captures many existing constrained learning problems, including constrained bandits, active learning with label budgets, online hypothesis testing with Type I error control, and model calibration. Central to our approach is the concept of upper counterfactual confidence bounds, which enables the design of practically efficient online algorithms with strong theoretical guarantees using any offline conditional density estimation oracle. To handle feasibility constraints in complex environments, we introduce a generalized notion of the eluder dimension, extending it from the classical setting based on square loss to a broader class of metric-like probability divergences. This allows us to capture the complexity of various density function classes and characterize the utility regret incurred due to feasibility constraint uncertainty. Our result offers a principled foundation for constrained sequential decision-making in both theory and practice.

We study the problem of estimating a distribution over a finite alphabet from an i.i.d. sample, with accuracy measured in relative entropy (Kullback-Leibler divergence). While optimal expected risk bounds are known, high-probability guarantees remain less well-understood. First, we analyze the classical Laplace (add-one) estimator, obtaining matching upper and lower bounds on its performance and showing its optimality among confidence-independent estimators. We then characterize the minimax-optimal high-probability risk, which is attained via a simple confidence-dependent smoothing technique. Interestingly, the optimal non-asymptotic risk exhibits an additional logarithmic factor over the ideal asymptotic risk. Next, motivated by scenarios where the alphabet exceeds the sample size, we investigate methods that adapt to the sparsity of the distribution at hand. We introduce an estimator using data-dependent smoothing, for which we establish a high-probability risk bound depending on two effective sparsity parameters. As part of the analysis, we also derive a sharp high-probability upper bound on the missing mass.

Label shift adaptation aims to recover target class priors when the labelled source distribution $P$ and the unlabelled target distribution $Q$ share $P(X \mid Y) = Q(X \mid Y)$ but $P(Y) \neq Q(Y)$. Classical black-box shift estimators invert an empirical confusion matrix of a frozen classifier, producing a brittle point estimate that ignores sampling noise and similarity among classes. We present Graph-Smoothed Bayesian BBSE (GS-B$^3$SE), a fully probabilistic alternative that places Laplacian-Gaussian priors on both target log-priors and confusion-matrix columns, tying them together on a label-similarity graph. The resulting posterior is tractable with HMC or a fast block Newton-CG scheme. We prove identifiability, $N^{-1/2}$ contraction, variance bounds that shrink with the graph's algebraic connectivity, and robustness to Laplacian misspecification. We also reinterpret GS-B$^3$SE through information geometry, showing that it generalizes existing shift estimators.

We address the problem of incremental sequence classification, where predictions are updated as new elements in the sequence are revealed. Drawing on temporal-difference learning from reinforcement learning, we identify a temporal-consistency condition that successive predictions should satisfy. We leverage this condition to develop a novel loss function for training incremental sequence classifiers. Through a concrete example, we demonstrate that optimizing this loss can offer substantial gains in data efficiency. We apply our method to text classification tasks and show that it improves predictive accuracy over competing approaches on several benchmark datasets. We further evaluate our approach on the task of verifying large language model generations for correctness in grade-school math problems. Our results show that models trained with our method are better able to distinguish promising generations from unpromising ones after observing only a few tokens.

Optimal decision-making under partial observability requires agents to balance reducing uncertainty (exploration) against pursuing immediate objectives (exploitation). In this paper, we introduce a novel policy optimization framework for continuous partially observable Markov decision processes (POMDPs) that explicitly addresses this challenge. Our method casts policy learning as probabilistic inference in a non-Markovian Feynman--Kac model that inherently captures the value of information gathering by anticipating future observations, without requiring extrinsic exploration bonuses or handcrafted heuristics. To optimize policies under this model, we develop a nested sequential Monte Carlo~(SMC) algorithm that efficiently estimates a history-dependent policy gradient under samples from the optimal trajectory distribution induced by the POMDP. We demonstrate the effectiveness of our algorithm across standard continuous POMDP benchmarks, where existing methods struggle to act under uncertainty.

We explore the use of conformal prediction to provide statistical uncertainty guarantees for runway detection in vision-based landing systems (VLS). Using fine-tuned YOLOv5 and YOLOv6 models on aerial imagery, we apply conformal prediction to quantify localization reliability under user-defined risk levels. We also introduce Conformal mean Average Precision (C-mAP), a novel metric aligning object detection performance with conformal guarantees. Our results show that conformal prediction can improve the reliability of runway detection by quantifying uncertainty in a statistically sound way, increasing safety on-board and paving the way for certification of ML system in the aerospace domain.

In this paper, we focus on the out-of-distribution (OOD) generalization of self-supervised learning (SSL). By analyzing the mini-batch construction during the SSL training phase, we first give one plausible explanation for SSL having OOD generalization. Then, from the perspective of data generation and causal inference, we analyze and conclude that SSL learns spurious correlations during the training process, which leads to a reduction in OOD generalization. To address this issue, we propose a post-intervention distribution (PID) grounded in the Structural Causal Model. PID offers a scenario where the spurious variable and label variable is mutually independent. Besides, we demonstrate that if each mini-batch during SSL training satisfies PID, the resulting SSL model can achieve optimal worst-case OOD performance. This motivates us to develop a batch sampling strategy that enforces PID constraints through the learning of a latent variable model. Through theoretical analysis, we demonstrate the identifiability of the latent variable model and validate the effectiveness of the proposed sampling strategy. Experiments conducted on various downstream OOD tasks demonstrate the effectiveness of the proposed sampling strategy.

Asymmetric, non-Gaussian probability distributions are often observed in the analysis of natural and engineering datasets. The lognormal distribution is a standard model for data with skewed frequency histograms and fat tails. However, the lognormal law severely restricts the asymptotic dependence of the probability density and the hazard function for high values. Herein we present a family of three-parameter non-Gaussian probability density functions that are based on generalized kappa-exponential and kappa-logarithm functions and investigate its mathematical properties. These kappa-lognormal densities represent continuous deformations of the lognormal with lighter right tails, controlled by the parameter kappa. In addition, bimodal distributions are obtained for certain parameter combinations. We derive closed-form analytic expressions for the main statistical functions of the kappa-lognormal distribution. For the moments, we derive bounds that are based on hypergeometric functions as well as series expansions. Explicit expressions for the gradient and Hessian of the negative log-likelihood are obtained to facilitate numerical maximum-likelihood estimates of the kappa-lognormal parameters from data. We also formulate a joint probability density function for kappa-lognormal stochastic processes by applying Jacobi's multivariate theorem to a latent Gaussian process. Estimation of the kappa-lognormal distribution based on synthetic and real data is explored. Furthermore, we investigate applications of kappa-lognormal processes with different covariance kernels in time series forecasting and spatial interpolation using warped Gaussian process regression. Our results are of practical interest for modeling skewed distributions in various scientific and engineering fields.

In electromyogram (EMG)-based motion recognition, a subject-specific classifier is typically trained with sufficient labeled data. However, this process demands extensive data collection over extended periods, burdening the subject. To address this, utilizing information from pre-training on multiple subjects for the training of the target subject could be beneficial. This paper proposes an inter-subject variance transfer learning method based on a Bayesian approach. This method is founded on the simple hypothesis that while the means of EMG features vary greatly across subjects, their variances may exhibit similar patterns. Our approach transfers variance information, acquired through pre-training on multiple source subjects, to a target subject within a Bayesian updating framework, thereby allowing accurate classification using limited target calibration data. A coefficient was also introduced to adjust the amount of information transferred for efficient transfer learning. Experimental evaluations using two EMG datasets demonstrated the effectiveness of our variance transfer strategy and its superiority compared to existing methods.

Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis--Hastings for sampling from mixtures of Gaussian distributions. We show that in fixed-dimensional settings, the algorithm's complexity scales polynomially with the separation between modes and logarithmically with $1/\varepsilon$, where $\varepsilon$ is the target accuracy in total variation distance.