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Stochastic Gradient Descent with Adaptive Data Stochastic gradient descent (SGD) is a central tool in modern optimization, but its classical theory relies on the assumption that data are independent of the decisions being optimized. In many operations research settings, this assumption fails: policies influence system dynamics, and the resulting data feed back into subsequent updates. In “Stochastic Gradient Descent with Adaptive Data,” Che, Dong, and Tong address this challenge by developing a general framework for analyzing SGD when data are generated adaptively by policy-dependent Markov processes. Their analysis shows that fully adaptive SGD can still attain convergence rates comparable to the classical i.i.d. setting, provided the underlying system satisfies mild ergodicity and continuity conditions. The theory is illustrated through canonical applications in operations research and reinforcement learning. Overall, the paper provides rigorous and reassuring theoretical foundations for deploying learning algorithms in dynamic environments where decisions and data are fundamentally intertwined. 

Abstract When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the$$\chi^2$$metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims. 

This article develops a conformal prediction method for classification tasks that can adapt to random label contamination in the calibration sample, often leading to more informative prediction sets with stronger coverage guarantees compared to existing approaches. This is obtained through a precise characterization of the coverage inflation (or deflation) suffered by standard conformal inferences in the presence of label contamination, which is then made actionable through a new calibration algorithm. Our solution can leverage different modelling assumptions about the contamination process, while requiring no knowledge of the underlying data distribution or of the inner workings of the classification model. The empirical performance of the proposed method is demonstrated through simulations and an application to object classification with the CIFAR-10H image data set. 

The inherent qualitative nature of textual data poses significant challenges for direct integration into statistical models. This paper presents a two-stage process for analyzing longitudinal textual data, offering a solution to this inherent challenge. The proposed model comprises (1) initial data preprocessing and sentiment extraction, followed by (2) applying a growth curve model to analyze the extracted sentiments directly. The paper also explores four distinct approaches for extracting sentiment scores in the dialogue, providing versatility to the proposed framework. The practical application of the proposed model is demonstrated through the analysis of an empirical longitudinal textual dataset. This framework offers a valuable contribution to the field by addressing the challenges associated with modeling qualitative textual data, providing a robust methodology for extracting and analyzing sentiments longitudinally.