Understanding temporal patterns in online search behavior is crucial for real-time marketing and trend forecasting. Google Trends offers a rich proxy for public interest, yet the high dimensionality and noise of its time-series data present challenges for effective clustering. This study evaluates three unsupervised clustering approaches, Symbolic Aggregate approXimation (SAX), enhanced SAX (eSAX), and Topological Data Analysis (TDA), applied to 20 Google Trends keywords representing major consumer categories. Our results show that while SAX and eSAX offer fast and interpretable clustering for stable time series, they struggle with volatility and complexity, often producing ambiguous ``catch-all'' clusters. TDA, by contrast, captures global structural features through persistent homology and achieves more balanced and meaningful groupings. We conclude with practical guidance for using symbolic and topological methods in consumer analytics and suggest that hybrid approaches combining both perspectives hold strong potential for future applications.

In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.

The mean shift (MS) is a non-parametric, density-based, iterative algorithm that has prominent usage in clustering and image segmentation. A rigorous proof for its convergence in full generality remains unknown. Two significant steps in this direction were taken in the paper \cite{Gh1}, which proved that for \textit{sufficiently large bandwidth}, the MS algorithm with the Gaussian kernel always converges in any dimension, and also by the same author in \cite{Gh2}, proved that MS always converges in one dimension for kernels with differentiable, strictly decreasing, convex profiles. In the more recent paper \cite{YT}, they have proved the convergence in more generality,\textit{ without any restriction on the bandwidth}, with the assumption that the KDE $f$ has a continuous Lipschitz gradient on the closure of the convex hull of the trajectory of the iterated sequence of the mode estimate, and also satisfies the Łojasiewicz property there. The main theoretical result of this paper is a generalization of those of \cite{Gh1}, where we show that (1) for\textit{ sufficiently large bandwidth} convergence is guaranteed in any dimension with \textit{any radially symmetric and strictly positive definite kernels}. The proof uses two alternate characterizations of radially symmetric positive definite smooth kernels by Schoenberg and Bernstein \cite{Fass}, and borrows some steps from the proofs in \cite{Gh1}. Although the authors acknowledge that the result in that paper is more restrictive than that of \cite{YT} due to the lower bandwidth limit, it uses a different set of assumptions than \cite{YT}, and the proof technique is different.

Solving conditional moment restrictions (CMRs) is a key problem considered in statistics, causal inference, and econometrics, where the aim is to solve for a function of interest that satisfies some conditional moment equalities. Specifically, many techniques for causal inference, such as instrumental variable (IV) regression and proximal causal learning (PCL), are CMR problems. Most CMR estimators use a two-stage approach, where the first-stage estimation is directly plugged into the second stage to estimate the function of interest. However, naively plugging in the first-stage estimator can cause heavy bias in the second stage. This is particularly the case for recently proposed CMR estimators that use deep neural network (DNN) estimators for both stages, where regularisation and overfitting bias is present. We propose DML-CMR, a two-stage CMR estimator that provides an unbiased estimate with fast convergence rate guarantees. We derive a novel learning objective to reduce bias and develop the DML-CMR algorithm following the double/debiased machine learning (DML) framework. We show that our DML-CMR estimator can achieve the minimax optimal convergence rate of $O(N^{-1/2})$ under parameterisation and mild regularity conditions, where $N$ is the sample size. We apply DML-CMR to a range of problems using DNN estimators, including IV regression and proximal causal learning on real-world datasets, demonstrating state-of-the-art performance against existing CMR estimators and algorithms tailored to those problems.

Educational disparities are rooted in and perpetuate social inequalities across multiple dimensions such as race, socioeconomic status, and geography. To reduce disparities, most intervention strategies focus on a single domain and frequently evaluate their effectiveness by using causal decomposition analysis. However, a growing body of research suggests that single-domain interventions may be insufficient for individuals marginalized on multiple fronts. While interventions across multiple domains are increasingly proposed, there is limited guidance on appropriate methods for evaluating their effectiveness. To address this gap, we develop an extended causal decomposition analysis that simultaneously targets multiple causally ordered intervening factors, allowing for the assessment of their synergistic effects. These scenarios often involve challenges related to model misspecification due to complex interactions among group categories, intervening factors, and their confounders with the outcome. To mitigate these challenges, we introduce a triply robust estimator that leverages machine learning techniques to address potential model misspecification. We apply our method to a cohort of students from the High School Longitudinal Study, focusing on math achievement disparities between Black, Hispanic, and White high schoolers. Specifically, we examine how two sequential interventions - equalizing the proportion of students who attend high-performing schools and equalizing enrollment in Algebra I by 9th grade across racial groups - may reduce these disparities.

Hyperparameter optimization (HPO) aims to identify an optimal hyperparameter configuration (HPC) such that the resulting model generalizes well to unseen data. As the expected generalization error cannot be optimized directly, it is estimated with a resampling strategy, such as holdout or cross-validation. This approach implicitly assumes that minimizing the validation error leads to improved generalization. However, since validation error estimates are inherently stochastic and depend on the resampling strategy, a natural question arises: Can excessive optimization of the validation error lead to overfitting at the HPO level, akin to overfitting in model training based on empirical risk minimization? In this paper, we investigate this phenomenon, which we term overtuning, a form of overfitting specific to HPO. Despite its practical relevance, overtuning has received limited attention in the HPO and AutoML literature. We provide a formal definition of overtuning and distinguish it from related concepts such as meta-overfitting. We then conduct a large-scale reanalysis of HPO benchmark data to assess the prevalence and severity of overtuning. Our results show that overtuning is more common than previously assumed, typically mild but occasionally severe. In approximately 10% of cases, overtuning leads to the selection of a seemingly optimal HPC with worse generalization error than the default or first configuration tried. We further analyze how factors such as performance metric, resampling strategy, dataset size, learning algorithm, and HPO method affect overtuning and discuss mitigation strategies. Our results highlight the need to raise awareness of overtuning, particularly in the small-data regime, indicating that further mitigation strategies should be studied.

We make two closely related theoretical contributions to the use of importance sampling schemes. First, for independent sampling, we prove that the minimax optimal trial distribution coincides with the target if and only if the target distribution has no atom with probability greater than $1/2$, where "minimax" means that the worst-case asymptotic variance of the self-normalized importance sampling estimator is minimized. When a large atom exists, it should be downweighted by the trial distribution. A similar phenomenon holds for a continuous target distribution concentrated on a small set. Second, we argue that it is often advantageous to run the Metropolis--Hastings algorithm with a tempered stationary distribution, $π(x)^β$, and correct for the bias by importance weighting. The dynamics of this "importance-tempered" sampling scheme can be described by a continuous-time Markov chain. We prove that for one-dimensional targets with polynomial tails, $π(x) \propto (1 + |x|)^{-γ}$, this chain is uniformly ergodic if and only if $1/γ< β< (γ- 2)/γ$. These results suggest that for target distributions with light or polynomial tails of order $γ> 3$, importance tempering can improve the precision of time-average estimators and essentially eliminate the need for burn-in.

Model regularization requires extensive manual tuning to balance complexity against overfitting. Cross-regularization resolves this tradeoff by directly adapting regularization parameters through validation gradients during training. The method splits parameter optimization - training data guides feature learning while validation data shapes complexity controls - converging provably to cross-validation optima. When implemented through noise injection in neural networks, this approach reveals striking patterns: unexpectedly high noise tolerance and architecture-specific regularization that emerges organically during training. Beyond complexity control, the framework integrates seamlessly with data augmentation, uncertainty calibration and growing datasets while maintaining single-run efficiency through a simple gradient-based approach.

In recent years, machine learning (ML) techniques have become a powerful tool for improving the accuracy of predictions and decision-making. Machine learning technologies have begun to penetrate all areas, including the real estate sector. Correct forecasting of real estate value plays an important role in the buyer-seller chain, because it ensures reasonableness of price expectations based on the offers available in the market and helps to avoid financial risks for both parties of the transaction. Accurate forecasting is also important for real estate investors to make an informed decision on a specific property. This study helps to gain a deeper understanding of how effective and accurate ensemble machine learning methods are in predicting real estate values. The results obtained in the work are quite accurate, as can be seen from the coefficient of determination (R^2), root mean square error (RMSE) and mean absolute error (MAE) calculated for each model. The Gradient Boosting Regressor model provides the highest accuracy, the Extra Trees Regressor, Hist Gradient Boosting Regressor and Random Forest Regressor models give good results. In general, ensemble machine learning techniques can be effectively used to solve real estate valuation. This work forms ideas for future research, which consist in the preliminary processing of the data set by searching and extracting anomalous values, as well as the practical implementation of the obtained results.

Differential privacy (DP) has become the standard for private data analysis. Certain machine learning applications only require privacy protection for specific protected attributes. Using naive variants of differential privacy in such use cases can result in unnecessary degradation of utility. In this work, we refine the definition of DP to create a more general and flexible framework that we call feature differential privacy (FDP). Our definition is simulation-based and allows for both addition/removal and replacement variants of privacy, and can handle arbitrary and adaptive separation of protected and non-protected features. We prove the properties of FDP, such as adaptive composition, and demonstrate its implications for limiting attribute inference attacks. We also propose a modification of the standard DP-SGD algorithm that satisfies FDP while leveraging desirable properties such as amplification via sub-sampling. We apply our framework to various machine learning tasks and show that it can significantly improve the utility of DP-trained models when public features are available. For example, we train diffusion models on the AFHQ dataset of animal faces and observe a drastic improvement in FID compared to DP, from 286.7 to 101.9 at $ε=8$, assuming that the blurred version of a training image is available as a public feature. Overall, our work provides a new approach to private data analysis that can help reduce the utility cost of DP while still providing strong privacy guarantees.