Advancing Semi-Supervised U-Statistics for Better Data Utilization

In many areas, getting data that is fully labeled can be very hard and costly. This situation leads to a big need for methods that can make good use of data that doesn't have labels. To help with this, we present a new approach called semi-supervised U-statistics. This method takes advantage of both labeled and unlabeled data and investigates how well it can perform in various situations.

#Importance of Semi-Supervised Learning

Semi-supervised learning is useful because it allows for more accurate predictions by making use of both labeled and unlabeled datasets. This is especially important in fields like healthcare, where annotating medical records is expensive and slow. Other applications, such as handwriting recognition and fraud detection, face similar challenges. By using semi-supervised learning, we can use a lot of unlabeled data to improve prediction accuracy.

Despite making progress in semi-supervised methods, much of the focus has been on classification tasks. Recently, attention has shifted to statistical estimation and inference in semi-supervised settings. The goal here is to understand how unlabeled data can be useful and improve traditional methods. Even though some progress has been made, there are still many unresolved problems that could benefit from using unlabeled data.

#U-Statistics and Their Challenge

U-statistics are a specific type of statistical estimator that can be hard to improve when we include unlabeled data in their construction. Some previous works investigated the idea of semi-supervised U-statistics but did not fully explore whether these methods can be optimal in all contexts. It's also unclear if improvements can be achieved when the underlying kernel of a U-statistic becomes degenerate.

#Connections to Missing Data Problems

To understand optimality properties, one can relate semi-supervised settings to missing data problems. In missing data, we assume that some data points are missing completely at random, which can help to analyze how semi-supervised learning might work. However, this comparison has its limitations, especially when making assumptions about the amount of labeled data in relation to unlabeled data.

#Contributions of Our Work

In our work, we aim to tackle the challenges in semi-supervised estimation and inference. We introduce a new class of semi-supervised estimators that enhance classical U-statistics, aiming to improve the statistical properties of these methods in various situations. Our main contributions can be summarized as follows:

#Semi-Supervised U-Statistics

We offer a new way to perform semi-supervised U-statistics that integrates extra information from unlabeled data. This allows for improved performance over traditional U-statistics. We present methods for implementing these estimators and identify conditions which help ensure that they have desirable statistical properties.

#Berry-Esseen Bounds

We quantify how well the proposed statistics approximate a normal distribution in finite samples. This involves studying Berry-Esseen bounds that demonstrate how the convergence rate of our estimators depends on prediction error. We show that our approach provides a better trade-off between validity and efficiency in certain cases.

#Minimax Lower Bounds

We establish lower bounds in semi-supervised settings that match the asymptotic mean squared error of our proposed estimators. This analysis allows us to demonstrate that our methods are asymptotically efficient.

#Degenerate U-Statistics and Adaptivity

We also look closely at cases where the kernel of the U-statistic is degenerate. We create a refined semi-supervised U-statistic that adapts to these situations, showing improvements over the classical U-statistic.

#Connection to Missing Data Problems

We discuss the relationship between semi-supervised learning and missing data frameworks, identifying situations where their minimax risks can converge. This connection allows for a richer understanding of how to utilize techniques from both fields.

#Related Work

Numerous studies have examined classic statistical problems in semi-supervised settings, leading to effective methods that enhance supervised approaches. Recent advancements have proposed semi-supervised mean estimators and explored the idea of empirical risk minimization by incorporating unlabeled data.

Our work fits within this growing body of research by presenting a broader framework for semi-supervised U-statistics. This includes the semi-supervised estimation methods discussed in previous studies, positioning our contributions as an essential addition to the literature.

#Problem Setup and Motivation

To introduce our semi-supervised U-statistics, we first define our problem setup clearly. We have a joint distribution with labeled and unlabeled samples drawn from it. The main goal is to estimate a parameter cleverly using both sets of data. Depending on the chosen functional, this problem can cover many important statistical parameters.

#Oracle Mean Estimation

We start with a straightforward case, focusing on estimating the population mean. We highlight that the sample mean has certain optimality properties but can be improved when extra information from covariates is included. This leads us to propose a new semi-supervised version of the U-statistic that effectively uses these additional covariates.

#Extension to a General Kernel

Next, we expand our semi-supervised mean estimator to a general kernel function. This step allows us to relate the new method back to U-statistics, applying similar reasoning that was used for the sample mean. By introducing our estimator, we aim to produce a more accurate estimate while still being unbiased.

#Practical Procedures for Semi-Supervised U-Statistics

#Procedure with Cross-Fitting

We then present two practical methods for implementing our semi-supervised U-statistic. The first method involves cross-fitting, where we partition the datasets and use one part to estimate parameters while the other part computes the U-statistic. This process is repeated with the roles of data being swapped, allowing for a final combined estimate, which enhances the overall estimation quality.

#Procedure without Sample Splitting

In our second approach, we analyze the total dataset without splitting it. While this method has its own requirements for theoretical guarantees, it can potentially improve small-sample performance. The focus here is on constructing a U-statistic that utilizes the entire dataset, offering an alternative that can be more efficient under certain conditions.

#Berry-Esseen Bounds

We now study Berry-Esseen bounds for our semi-supervised U-statistics. A key aspect of this analysis is demonstrating how the convergence rate to a normal distribution relies on different variables and estimates. This is important because it provides insights into the distributional properties of our proposed method.

#Cross-Fit Estimator

We derive a Berry-Esseen bound for the cross-fit estimator, analyzing how well our method approximates a normal distribution. This involves looking at various moments and ensuring that our estimators converge appropriately.

#Single-Split Estimator

We also investigate a single-split version of our semi-supervised U-statistic. This method offers different performance characteristics compared to the cross-fit estimator, and highlights a trade-off between validity and efficiency when constructing confidence intervals.

#Minimax Lower Bounds

In this section, we derive lower bounds for estimating parameters in semi-supervised settings. The approach we take helps clarify the challenges we face in this domain, and provides a structured way to analyze and compare our estimators.

#Van Trees Inequality

By adapting the well-known van Trees inequality, we establish a framework for analyzing the minimax risk under semi-supervised settings. This crucial step allows us to present asymptotically tight lower bounds for the risks we consider.

#Degenerate U-Statistics and Adaptivity

Next, we address the scenario where the kernel of the U-statistic is degenerate. In such cases, we propose a refined version of the semi-supervised U-statistic that can adjust to the degeneracy and enhance performance. By focusing on specific cases of bivariate kernels, we demonstrate improvements across different regimes.

#Practical Applications: Estimating Parameters

#Parameter Estimation

We show how our semi-supervised U-statistic framework can be applied to estimate parameters effectively. By providing a clear method for how to carry out these estimations, we help bridge the gap between theory and practice.

#Simulation Studies

To bolster our theoretical findings, we conduct simulation studies. These studies validate our proposed methods and demonstrate their performance against existing techniques. This empirical evidence is crucial in understanding the practical implications of our work.

#Semi-Supervised Nonparametric Tests

We further explore practical applications by developing semi-supervised tests, such as Kendall's tau and Wilcoxon signed rank tests. These tests are designed to assess independence and performance considerably surpass classical methods.

#Conclusion

In summary, our study introduces semi-supervised U-statistics that meaningfully incorporate unlabeled data to improve classical methods. By leveraging techniques like cross-fitting, we show that our approach can achieve strong performance under various conditions. Our findings have significant implications for statistical estimation and inference, expanding the range of applicable scenarios.

#Future Work

There is much room for future exploration in this area. Possible extensions could involve different forms of U-statistics and addressing the computational challenges linked to more complex situations. Additionally, refining adaptive results for higher-order kernels could yield benefits for many inference methods. We view the connection between semi-supervised learning and missing data as a rich area ripe for further investigation.

#Acknowledgments

We express gratitude to those who provided insight and feedback on our work. Their contributions have been vital in shaping the ideas presented here.