A Statistical Framework of Watermarks for Large Language Models: Pivot, Detection Efficiency and Optimal Rules

Since ChatGPT was introduced in November 2022, embedding (nearly) unnoticeable statistical signals into text generated by large language models (LLMs), also known as watermarking, has been used as a principled approach to provable detection of LLM-generated text from its human-written counterpart. In this paper, we introduce a general and flexible framework for reasoning about the statistical efficiency of watermarks and designing powerful detection rules. Inspired by the hypothesis testing formulation of watermark detection, our framework starts by selecting a pivotal statistic of the text and a secret key -- provided by the LLM to the verifier -- to enable controlling the false positive rate (the error of mistakenly detecting human-written text as LLM-generated). Next, this framework allows one to evaluate the power of watermark detection rules by obtaining a closed-form expression of the asymptotic false negative rate (the error of incorrectly classifying LLM-generated text as human-written). Our framework further reduces the problem of determining the optimal detection rule to solving a minimax optimization program. We apply this framework to two representative watermarks -- one of which has been internally implemented at OpenAI -- and obtain several findings that can be instrumental in guiding the practice of implementing watermarks. In particular, we derive optimal detection rules for these watermarks under our framework. These theoretically derived detection rules are demonstrated to be competitive and sometimes enjoy a higher power than existing detection approaches through numerical experiments.

Fair Canonical Correlation Analysis

This paper investigates fairness and bias in Canonical Correlation Analysis (CCA), a widely used statistical technique for examining the relationship between two sets of variables. We present a framework that alleviates unfairness by minimizing the correlation disparity error associated with protected attributes. Our approach enables CCA to learn global projection matrices from all data points while ensuring that these matrices yield comparable correlation levels to group-specific projection matrices. Experimental evaluation on both synthetic and real-world datasets demonstrates the efficacy of our method in reducing correlation disparity error without compromising CCA accuracy.

MISNN: Multiple Imputation via Semi-parametric Neural Networks

Multiple imputation (MI) has been widely applied to missing value problems in biomedical, social and econometric research, in order to avoid improper inference in the downstream data analysis. In the presence of high-dimensional data, imputation models that include feature selection, especially $\ell_1$ regularized regression (such as Lasso, adaptive Lasso, and Elastic Net), are common choices to prevent the model from underdetermination. However, conducting MI with feature selection is difficult: existing methods are often computationally inefficient and poor in performance. We propose MISNN, a novel and efficient algorithm that incorporates feature selection for MI. Leveraging the approximation power of neural networks, MISNN is a general and flexible framework, compatible with any feature selection method, any neural network architecture, high/low-dimensional data and general missing patterns. Through empirical experiments, MISNN has demonstrated great advantages over state-of-the-art imputation methods (e.g. Bayesian Lasso and matrix completion), in terms of imputation accuracy, statistical consistency and computation speed.

Multiple Imputation with Neural Network Gaussian Process for High-dimensional Incomplete Data

Missing data are ubiquitous in real world applications and, if not adequately handled, may lead to the loss of information and biased findings in downstream analysis. Particularly, high-dimensional incomplete data with a moderate sample size, such as analysis of multi-omics data, present daunting challenges. Imputation is arguably the most popular method for handling missing data, though existing imputation methods have a number of limitations. Single imputation methods such as matrix completion methods do not adequately account for imputation uncertainty and hence would yield improper statistical inference. In contrast, multiple imputation (MI) methods allow for proper inference but existing methods do not perform well in high-dimensional settings. Our work aims to address these significant methodological gaps, leveraging recent advances in neural network Gaussian process (NNGP) from a Bayesian viewpoint. We propose two NNGP-based MI methods, namely MI-NNGP, that can apply multiple imputations for missing values from a joint (posterior predictive) distribution. The MI-NNGP methods are shown to significantly outperform existing state-of-the-art methods on synthetic and real datasets, in terms of imputation error, statistical inference, robustness to missing rates, and computation costs, under three missing data mechanisms, MCAR, MAR, and MNAR.

CEDAR: Communication Efficient Distributed Analysis for Regressions

Electronic health records (EHRs) offer great promises for advancing precision medicine and, at the same time, present significant analytical challenges. Particularly, it is often the case that patient-level data in EHRs cannot be shared across institutions (data sources) due to government regulations and/or institutional policies. As a result, there are growing interests about distributed learning over multiple EHRs databases without sharing patient-level data. To tackle such challenges, we propose a novel communication efficient method that aggregates the local optimal estimates, by turning the problem into a missing data problem. In addition, we propose incorporating posterior samples of remote sites, which can provide partial information on the missing quantities and improve efficiency of parameter estimates while having the differential privacy property and thus reducing the risk of information leaking. The proposed approach, without sharing the raw patient level data, allows for proper statistical inference and can accommodate sparse regressions. We provide theoretical investigation for the asymptotic properties of the proposed method for statistical inference as well as differential privacy, and evaluate its performance in simulations and real data analyses in comparison with several recently developed methods.

Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation

Estimating treatment effects is of great importance for many biomedical applications with observational data. Particularly, interpretability of the treatment effects is preferable for many biomedical researchers. In this paper, we first give a theoretical analysis and propose an upper bound for the bias of average treatment effect estimation under the strong ignorability assumption. The proposed upper bound consists of two parts: training error for factual outcomes, and the distance between treated and control distributions. We use the Weighted Energy Distance (WED) to measure the distance between two distributions. Motivated by the theoretical analysis, we implement this upper bound as an objective function being minimized by leveraging a novel additive neural network architecture, which combines the expressivity of deep neural network, the interpretability of generalized additive model, the sufficiency of the balancing score for estimation adjustment, and covariate balancing properties of treated and control distributions, for estimating average treatment effects from observational data. Furthermore, we impose a so-called weighted regularization procedure based on non-parametric theory, to obtain some desirable asymptotic properties. The proposed method is illustrated by re-examining the benchmark datasets for causal inference, and it outperforms the state-of-art.

Multiple Imputation via Generative Adversarial Network for High-dimensional Blockwise Missing Value Problems

Missing data are present in most real world problems and need careful handling to preserve the prediction accuracy and statistical consistency in the downstream analysis. As the gold standard of handling missing data, multiple imputation (MI) methods are proposed to account for the imputation uncertainty and provide proper statistical inference. In this work, we propose Multiple Imputation via Generative Adversarial Network (MI-GAN), a deep learning-based (in specific, a GAN-based) multiple imputation method, that can work under missing at random (MAR) mechanism with theoretical support. MI-GAN leverages recent progress in conditional generative adversarial neural works and shows strong performance matching existing state-of-the-art imputation methods on high-dimensional datasets, in terms of imputation error. In particular, MI-GAN significantly outperforms other imputation methods in the sense of statistical inference and computational speed.

Assessing Fairness in the Presence of Missing Data

Missing data are prevalent and present daunting challenges in real data analysis. While there is a growing body of literature on fairness in analysis of fully observed data, there has been little theoretical work on investigating fairness in analysis of incomplete data. In practice, a popular analytical approach for dealing with missing data is to use only the set of complete cases, i.e., observations with all features fully observed to train a prediction algorithm. However, depending on the missing data mechanism, the distribution of complete cases and the distribution of the complete data may be substantially different. When the goal is to develop a fair algorithm in the complete data domain where there are no missing values, an algorithm that is fair in the complete case domain may show disproportionate bias towards some marginalized groups in the complete data domain. To fill this significant gap, we study the problem of estimating fairness in the complete data domain for an arbitrary model evaluated merely using complete cases. We provide upper and lower bounds on the fairness estimation error and conduct numerical experiments to assess our theoretical results. Our work provides the first known theoretical results on fairness guarantee in analysis of incomplete data.

Fairness in Missing Data Imputation

Missing data are ubiquitous in the era of big data and, if inadequately handled, are known to lead to biased findings and have deleterious impact on data-driven decision makings. To mitigate its impact, many missing value imputation methods have been developed. However, the fairness of these imputation methods across sensitive groups has not been studied. In this paper, we conduct the first known research on fairness of missing data imputation. By studying the performance of imputation methods in three commonly used datasets, we demonstrate that unfairness of missing value imputation widely exists and may be associated with multiple factors. Our results suggest that, in practice, a careful investigation of related factors can provide valuable insights on mitigating unfairness associated with missing data imputation.

Differentially Private Bayesian Neural Networks on Accuracy, Privacy and Reliability

Bayesian neural network (BNN) allows for uncertainty quantification in prediction, offering an advantage over regular neural networks that has not been explored in the differential privacy (DP) framework. We fill this important gap by leveraging recent development in Bayesian deep learning and privacy accounting to offer a more precise analysis of the trade-off between privacy and accuracy in BNN. We propose three DP-BNNs that characterize the weight uncertainty for the same network architecture in distinct ways, namely DP-SGLD (via the noisy gradient method), DP-BBP (via changing the parameters of interest) and DP-MC Dropout (via the model architecture). Interestingly, we show a new equivalence between DP-SGD and DP-SGLD, implying that some non-Bayesian DP training naturally allows for uncertainty quantification. However, the hyperparameters such as learning rate and batch size, can have different or even opposite effects in DP-SGD and DP-SGLD. Extensive experiments are conducted to compare DP-BNNs, in terms of privacy guarantee, prediction accuracy, uncertainty quantification, calibration, computation speed, and generalizability to network architecture. As a result, we observe a new tradeoff between the privacy and the reliability. When compared to non-DP and non-Bayesian approaches, DP-SGLD is remarkably accurate under strong privacy guarantee, demonstrating the great potential of DP-BNN in real-world tasks.