Minimax Optimal Fair Classification with Bounded Demographic Disparity

Mitigating the disparate impact of statistical machine learning methods is crucial for ensuring fairness. While extensive research aims to reduce disparity, the effect of using a \emph{finite dataset} -- as opposed to the entire population -- remains unclear. This paper explores the statistical foundations of fair binary classification with two protected groups, focusing on controlling demographic disparity, defined as the difference in acceptance rates between the groups. Although fairness may come at the cost of accuracy even with infinite data, we show that using a finite sample incurs additional costs due to the need to estimate group-specific acceptance thresholds. We study the minimax optimal classification error while constraining demographic disparity to a user-specified threshold. To quantify the impact of fairness constraints, we introduce a novel measure called \emph{fairness-aware excess risk} and derive a minimax lower bound on this measure that all classifiers must satisfy. Furthermore, we propose FairBayes-DDP+, a group-wise thresholding method with an offset that we show attains the minimax lower bound. Our lower bound proofs involve several innovations. Experiments support that FairBayes-DDP+ controls disparity at the user-specified level, while being faster and having a more favorable fairness-accuracy tradeoff than several baselines.

Approximation of RKHS Functionals by Neural Networks

Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we study the approximation of functionals on reproducing kernel Hilbert spaces (RKHS's) using neural networks. We establish the universality of the approximation of functionals on the RKHS's. Specifically, we derive explicit error bounds for those induced by inverse multiquadric, Gaussian, and Sobolev kernels. Moreover, we apply our findings to functional regression, proving that neural networks can accurately approximate the regression maps in generalized functional linear models. Existing works on functional learning require integration-type basis function expansions with a set of pre-specified basis functions. By leveraging the interpolating orthogonal projections in RKHS's, our proposed network is much simpler in that we use point evaluations to replace basis function expansions.

FairRR: Pre-Processing for Group Fairness through Randomized Response

The increasing usage of machine learning models in consequential decision-making processes has spurred research into the fairness of these systems. While significant work has been done to study group fairness in the in-processing and post-processing setting, there has been little that theoretically connects these results to the pre-processing domain. This paper proposes that achieving group fairness in downstream models can be formulated as finding the optimal design matrix in which to modify a response variable in a Randomized Response framework. We show that measures of group fairness can be directly controlled for with optimal model utility, proposing a pre-processing algorithm called FairRR that yields excellent downstream model utility and fairness.

Rate-Optimal Rank Aggregation with Private Pairwise Rankings

In various real-world scenarios like recommender systems and political surveys, pairwise rankings are commonly collected and utilized for rank aggregation to obtain an overall ranking of items. However, preference rankings can reveal individuals' personal preferences, underscoring the need to protect them before releasing for downstream analysis. In this paper, we address the challenge of preserving privacy while ensuring the utility of rank aggregation based on pairwise rankings generated from the Bradley-Terry-Luce (BTL) model. Using the randomized response mechanism to perturb raw pairwise rankings is a common privacy protection strategy used in practice, but a critical challenge arises because the privatized rankings no longer adhere to the BTL model, resulting in significant bias in downstream rank aggregation tasks. Motivated from this, we propose a debiased randomized response mechanism to protect the raw pairwise rankings, ensuring consistent estimation of true preferences and rankings in downstream rank aggregation. Theoretically, we offer insights into the relationship between overall privacy guarantees and estimation errors from private ranking data, and establish minimax rates for estimation errors. This enables the determination of optimal privacy guarantees that balance consistency in rank aggregation with robust privacy protection. We also investigate convergence rates of expected ranking errors for partial and full ranking recovery, quantifying how privacy protection influences the specification of top-$K$ item sets and complete rankings. Our findings are validated through extensive simulations and a real application.

Bayes-Optimal Fair Classification with Linear Disparity Constraints via Pre-, In-, and Post-processing

Machine learning algorithms may have disparate impacts on protected groups. To address this, we develop methods for Bayes-optimal fair classification, aiming to minimize classification error subject to given group fairness constraints. We introduce the notion of \emph{linear disparity measures}, which are linear functions of a probabilistic classifier; and \emph{bilinear disparity measures}, which are also linear in the group-wise regression functions. We show that several popular disparity measures -- the deviations from demographic parity, equality of opportunity, and predictive equality -- are bilinear. We find the form of Bayes-optimal fair classifiers under a single linear disparity measure, by uncovering a connection with the Neyman-Pearson lemma. For bilinear disparity measures, Bayes-optimal fair classifiers become group-wise thresholding rules. Our approach can also handle multiple fairness constraints (such as equalized odds), and the common scenario when the protected attribute cannot be used at the prediction phase. Leveraging our theoretical results, we design methods that learn fair Bayes-optimal classifiers under bilinear disparity constraints. Our methods cover three popular approaches to fairness-aware classification, via pre-processing (Fair Up- and Down-Sampling), in-processing (Fair Cost-Sensitive Classification) and post-processing (a Fair Plug-In Rule). Our methods control disparity directly while achieving near-optimal fairness-accuracy tradeoffs. We show empirically that our methods compare favorably to existing algorithms.

Benefits of Transformer: In-Context Learning in Linear Regression Tasks with Unstructured Data

In practice, it is observed that transformer-based models can learn concepts in context in the inference stage. While existing literature, e.g., \citet{zhang2023trained,huang2023context}, provide theoretical explanations on this in-context learning ability, they assume the input $x_i$ and the output $y_i$ for each sample are embedded in the same token (i.e., structured data). However, in reality, they are presented in two tokens (i.e., unstructured data \cite{wibisono2023role}). In this case, this paper conducts experiments in linear regression tasks to study the benefits of the architecture of transformers and provides some corresponding theoretical intuitions to explain why the transformer can learn from unstructured data. We study the exact components in a transformer that facilitate the in-context learning. In particular, we observe that (1) a transformer with two layers of softmax (self-)attentions with look-ahead attention mask can learn from the prompt if $y_i$ is in the token next to $x_i$ for each example; (2) positional encoding can further improve the performance; and (3) multi-head attention with a high input embedding dimension has a better prediction performance than single-head attention.

Better Representations via Adversarial Training in Pre-Training: A Theoretical Perspective

Pre-training is known to generate universal representations for downstream tasks in large-scale deep learning such as large language models. Existing literature, e.g., \cite{kim2020adversarial}, empirically observe that the downstream tasks can inherit the adversarial robustness of the pre-trained model. We provide theoretical justifications for this robustness inheritance phenomenon. Our theoretical results reveal that feature purification plays an important role in connecting the adversarial robustness of the pre-trained model and the downstream tasks in two-layer neural networks. Specifically, we show that (i) with adversarial training, each hidden node tends to pick only one (or a few) feature; (ii) without adversarial training, the hidden nodes can be vulnerable to attacks. This observation is valid for both supervised pre-training and contrastive learning. With purified nodes, it turns out that clean training is enough to achieve adversarial robustness in downstream tasks.

A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Models

In this article, we review the literature on statistical theories of neural networks from three perspectives. In the first part, results on excess risks for neural networks are reviewed in the nonparametric framework of regression or classification. These results rely on explicit constructions of neural networks, leading to fast convergence rates of excess risks, in that tools from the approximation theory are adopted. Through these constructions, the width and depth of the networks can be expressed in terms of sample size, data dimension, and function smoothness. Nonetheless, their underlying analysis only applies to the global minimizer in the highly non-convex landscape of deep neural networks. This motivates us to review the training dynamics of neural networks in the second part. Specifically, we review papers that attempt to answer ``how the neural network trained via gradient-based methods finds the solution that can generalize well on unseen data.'' In particular, two well-known paradigms are reviewed: the Neural Tangent Kernel (NTK) paradigm, and Mean-Field (MF) paradigm. In the last part, we review the most recent theoretical advancements in generative models including Generative Adversarial Networks (GANs), diffusion models, and in-context learning (ICL) in the Large Language Models (LLMs). The former two models are known to be the main pillars of the modern generative AI era, while ICL is a strong capability of LLMs in learning from a few examples in the context. Finally, we conclude the paper by suggesting several promising directions for deep learning theory.

Downstream Task-Oriented Generative Model Selections on Synthetic Data Training for Fraud Detection Models

Devising procedures for downstream task-oriented generative model selections is an unresolved problem of practical importance. Existing studies focused on the utility of a single family of generative models. They provided limited insights on how synthetic data practitioners select the best family generative models for synthetic training tasks given a specific combination of machine learning model class and performance metric. In this paper, we approach the downstream task-oriented generative model selections problem in the case of training fraud detection models and investigate the best practice given different combinations of model interpretability and model performance constraints. Our investigation supports that, while both Neural Network(NN)-based and Bayesian Network(BN)-based generative models are both good to complete synthetic training task under loose model interpretability constrain, the BN-based generative models is better than NN-based when synthetic training fraud detection model under strict model interpretability constrain. Our results provides practical guidance for machine learning practitioner who is interested in replacing their training dataset from real to synthetic, and shed lights on more general downstream task-oriented generative model selection problems.