This paper introduces a Wasserstein Distributionally Robust Optimization framework for nonparametric regression, using Lipschitz-constrained feedforward neural networks to derive non-asymptotic error bounds for local worst-case risk under model misspecification, demonstrating robustness through simulations and MNIST dataset application.
Nonparametric Regression, Robustness, Neural Networks, Model Uncertainty, Generalization Bounds
Changyu Liu, Yuling Jiao, Junhui Wang, Jian Huang
The Hong Kong Polytechnic University, Wuhan University, The Chinese University of Hong Kong
Generated by grok-3
Background Problem
The paper addresses the challenge of model uncertainty in nonparametric regression, where traditional estimators may fail due to incorrect modeling assumptions, data contamination, or distributional inconsistencies between training and target data. It aims to enhance robustness by developing a Wasserstein Distributionally Robust Optimization (WDRO) framework that minimizes the local worst-case risk within a predefined ambiguity set, thereby tackling generalization issues under potential model misspecification where the estimation function space does not fully capture the target function.
Method
The core method is a distributionally robust nonparametric estimator based on Wasserstein Distributionally Robust Optimization (WDRO). It formulates the problem as minimizing the local worst-case risk, defined as the supremum of expected loss over distributions within a δ-Wasserstein ball centered at the true data distribution. The estimation function space is constructed using feedforward neural networks (FNNs) with Lipschitz constraints to ensure controlled complexity and approximation ability. Key steps include: (1) defining the ambiguity set using Wasserstein distance of order k, (2) reformulating the worst-case risk via strong duality into a tractable optimization problem, and (3) deriving non-asymptotic error bounds for excess local worst-case risk and natural risk under model misspecification. Additionally, a Lagrangian relaxation approach is proposed to improve computational efficiency by transforming the constrained optimization into a penalized form.
Experiment
The experiments evaluate the proposed WDRO-based nonparametric estimator through simulation studies and an application to the MNIST dataset. Simulations cover various regression tasks (robust, quantile, least squares) and classification, testing the estimator under different levels of uncertainty (δ). The setup includes comparisons of error rates with theoretical bounds, showing that the achieved rates (e.g., n^{-α/(2d+3α)} for Lipschitz losses and n^{-2α/(2d+5α)} for quadratic loss) align with expectations, though slightly slower than unconstrained FNN rates due to Lipschitz constraints. The MNIST application demonstrates robustness to distributional shifts, but lacks comprehensive comparisons with other robust methods or detailed analysis of failure modes. While the experimental design is reasonable for validating theoretical bounds, it is not exhaustive, as it misses broader benchmarks and real-world complexities, potentially overestimating practical effectiveness.
Further Thoughts
The paper’s focus on Wasserstein Distributionally Robust Optimization opens up interesting avenues for integrating with other robust learning paradigms, such as adversarial training or domain adaptation, where distributional shifts are also critical. I am particularly curious about how the proposed method would perform in federated learning settings, where data heterogeneity across clients mirrors the distributional inconsistencies addressed here—could WDRO provide a unified framework for robustness in such decentralized environments? Additionally, the reliance on Lipschitz-constrained FNNs, while theoretically sound for controlling complexity, might be restrictive in capturing highly non-smooth functions; exploring hybrid approaches with adaptive constraints or alternative architectures like transformers could be a fruitful direction. Lastly, the limited empirical scope suggests a need for broader testing against state-of-the-art robust methods (e.g., adversarial training or other DRO variants) on diverse datasets beyond MNIST, such as natural language or time-series data, to truly gauge practical impact and uncover potential limitations in high-dimensional, unstructured settings.