Xuelin Zhang
Publications
On the Stability and Generalization of First-order Bilevel Minimax Optimization
Bilevel optimization and bilevel minimax optimization have recently emerged as unifying frameworks for a range of machine-learning tasks, including hyperparameter optimization and reinforcement learning. The existing literature focuses on empirical efficiency and convergence guarantees, leaving a critical theoretical gap in understanding how well these algorithms generalize. To bridge this gap, we provide the first systematic generalization analysis for first-order gradient-based bilevel minimax solvers with lower-level minimax problems. Specifically, by leveraging algorithmic stability arguments, we derive fine-grained generalization bounds for three representative algorithms, including single-timescale stochastic gradient descent-ascent, and two variants of two-timescale stochastic gradient descent-ascent. Our results reveal a precise trade-off among algorithmic stability, generalization gaps, and practical settings. Furthermore, extensive empirical evaluations corroborate our theoretical insights on realistic optimization tasks with bilevel minimax structures.
Meta Additive Model: Interpretable Sparse Learning With Auto Weighting
Sparse additive models have attracted much attention in high-dimensional data analysis due to their flexible representation and strong interpretability. However, most existing models are limited to single-level learning under the mean-squared error criterion, whose empirical performance can degrade significantly in the presence of complex noise, such as non-Gaussian perturbations, outliers, noisy labels, and imbalanced categories. The sample reweighting strategy is widely used to reduce the model's sensitivity to atypical data; however, it typically requires prespecifying the weighting functions and manually selecting additional hyperparameters. To address this issue, we propose a new meta additive model (MAM) based on the bilevel optimization framework, which learns data-driven weighting of individual losses by parameterizing the weighting function via an MLP trained on meta data. MAM is capable of a variety of learning tasks, including variable selection, robust regression estimation, and imbalanced classification. Theoretically, MAM provides guarantees on convergence in computation, algorithmic generalization, and variable selection consistency under mild conditions. Empirically, MAM outperforms several state-of-the-art additive models on both synthetic and real-world data under various data corruptions.
S2MAM: Semi-supervised Meta Additive Model for Robust Estimation and Variable Selection
Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new \textit{Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.
Fine-grained Analysis of Stability and Generalization for Stochastic Bilevel Optimization
Stochastic bilevel optimization (SBO) has been integrated into many machine learning paradigms recently, including hyperparameter optimization, meta learning, and reinforcement learning. Along with the wide range of applications, there have been numerous studies on the computational behavior of SBO. However, the generalization guarantees of SBO methods are far less understood from the lens of statistical learning theory. In this paper, we provide a systematic generalization analysis of the first-order gradient-based bilevel optimization methods. Firstly, we establish the quantitative connections between the on-average argument stability and the generalization gap of SBO methods. Then, we derive the upper bounds of on-average argument stability for single-timescale stochastic gradient descent (SGD) and two-timescale SGD, where three settings (nonconvex-nonconvex (NC-NC), convex-convex (C-C), and strongly-convex-strongly-convex (SC-SC)) are considered respectively. Experimental analysis validates our theoretical findings. Compared with the previous algorithmic stability analysis, our results do not require reinitializing the inner-level parameters at each iteration and are applicable to more general objective functions.
Beyond False Discovery Rate: A Stepdown Group SLOPE Approach for Grouped Variable Selection
High-dimensional feature selection is routinely required to balance statistical power with strict control of multiple-error metrics such as the k-Family-Wise Error Rate (k-FWER) and the False Discovery Proportion (FDP), yet some existing frameworks are confined to the narrower goal of controlling the expected False Discovery Rate (FDR) and can not exploit the group-structure of the covariates, such as Sorted L-One Penalized Estimation (SLOPE). We introduce the Group Stepdown SLOPE, a unified optimization procedure which is capable of embedding the Lehmann-Romano stepdown rules into SLOPE to achieve finite-sample guarantees under k-FWER and FDP thresholds. Specifically, we derive closed-form regularization sequences under orthogonal designs that provably bound k-FWER and FDP at user-specified levels, and extend these results to grouped settings via gk-SLOPE and gF-SLOPE, which control the analogous group-level errors gk-FWER and gFDP. For non-orthogonal general designs, we provide a calibrated data-driven sequence inspired by Gaussian approximation and Monte-Carlo correction, preserving convexity and scalability. Extensive simulations are conducted across sparse, correlated, and group-structured regimes. Empirical results corroborate our theoretical findings that the proposed methods achieve nominal error control, while yielding markedly higher power than competing stepdown procedures, thereby confirming the practical value of the theoretical advances.