arXiv subjects

Qihang Lin

Publications and source records attributed to Qihang Lin.

At least 19 recordsLinked to original sources

arXiv2026

Constraint Qualifications and Regularity Conditions for Nonlinear Constraint Systems

Constraint qualifications (CQs) and regularity conditions (RCs) play a fundamental role in establishing global iteration and oracle complexity results for first-order methods on solving functional constrained optimization problems. Various conditions have been proposed and employed in the literature, with their validity established for different classes of applications. However, the use of different conditions across existing studies makes it difficult to directly compare their theoretical results and underlying assumptions. In this report, we systematically review several widely used CQs and RCs and investigate their relationships. In particular, we establish implications among these conditions and identify cases in which the implications are strict, thereby clarifying their relative strengths. Our results provide a unified perspective on existing regularity assumptions and offer guidance for selecting appropriate conditions in the complexity analysis of first-order methods.

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arXiv2026

Learning the Robustness Mechanism with Bilevel Optimization

We propose a distributionally robust learning framework where parameters defining the robustness mechanism are learned from held-out data instead of extensively tuned. Using bilevel optimization with both upper and lower level minimax problems, we create two instances of our framework to tackle setups with and without group labels in the training set. Theoretically, we provide sample complexity analysis for our robustness mechanism learning paradigm, showing that it achieves generalization guarantees comparable to exhaustive grid search while being more computationally efficient. Empirically, we evaluate our framework under a challenging setup when both intra-group and inter-group test distribution shifts occur at the same time, thereby demonstrating the efficacy and scalability of our method.

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arXiv2026

Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems

We study a class of bilevel optimization problems in which both the upper- and lower-level problems have minimax structures. This setting captures a broad range of emerging applications. Despite the extensive literature on bilevel optimization and minimax optimization separately, existing methods mainly focus on bilevel optimization with lower-level minimization problems, often under strong convexity assumptions, and are not directly applicable to the minimax lower-level setting considered here. To address this gap, we develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem. In the deterministic setting, we establish that the proposed method finds an $ε$-KKT point with $\tilde{O}(ε^{-4})$ oracle complexity. We further show that bilevel problems with convex constrained lower-level minimization can be reformulated as special cases of our framework via Lagrangian duality, leading to an $\tilde{O}(ε^{-4})$ complexity bound that improves upon the existing $\tilde{O}(ε^{-7})$ result. Finally, we extend our approach to the stochastic setting, where only stochastic gradient oracles are available, and prove that the proposed stochastic method finds a nearly $ε$-KKT point with $\tilde{O}(ε^{-9})$ oracle complexity.

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arXiv2026

Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints

Fairness in machine learning has become a critical concern. Existing approaches often focus on achieving full fairness across all score ranges generated by predictive models, ensuring fairness in both high- and low-percentile populations. However, this stringent requirement can compromise predictive performance and may not align with the practical fairness concerns of stakeholders. In this work, we propose a novel framework for building partially fair machine learning models that enforce fairness only within a specific percentile interval of interest while maintaining flexibility in other regions. We introduce statistical metrics to evaluate partial fairness within a given percentile interval. To achieve partial fairness, we propose an in-processing method by formulating the model training problem as constrained optimization with difference-of-convex constraints, which can be solved by an inexact difference-of-convex algorithm (IDCA). We provide the complexity analysis of IDCA for finding a nearly KKT point. Through numerical experiments on real-world datasets, we demonstrate that our framework achieves high predictive performance while enforcing partial fairness where it matters most.

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arXiv2026

Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints

Constrained optimization with multiple functional inequality constraints has significant applications in machine learning. This paper examines a crucial subset of such problems where both the objective and constraint functions are weakly convex. Existing methods often face limitations, including slow convergence rates or reliance on double-loop algorithmic designs. To overcome these challenges, we introduce a novel single-loop penalty-based stochastic algorithm. Following the classical exact penalty method, our approach employs a {\bf hinge-based penalty}, which permits the use of a constant penalty parameter, enabling us to achieve a {\bf state-of-the-art complexity} for finding an approximate Karush-Kuhn-Tucker (KKT) solution. We further extend our algorithm to address finite-sum coupled compositional objectives, which are prevalent in artificial intelligence applications, establishing improved complexity over existing approaches. Finally, we validate our method through experiments on fair learning with receiver operating characteristic (ROC) fairness constraints and continual learning with non-forgetting constraints.

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arXiv2026

Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions

In this paper, we investigate how structural properties of the constraint system impact the oracle complexity of smooth non-convex optimization problems with convex inequality constraints over a simple polytope. In particular, we show that, under a local error bound condition with exponent $d\in[1,2]$ on constraint functions, an inexact Moreau envelope Lagrangian method can attain an $ε$-Karush--Kuhn--Tucker point with $\tilde O(ε^{-2d})$ gradient oracle complexity. When $d=1$, this result matches the best-known complexity in literature up to logarithmic factors. Importantly, the assumed error bound condition with any $d\in[1,2]$ is strictly weaker than the local linear independence constraint qualification that is required to achieve the best-known complexity. Our results clarify the interplay between error bound conditions of constraints and algorithmic complexity, and extend complexity guarantees to a broader class of constrained non-convex problems.

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arXiv2025

Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization

Finite-sum Coupled Compositional Optimization (FCCO), characterized by its coupled compositional objective structure, emerges as an important optimization paradigm for addressing a wide range of machine learning problems. In this paper, we focus on a challenging class of non-convex non-smooth FCCO, where the outer functions are non-smooth weakly convex or convex and the inner functions are smooth or weakly convex. Existing state-of-the-art result face two key limitations: (1) a high iteration complexity of $O(1/ε^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. Our main contributions are two fold: (i) We propose stochastic momentum methods tailored for non-smooth FCCO that come with provable convergence guarantees; (ii) We establish a new state-of-the-art iteration complexity of $O(1/ε^5)$. Moreover, we apply our algorithms to multiple inequality constrained non-convex optimization problems involving smooth or weakly convex functional inequality constraints. By optimizing a smoothed hinge penalty based formulation, we achieve a new state-of-the-art complexity of $O(1/ε^5)$ for finding an (nearly) $ε$-level KKT solution. Experiments on three tasks demonstrate the effectiveness of the proposed algorithms.

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arXiv2025

An Adaptive Parameter-free and Projection-free Restarting Level Set Method for Constrained Convex Optimization Under the Error Bound Condition

Recent efforts to accelerate first-order methods have focused on convex optimization problems that satisfy a geometric property known as error-bound condition, which covers a broad class of problems, including piece-wise linear programs and strongly convex programs. Parameter-free first-order methods that employ projection-free updates have the potential to broaden the benefit of acceleration. Such a method has been developed for unconstrained convex optimization but is lacking for general constrained convex optimization. We propose a parameter-free level-set method for the latter constrained case based on projection-free subgradient method that exhibits accelerated convergence for problems that satisfy an error-bound condition. Our method maintains a separate copy of the level-set sub-problem for each level parameter value and restarts the computation of these copies based on objective function progress. Applying such a restarting scheme in a level-set context is novel and results in an algorithm that dynamically adapts the precision of each copy. This property is key to extending prior restarting methods based on static precision that have been proposed for unconstrained convex optimization to handle constraints. We report promising numerical performance relative to benchmark methods.

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arXiv2025

Learning to Rank with Top-$K$ Fairness

Fairness in ranking models is crucial, as disparities in exposure can disproportionately affect protected groups. Most fairness-aware ranking systems focus on ensuring comparable average exposure for groups across the entire ranked list, which may not fully address real-world concerns. For example, when a ranking model is used for allocating resources among candidates or disaster hotspots, decision-makers often prioritize only the top-$K$ ranked items, while the ranking beyond top-$K$ becomes less relevant. In this paper, we propose a list-wise learning-to-rank framework that addresses the issues of inequalities in top-$K$ rankings at training time. Specifically, we propose a top-$K$ exposure disparity measure that extends the classic exposure disparity metric in a ranked list. We then learn a ranker to balance relevance and fairness in top-$K$ rankings. Since direct top-$K$ selection is computationally expensive for a large number of items, we transform the non-differentiable selection process into a differentiable objective function and develop efficient stochastic optimization algorithms to achieve both high accuracy and sufficient fairness. Extensive experiments demonstrate that our method outperforms existing methods.

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arXiv2025

Lower Complexity Bounds of First-order Methods for Affinely Constrained Composite Non-convex Problems

Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these methods. However, little can be claimed about their optimality as no lower bound is known, except for a few special \emph{smooth non-convex} cases. In this paper, we make the first attempt to establish lower complexity bounds of FOMs for solving a class of composite non-convex non-smooth optimization with linear constraints. Assuming two different first-order oracles, we establish lower complexity bounds of FOMs to produce a (near) $ε$-stationary point of a problem (and its reformulation) in the considered problem class, for any given tolerance $ε>0$. Our lower bounds indicate that the existence of a non-smooth convex regularizer can evidently increase the difficulty of an affinely constrained regularized problem over its nonregularized counterpart. In addition, we show that our lower bound of FOMs with the second oracle is tight, with a difference of up to a logarithmic factor from an upper complexity bound established in the extended arXiv version of this paper.

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arXiv2025

A Retention-Centric Framework for Continual Learning with Guaranteed Model Developmental Safety

In real-world applications, learning-enabled systems often undergo iterative model development to address challenging or emerging tasks, which involve collecting new data, training a new model and validating the model. This continual model development process raises a significant issue that acquiring new or improving existing capabilities may inadvertently lose good capabilities of the old model, also known as catastrophic forgetting. While existing continual learning aims to mitigate catastrophic forgetting by trading off performance on previous tasks and new tasks to ensure good average performance, it often falls short in cost-sensitive applications, where failing to preserve essential established capabilities introduces unforeseen costs and risks and substantial expenses for re-improving these capabilities. To address this issue, we impose a requirement on learning systems to ensure that a new model strictly retains important capabilities of the old model while improving target-task performance, which we term model developmental safety. To ensure model developmental safety, we propose a retention-centric framework with data-dependent constraints, and study how to continually develop a pretrained CLIP model for acquiring new or improving existing capabilities of image classification. We propose an efficient constrained optimization algorithm with theoretical guarantees and use its insights to finetune the CLIP model with task-dependent heads for promoting the model developmental safety. Experiments on autonomous driving and scene recognition datasets validate the efficacy of our method.

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arXiv2025

A Near-optimal Method for Linearly Constrained Composite Non-convex Non-smooth Problems

We study first-order methods (FOMs) for solving \emph{composite nonconvex nonsmooth} optimization with linear constraints. Recently, the lower complexity bounds of FOMs on finding an ($\varepsilon,\varepsilon$)-KKT point of the considered problem is established in \cite{liu2025lowercomplexityboundsfirstorder}. However, optimization algorithms that achieve this lower bound had not been developed. In this paper, we propose an inexact proximal gradient method, where subproblems are solved using a recovering primal-dual procedure. Without making the bounded domain assumption, we establish that the oracle complexity of the proposed method, for finding an ($\varepsilon,\varepsilon$)-KKT point of the considered problem, matches the lower bounds up to a logarithmic factor. Consequently, in terms of the complexity, our algorithm outperforms all existing methods. We demonstrate the advantages of our proposed algorithm over the (linearized) alternating direction method of multipliers and the (proximal) augmented Lagrangian method in the numerical experiments.

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arXiv2025

Multi-Output Distributional Fairness via Post-Processing

The post-processing approaches are becoming prominent techniques to enhance machine learning models' fairness because of their intuitiveness, low computational cost, and excellent scalability. However, most existing post-processing methods are designed for task-specific fairness measures and are limited to single-output models. In this paper, we introduce a post-processing method for multi-output models, such as the ones used for multi-task/multi-class classification and representation learning, to enhance a model's distributional parity, a task-agnostic fairness measure. Existing methods for achieving distributional parity rely on the (inverse) cumulative density function of a model's output, restricting their applicability to single-output models. Extending previous works, we propose to employ optimal transport mappings to move a model's outputs across different groups towards their empirical Wasserstein barycenter. An approximation technique is applied to reduce the complexity of computing the exact barycenter and a kernel regression method is proposed to extend this process to out-of-sample data. Our empirical studies evaluate the proposed approach against various baselines on multi-task/multi-class classification and representation learning tasks, demonstrating the effectiveness of the proposed approach.

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arXiv2024

A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional Optimization

This note studies numerical methods for solving compositional optimization problems, where the inner function is smooth, and the outer function is Lipschitz continuous, non-smooth, and non-convex but exhibits one of two special structures that enable the design of efficient first-order methods. In the first structure, the outer function allows for an easily solvable proximal mapping. We demonstrate that, in this case, a smoothing compositional gradient method can find a $(δ,ε)$-stationary point--specifically defined for compositional optimization--in $O(1/(δε^2))$ iterations. In the second structure, the outer function is expressed as a difference-of-convex function, where each convex component is simple enough to allow an efficiently solvable proximal linear subproblem. In this case, we show that a prox-linear method can find a nearly $ε$-critical point in $O(1/ε^2)$ iterations.

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arXiv2024

FedPAE: Peer-Adaptive Ensemble Learning for Asynchronous and Model-Heterogeneous Federated Learning

Federated learning (FL) enables multiple clients with distributed data sources to collaboratively train a shared model without compromising data privacy. However, existing FL paradigms face challenges due to heterogeneity in client data distributions and system capabilities. Personalized federated learning (pFL) has been proposed to mitigate these problems, but often requires a shared model architecture and a central entity for parameter aggregation, resulting in scalability and communication issues. More recently, model-heterogeneous FL has gained attention due to its ability to support diverse client models, but existing methods are limited by their dependence on a centralized framework, synchronized training, and publicly available datasets. To address these limitations, we introduce Federated Peer-Adaptive Ensemble Learning (FedPAE), a fully decentralized pFL algorithm that supports model heterogeneity and asynchronous learning. Our approach utilizes a peer-to-peer model sharing mechanism and ensemble selection to achieve a more refined balance between local and global information. Experimental results show that FedPAE outperforms existing state-of-the-art pFL algorithms, effectively managing diverse client capabilities and demonstrating robustness against statistical heterogeneity.

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arXiv2024

Deterministic and Stochastic Accelerated Gradient Method for Convex Semi-Infinite Optimization

This paper explores numerical methods for solving a convex differentiable semi-infinite program. We introduce a primal-dual gradient method which performs three updates iteratively: a momentum gradient ascend step to update the constraint parameters, a momentum gradient ascend step to update the dual variables, and a gradient descend step to update the primal variables. Our approach also extends to scenarios where gradients and function values are accessible solely through stochastic oracles. This method extends the recent primal-dual methods, for example, Hamedani and Aybat (2021); Boob et al. (2022), for optimization with a finite number of constraints. We show the iteration complexity of the proposed method for finding an $ε$-optimal solution under different convexity and concavity assumptions on the functions.

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arXiv2024

Provable Optimization for Adversarial Fair Self-supervised Contrastive Learning

This paper studies learning fair encoders in a self-supervised learning (SSL) setting, in which all data are unlabeled and only a small portion of them are annotated with sensitive attribute. Adversarial fair representation learning is well suited for this scenario by minimizing a contrastive loss over unlabeled data while maximizing an adversarial loss of predicting the sensitive attribute over the data with sensitive attribute. Nevertheless, optimizing adversarial fair representation learning presents significant challenges due to solving a non-convex non-concave minimax game. The complexity deepens when incorporating a global contrastive loss that contrasts each anchor data point against all other examples. A central question is ``{\it can we design a provable yet efficient algorithm for solving adversarial fair self-supervised contrastive learning}?'' Building on advanced optimization techniques, we propose a stochastic algorithm dubbed SoFCLR with a convergence analysis under reasonable conditions without requring a large batch size. We conduct extensive experiments to demonstrate the effectiveness of the proposed approach for downstream classification with eight fairness notions.

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arXiv2023

Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations.

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