arXiv subjects

Hong T. M. Chu

Publications and source records attributed to Hong T. M. Chu.

9 recordsLinked to original sources

arXiv2026

Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis

Distributionally Robust (DR) optimization aims to certify worst-case risk within a Wasserstein uncertainty set. Current certifications typically rely either on global Lipschitz bounds, which are often conservative, or on local gradient information, which provides only a first-order approximation. This paper introduces a novel geometric framework based on the least concave majorants of the growth rate functions. Our proposed concave certificate establishes a tight bound on DR risk that remains applicable to non-Lipschitz and non-differentiable losses. We extend this framework to complexity analysis, introducing the worst-case generalization bound that complements the standard statistical generalization bound. Furthermore, we utilize this certificate to bound the gap between adversarial and empirical Rademacher complexity, demonstrating that dependencies on input diameter, network width, and depth can be eliminated. For practical application in deep learning, we introduce the adversarial score as a tractable relaxation of the concave certificate that enables efficient and layer-wise analysis of neural networks. We validate our theoretical results in various numerical experiments on classification and regression tasks using real-world data.

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arXiv2026

Tight Robustness Certificates and Wasserstein Distributional Attacks for Deep Neural Networks

Wasserstein distributionally robust optimization (WDRO) provides a framework for adversarial robustness, yet existing methods based on global Lipschitz continuity or strong duality often yield loose upper bounds or require prohibitive computation. We address these limitations with a primal approach and adopt a notion of exact Lipschitz certificates to tighten this upper bound of WDRO. For ReLU networks, we leverage the piecewise-affine structure on activation cells to obtain an exact tractable characterization of the corresponding WDRO problem. We further extend our analysis to modern architectures with smooth activations (e.g., GELU, SiLU), such as Transformers. Additionally, we propose novel Wasserstein Distributional Attacks (WDA, WDA++) that construct candidates for the worst-case distribution. Compared to existing attacks that are restricted to point-wise perturbations, our methods offer greater flexibility in the number and location of attack points. Extensive evaluations demonstrate that our proposed framework achieves competitive robust accuracy against state-of-the-art baselines while offering tighter certificates than existing methods. Our code is available at https://github.com/OLab-Repo/WDA.

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arXiv2025

Time-integrated Optimal Transport: A Robust Minimax Framework

Comparing time series in a principled manner requires capturing both temporal alignment and distributional similarity of features. Optimal transport (OT) has recently emerged as a powerful tool for this task, but existing OT-based approaches often depend on manually selected balancing parameters and can be computationally intensive. In this work, we introduce the Time-integrated Optimal Transport (TiOT) framework, which integrates temporal and feature components into a unified objective and yields a well-defined metric on the space of probability measures. This metric preserves fundamental properties of the Wasserstein distance, while avoiding the need for parameter tuning. To address the corresponding computational challenges, we introduce an entropic regularized approximation of TiOT, which can be efficiently solved using a block coordinate descent algorithm. Extensive experiments on both synthetic and real-world time series datasets demonstrate that our approach achieves improved accuracy and stability while maintaining comparable efficiency.

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arXiv2025

A Robust Optimization Model for Cost-Efficient and Fast Electric Vehicle Charging with L2-norm Uncertainty

In this paper, we propose a robust optimization model that addresses both the cost-efficiency and fast charging requirements for electric vehicles (EVs) at charging stations. By combining elements from traditional cost-minimization models and a fast charging objective, we construct an optimization model that balances user costs with rapid power allocation. Additionally, we incorporate L2-norm uncertainty into the charging cost, ensuring that the model remains resilient under cost fluctuations. The proposed model is tested under real-world scenarios and demonstrates its potential for efficient and flexible EV charging solutions.

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arXiv2024

A Corrected Inexact Proximal Augmented Lagrangian Method with a Relative Error Criterion for a Class of Group-quadratic Regularized Optimal Transport Problems

The optimal transport (OT) problem and its related problems have attracted significant attention and have been extensively studied in various applications. In this paper, we focus on a class of group-quadratic regularized OT problems which aim to find solutions with specialized structures that are advantageous in practical scenarios. To solve this class of problems, we propose a corrected inexact proximal augmented Lagrangian method (ciPALM), with the subproblems being solved by the semi-smooth Newton ({\sc Ssn}) method. We establish that the proposed method exhibits appealing convergence properties under mild conditions. Moreover, our ciPALM distinguishes itself from the recently developed semismooth Newton-based inexact proximal augmented Lagrangian ({\sc Snipal}) method for linear programming. Specifically, {\sc Snipal} uses an absolute error criterion for the approximate minimization of the subproblem for which a summable sequence of tolerance parameters needs to be pre-specified for practical implementations. In contrast, our ciPALM adopts a relative error criterion with a \textit{single} tolerance parameter, which would be more friendly to tune from computational and implementation perspectives. These favorable properties position our ciPALM as a promising candidate for tackling large-scale problems. Various numerical studies validate the effectiveness of employing a relative error criterion for the inexact proximal augmented Lagrangian method, and also demonstrate that our ciPALM is competitive for solving large-scale group-quadratic regularized OT problems.

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arXiv2024

Implicit Regularization via Spectral Neural Networks and Non-linear Matrix Sensing

The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural nets, even without any explicit regularizer in the loss function, converges to the solution of a regularized learning problem. However, known results attempting to theoretically explain this phenomenon focus overwhelmingly on the setting of linear neural nets, and the simplicity of the linear structure is particularly crucial to existing arguments. In this paper, we explore this problem in the context of more realistic neural networks with a general class of non-linear activation functions, and rigorously demonstrate the implicit regularization phenomenon for such networks in the setting of matrix sensing problems, together with rigorous rate guarantees that ensure exponentially fast convergence of gradient descent.In this vein, we contribute a network architecture called Spectral Neural Networks (abbrv. SNN) that is particularly suitable for matrix learning problems. Conceptually, this entails coordinatizing the space of matrices by their singular values and singular vectors, as opposed to by their entries, a potentially fruitful perspective for matrix learning. We demonstrate that the SNN architecture is inherently much more amenable to theoretical analysis than vanilla neural nets and confirm its effectiveness in the context of matrix sensing, via both mathematical guarantees and empirical investigations. We believe that the SNN architecture has the potential to be of wide applicability in a broad class of matrix learning scenarios.

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arXiv2024

Wasserstein distributionally robust optimization and its tractable regularization formulations

We study a variety of Wasserstein distributionally robust optimization (WDRO) problems where the distributions in the ambiguity set are chosen by constraining their Wasserstein discrepancies to the empirical distribution. Using the notion of weak Lipschitz property, we derive lower and upper bounds of the corresponding worst-case loss quantity and propose sufficient conditions under which this quantity coincides with its regularization scheme counterpart. Our constructive methodology and elementary analysis also directly characterize the closed-form of the approximate worst-case distribution. Extensive applications show that our theoretical results are applicable to various problems, including regression, classification and risk measure problems.

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arXiv2023

On Regularized Square-root Regression Problems: Distributionally Robust Interpretation and Fast Computations

Square-root (loss) regularized models have recently become popular in linear regression due to their nice statistical properties. Moreover, some of these models can be interpreted as the distributionally robust optimization counterparts of the traditional least-squares regularized models. In this paper, we give a unified proof to show that any square-root regularized model whose penalty function being the sum of a simple norm and a seminorm can be interpreted as the distributionally robust optimization (DRO) formulation of the corresponding least-squares problem. In particular, the optimal transport cost in the DRO formulation is given by a certain dual form of the penalty. To solve the resulting square-root regularized model whose loss function and penalty function are both nonsmooth, we design a proximal point dual semismooth Newton algorithm and demonstrate its efficiency when the penalty is the sparse group Lasso penalty or the fused Lasso penalty. Extensive experiments demonstrate that our algorithm is highly efficient for solving the square-root sparse group Lasso problems and the square-root fused Lasso problems.

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arXiv2022

An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving large-scale CMOT problems. The experiments on the discrete tomography problem also highlight the potential modeling power of our model.

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