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Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require…

Statistics Theory · Mathematics 2025-10-22 Jonathan Chirinos Rodriguez , Ernesto De Vito , Cesare Molinari , Lorenzo Rosasco , Silvia Villa

The ability to generalize under distributional shifts is essential to reliable machine learning, while models optimized with empirical risk minimization usually fail on non-$i.i.d$ testing data. Recently, invariant learning methods for…

Machine Learning · Computer Science 2021-10-26 Jiashuo Liu , Zheyuan Hu , Peng Cui , Bo Li , Zheyan Shen

Invariant risk minimization (IRM) (Arjovsky et al., 2019) is a recently proposed framework designed for learning predictors that are invariant to spurious correlations across different training environments. Yet, despite its theoretical…

Machine Learning · Statistics 2020-07-07 Yo Joong Choe , Jiyeon Ham , Kyubyong Park

The EM (Expectation-Maximization) algorithm is regarded as an MM (Majorization-Minimization) algorithm for maximum likelihood estimation of statistical models. Expanding this view, this paper demonstrates that by choosing an appropriate…

Optimization and Control · Mathematics 2026-02-12 Kensuke Asai , Jun-ya Gotoh

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of…

Numerical Analysis · Mathematics 2019-11-28 N. Benjamin Erichson , Lionel Mathelin , Steven L. Brunton , J. Nathan Kutz

Models trained with empirical risk minimization (ERM) are known to learn to rely on spurious features, i.e., their prediction is based on undesired auxiliary features which are strongly correlated with class labels but lack causal…

Machine Learning · Computer Science 2024-01-11 Phuong Quynh Le , Jörg Schlötterer , Christin Seifert

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…

Numerical Analysis · Mathematics 2025-05-09 Jamie M. Taylor , David Pardo

Machine learning algorithms with empirical risk minimization usually suffer from poor generalization performance due to the greedy exploitation of correlations among the training data, which are not stable under distributional shifts.…

Machine Learning · Computer Science 2021-06-18 Jiashuo Liu , Zheyuan Hu , Peng Cui , Bo Li , Zheyan Shen

Distributional reinforcement learning (DRL) enhances the understanding of the effects of the randomness in the environment by letting agents learn the distribution of a random return, rather than its expected value as in standard…

Optimization and Control · Mathematics 2024-03-26 Zifan Wang , Yulong Gao , Siyi Wang , Michael M. Zavlanos , Alessandro Abate , Karl H. Johansson

Effective regularization techniques are highly desired in deep learning for alleviating overfitting and improving generalization. This work proposes a new regularization scheme, based on the understanding that the flat local minima of the…

Machine Learning · Computer Science 2021-05-10 Yaowei Zheng , Richong Zhang , Yongyi Mao

This article provides, through theoretical analysis, an in-depth understanding of the classification performance of the empirical risk minimization framework, in both ridge-regularized and unregularized cases, when high dimensional data are…

Machine Learning · Statistics 2020-11-26 Xiaoyi Mai , Zhenyu Liao

We design simple screening tests to automatically discard data samples in empirical risk minimization without losing optimization guarantees. We derive loss functions that produce dual objectives with a sparse solution. We also show how to…

Machine Learning · Computer Science 2020-06-15 Grégoire Mialon , Alexandre d'Aspremont , Julien Mairal

Population risk is always of primary interest in machine learning; however, learning algorithms only have access to the empirical risk. Even for applications with nonconvex nonsmooth losses (such as modern deep networks), the population…

Machine Learning · Computer Science 2018-10-19 Chi Jin , Lydia T. Liu , Rong Ge , Michael I. Jordan

Convex risk measures play a foundational role in the area of stochastic optimization. However, in contrast to risk neutral models, their applications are still limited due to the lack of efficient solution methods. In particular, the mean…

Optimization and Control · Mathematics 2024-12-30 Zhichao Jia , Guanghui Lan , Zhe Zhang

This paper considers batch Reinforcement Learning (RL) with general value function approximation. Our study investigates the minimal assumptions to reliably estimate/minimize Bellman error, and characterizes the generalization performance…

Machine Learning · Computer Science 2021-03-26 Yaqi Duan , Chi Jin , Zhiyuan Li

Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution $\mathsf{P}$ and returns a hypothesis $f$ chosen from a fixed class…

Machine Learning · Computer Science 2014-11-25 Nishant A. Mehta , Robert C. Williamson

Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed…

Machine Learning · Computer Science 2024-01-24 Alexandre d'Aspremont , Cristóbal Guzmán , Clément Lezane

Risk-sensitive planning aims to identify policies maximizing some tail-focused metrics in Markov Decision Processes (MDPs). Such an optimization task can be very costly for the most widely used and interpretable metrics such as threshold…

Machine Learning · Statistics 2025-07-09 Alexandre Marthe , Samuel Bounan , Aurélien Garivier , Claire Vernade

We study conditional risk minimization (CRM), i.e. the problem of learning a hypothesis of minimal risk for prediction at the next step of sequentially arriving dependent data. Despite it being a fundamental problem, successful learning in…

Machine Learning · Statistics 2018-11-06 Alexander Zimin , Christoph Lampert

We study the problem of excess risk evaluation for empirical risk minimization (ERM) under convex losses. We show that by leveraging the idea of wild refitting, one can upper bound the excess risk through the so-called "wild optimism,"…

Machine Learning · Computer Science 2026-03-26 Haichen Hu , David Simchi-Levi