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The goal of this tutorial is to introduce key models, algorithms, and open questions related to the use of optimization methods for solving problems arising in machine learning. It is written with an INFORMS audience in mind, specifically…

Machine Learning · Statistics 2017-07-03 Frank E. Curtis , Katya Scheinberg

Modern deep models are often pretrained on large-scale data with missing labels using composite objectives, where the relative weights of multiple loss terms act as hyperparameters. Tuning these weights with random search or Bayesian…

Machine Learning · Computer Science 2026-05-11 Ivan Karpukhin , Andrey Savchenko

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…

Machine Learning · Computer Science 2026-04-07 Xuelin Zhang , Hong Chen , Bin Gu , Tieliang Gong , Feng Zheng

We propose a federated learning method with weighted nodes in which the weights can be modified to optimize the model's performance on a separate validation set. The problem is formulated as a bilevel optimization where the inner problem is…

Machine Learning · Computer Science 2022-10-11 Yankun Huang , Qihang Lin , Nick Street , Stephen Baek

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two…

Machine Learning · Computer Science 2024-11-26 Congliang Chen , Li Shen , Zhiqiang Xu , Wei Liu , Zhi-Quan Luo , Peilin Zhao

Bilevel optimization (BLO) offers a principled framework for hierarchical decision-making and has been widely applied in machine learning tasks such as hyperparameter optimization and meta-learning. While existing BLO methods are mostly…

Optimization and Control · Mathematics 2025-10-20 Zhuo Chen , Xinjian Xu , Shihui Ying , Tieyong Zeng

Gradient-based bilevel optimisation is a powerful technique with applications in hyperparameter optimisation, task adaptation, algorithm discovery, meta-learning more broadly, and beyond. It often requires differentiating through the…

Machine Learning · Computer Science 2025-06-11 Iurii Kemaev , Dan A Calian , Luisa M Zintgraf , Gregory Farquhar , Hado van Hasselt

Even for the gradient descent (GD) method applied to neural network training, understanding its optimization dynamics, including convergence rate, iterate trajectories, function value oscillations, and especially its implicit acceleration,…

Machine Learning · Computer Science 2026-05-22 Alexander Tyurin

One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…

Statistics Theory · Mathematics 2021-01-08 Neil K. Chada , Claudia Schillings , Xin T. Tong , Simon Weissmann

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal…

Machine Learning · Computer Science 2025-05-29 Ruth Wan Theng Chew , Quoc Phong Nguyen , Bryan Kian Hsiang Low

Reinforcement learning techniques achieved human-level performance in several tasks in the last decade. However, in recent years, the need for interpretability emerged: we want to be able to understand how a system works and the reasons…

Machine Learning · Computer Science 2023-01-13 Leonardo Lucio Custode , Giovanni Iacca

Bilevel optimization is a popular hierarchical model in machine learning, and has been widely applied to many machine learning tasks such as meta learning, hyperparameter learning and policy optimization. Although many bilevel optimization…

Machine Learning · Computer Science 2022-11-15 Feihu Huang

Hyperparameter optimization can be formulated as a bilevel optimization problem, where the optimal parameters on the training set depend on the hyperparameters. We aim to adapt regularization hyperparameters for neural networks by fitting…

Machine Learning · Computer Science 2019-03-08 Matthew MacKay , Paul Vicol , Jon Lorraine , David Duvenaud , Roger Grosse

The mathematical program with equilibrium constraints (MPEC) is a powerful yet challenging class of constrained optimization problems, where the constraints are characterized by a parametrized variational inequality (VI) problem. While…

Optimization and Control · Mathematics 2026-05-19 Mohammadjavad Ebrahimi , Uday V. Shanbhag , Farzad Yousefian

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems,…

Optimization and Control · Mathematics 2025-12-04 James Kotary , Himanshu Sharma , Ethan King , Draguna Vrabie , Ferdinando Fioretto , Jan Drgona

With the development of large-scale models, traditional distributed bilevel optimization algorithms cannot be applied directly in low-resource clients. The key reason lies in the excessive computation involved in optimizing both the lower-…

Distributed, Parallel, and Cluster Computing · Computer Science 2026-01-01 Mingyi Li , Xiao Zhang , Ruisheng Zheng , Hongjian Shi , Yuan Yuan , Xiuzhen Cheng , Dongxiao Yu

We propose efficient methods for solving stochastic simple bilevel optimization problems with convex inner levels, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic…

Optimization and Control · Mathematics 2025-11-25 Khanh-Hung Giang-Tran , Soroosh Shafiee , Nam Ho-Nguyen

We study the implicit bias of generic optimization methods, such as mirror descent, natural gradient descent, and steepest descent with respect to different potentials and norms, when optimizing underdetermined linear regression or…

Machine Learning · Statistics 2020-06-24 Suriya Gunasekar , Jason Lee , Daniel Soudry , Nathan Srebro

Data rebalancing techniques, including oversampling and undersampling, are a common approach to addressing the challenges of imbalanced data. To tackle unresolved problems related to both oversampling and undersampling, we propose a new…

Machine Learning · Computer Science 2025-07-11 Karen Medlin , Sven Leyffer , Krishnan Raghavan

In this paper we consider the problem of learning variational models in the context of supervised learning via risk minimization. Our goal is to provide a deeper understanding of the two approaches of learning of variational models via…

Machine Learning · Statistics 2023-09-07 Christoph Brauer , Niklas Breustedt , Timo de Wolff , Dirk A. Lorenz