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Bilevel Optimization Programming is used to model complex and conflicting interactions between agents, for example in Robust AI or Privacy-preserving AI. Integrating bilevel mathematical programming within deep learning is thus an essential…

Machine Learning · Computer Science 2023-03-01 Francesco Alesiani

Many optimization problems require hyperparameters, i.e., parameters that must be pre-specified in advance, such as regularization parameters and parametric regularizers in variational regularization methods for inverse problems, and…

Optimization and Control · Mathematics 2025-10-09 Matthias J. Ehrhardt , Silvia Gazzola , Sebastian J. Scott

We study two procedures (reverse-mode and forward-mode) for computing the gradient of the validation error with respect to the hyperparameters of any iterative learning algorithm such as stochastic gradient descent. These procedures mirror…

Machine Learning · Statistics 2017-12-13 Luca Franceschi , Michele Donini , Paolo Frasconi , Massimiliano Pontil

Bilevel learning has gained prominence in machine learning, inverse problems, and imaging applications, including hyperparameter optimization, learning data-adaptive regularizers, and optimizing forward operators. The large-scale nature of…

Optimization and Control · Mathematics 2025-05-20 Mohammad Sadegh Salehi , Subhadip Mukherjee , Lindon Roberts , Matthias J. Ehrhardt

We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit…

Optimization and Control · Mathematics 2022-07-12 Michael Arbel , Julien Mairal

The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the…

Machine Learning · Computer Science 2021-10-26 Fan Bao , Guoqiang Wu , Chongxuan Li , Jun Zhu , Bo Zhang

Although application examples of multilevel optimization have already been discussed since the 1990s, the development of solution methods was almost limited to bilevel cases due to the difficulty of the problem. In recent years, in machine…

Optimization and Control · Mathematics 2021-10-27 Ryo Sato , Mirai Tanaka , Akiko Takeda

We design and analyze a novel accelerated gradient-based algorithm for a class of bilevel optimization problems. These problems have various applications arising from machine learning and image processing, where optimal solutions of the two…

Optimization and Control · Mathematics 2023-11-20 Sepideh Samadi , Daniel Burbano , Farzad Yousefian

Due to the hierarchical structure of many machine learning problems, bilevel programming is becoming more and more important recently, however, the complicated correlation between the inner and outer problem makes it extremely challenging…

Machine Learning · Computer Science 2020-09-03 Junyi Li , Bin Gu , Heng Huang

Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization…

Machine Learning · Computer Science 2021-08-03 Kaiyi Ji

Functional bilevel methods estimate a lower-level function and plug it into a hypergradient, but this plug-in gradient can retain first-order bias when the lower-level problem is learned nonparametrically. To remove this bias, we develop a…

Machine Learning · Statistics 2026-05-21 Fares El Khoury , Houssam Zenati , Nathan Kallus , Michael Arbel , Aurélien Bibaut

In the recent years, various gradient descent algorithms including the methods of gradient descent, gradient descent with momentum, adaptive gradient (AdaGrad), root-mean-square propagation (RMSProp) and adaptive moment estimation (Adam)…

Machine Learning · Computer Science 2024-09-19 Abel C. H. Chen

Most models in machine learning contain at least one hyperparameter to control for model complexity. Choosing an appropriate set of hyperparameters is both crucial in terms of model accuracy and computationally challenging. In this work we…

Machine Learning · Statistics 2022-11-22 Fabian Pedregosa

Hyperparameter tuning is an active area of research in machine learning, where the aim is to identify the optimal hyperparameters that provide the best performance on the validation set. Hyperparameter tuning is often achieved using naive…

Machine Learning · Computer Science 2020-07-23 Ankur Sinha , Tanmay Khandait , Raja Mohanty

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

Bilevel optimization has found successful applications in various machine learning problems, including hyper-parameter optimization, data cleaning, and meta-learning. However, its huge computational cost presents a significant challenge for…

Machine Learning · Computer Science 2024-11-05 Xiaoyu Wang , Rui Pan , Renjie Pi , Jipeng Zhang

Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but…

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to…

Optimization and Control · Mathematics 2024-05-08 Ensio Suonperä , Tuomo Valkonen

Gradient-based optimization has been critical to the success of machine learning, updating a single set of parameters to minimize a single loss. A growing number of applications rely on a generalization of this, where we have a bilevel or…

Machine Learning · Computer Science 2024-07-02 Jonathan Lorraine

Bilevel reinforcement learning (RL), which features intertwined two-level problems, has attracted growing interest recently. The inherent non-convexity of the lower-level RL problem is, however, to be an impediment to developing bilevel…

Optimization and Control · Mathematics 2025-02-28 Yan Yang , Bin Gao , Ya-xiang Yuan