# Will Bilevel Optimizers Benefit from Loops

arXiv: https://arxiv.org/abs/2205.14224 · HTML: https://mlsanity.com/paper/2205.14224/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 27 May 2022.

## Builds on (most cited first, 27 of 27)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton](https://mlsanity.com/paper/2006.04045/) (2020)
- [Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters](https://mlsanity.com/paper/2006.09486/) (2020)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization](https://mlsanity.com/paper/2106.07991/) (2021)
- [A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems](https://mlsanity.com/paper/2010.15768/) (2020)
- [A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse](https://mlsanity.com/paper/2112.04660/) (2021)
- [iDARTS: Differentiable Architecture Search with Stochastic Implicit Gradients](https://mlsanity.com/paper/2106.10784/) (2021)
- [Lower Bounds and Accelerated Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2102.03926/) (2021)
- [Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning](https://mlsanity.com/paper/2002.07836/) (2020)
- [Improved Bilevel Model: Fast and Optimal Algorithm with Theoretical Guarantee](https://mlsanity.com/paper/2009.00690/) (2020)
- [Efficiently Escaping Saddle Points in Bilevel Optimization](https://mlsanity.com/paper/2202.03684/) (2022)

## Built on by (most cited first, 30 of 36)

- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [SEAL: Safety-enhanced Aligned LLM Fine-tuning via Bilevel Data Selection](https://mlsanity.com/paper/2410.07471/) (2024)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [Averaged Method of Multipliers for Bi-Level Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2302.03407/) (2023)
- [Decentralized Stochastic Bilevel Optimization with Improved per-Iteration Complexity](https://mlsanity.com/paper/2210.12839/) (2022)
- [Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles](https://mlsanity.com/paper/2306.14853/) (2023)
- [Complexity of Single Loop Algorithms for Nonlinear Programming with Stochastic Objective and Constraints](https://mlsanity.com/paper/2311.00678/) (2023)
- [Constrained Bi-Level Optimization: Proximal Lagrangian Value function Approach and Hessian-free Algorithm](https://mlsanity.com/paper/2401.16164/) (2024)
- [Moreau Envelope for Nonconvex Bi-Level Optimization: A Single-loop and Hessian-free Solution Strategy](https://mlsanity.com/paper/2405.09927/) (2024)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [Value Function Based Difference-of-Convex Algorithm for Bilevel Hyperparameter Selection Problems](https://mlsanity.com/paper/2206.05976/) (2022)
- [Accelerating Inexact HyperGradient Descent for Bilevel Optimization](https://mlsanity.com/paper/2307.00126/) (2023)
- [On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level](https://mlsanity.com/paper/2303.03944/) (2023)
- [Alternating Implicit Projected SGD and Its Efficient Variants for Equality-constrained Bilevel Optimization](https://mlsanity.com/paper/2211.07096/) (2022)
- [Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity](https://mlsanity.com/paper/2405.19697/) (2024)
- [Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs](https://mlsanity.com/paper/2306.16761/) (2023)
- [Non-Convex Bilevel Optimization with Time-Varying Objective Functions](https://mlsanity.com/paper/2308.03811/) (2023)
- [A Single-Loop Algorithm for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2311.08945/) (2023)
- [A Primal-Dual-Assisted Penalty Approach to Bilevel Optimization with Coupled Constraints](https://mlsanity.com/paper/2406.10148/) (2024)
- [Network Utility Maximization with Unknown Utility Functions: A Distributed, Data-Driven Bilevel Optimization Approach](https://mlsanity.com/paper/2301.01801/) (2023)
- [Tuning-Free Bilevel Optimization: New Algorithms and Convergence Analysis](https://mlsanity.com/paper/2410.05140/) (2024)
- [BiSLS/SPS: Auto-tune Step Sizes for Stable Bi-level Optimization](https://mlsanity.com/paper/2305.18666/) (2023)
- [LancBiO: dynamic Lanczos-aided bilevel optimization via Krylov subspace](https://mlsanity.com/paper/2404.03331/) (2024)
- [Efficient Curvature-Aware Hypergradient Approximation for Bilevel Optimization](https://mlsanity.com/paper/2505.02101/) (2025)
- [qNBO: quasi-Newton Meets Bilevel Optimization](https://mlsanity.com/paper/2502.01076/) (2025)
- [Federated Learning Can Find Friends That Are Advantageous](https://mlsanity.com/paper/2402.05050/) (2024)
- [Understanding the Statistical Accuracy-Communication Trade-off in Personalized Federated Learning with Minimax Guarantees](https://mlsanity.com/paper/2410.08934/) (2024)
- [Fully First-Order Methods for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2410.19319/) (2024)
- [Exploring the Generalization Capabilities of AID-based Bi-level Optimization](https://mlsanity.com/paper/2411.16081/) (2024)
- [DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2606.21153/) (2026)

## Abstract (arXiv)

Bilevel optimization has arisen as a powerful tool for solving a variety of machine learning problems. Two current popular bilevel optimizers AID-BiO and ITD-BiO naturally involve solving one or two sub-problems, and consequently, whether we solve these problems with loops (that take many iterations) or without loops (that take only a few iterations) can significantly affect the overall computational efficiency. Existing studies in the literature cover only some of those implementation choices, and the complexity bounds available are not refined enough to enable rigorous comparison among different implementations. In this paper, we first establish unified convergence analysis for both AID-BiO and ITD-BiO that are applicable to all implementation choices of loops. We then specialize our results to characterize the computational complexity for all implementations, which enable an explicit comparison among them. Our result indicates that for AID-BiO, the loop for estimating the optimal point of the inner function is beneficial for overall efficiency, although it causes higher complexity for each update step, and the loop for approximating the outer-level Hessian-inverse-vector product reduces the gradient complexity. For ITD-BiO, the two loops always coexist, and our convergence upper and lower bounds show that such loops are necessary to guarantee a vanishing convergence error, whereas the no-loop scheme suffers from an unavoidable non-vanishing convergence error. Our numerical experiments further corroborate our theoretical results.