# A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton

arXiv: https://arxiv.org/abs/2006.04045 · HTML: https://mlsanity.com/paper/2006.04045/

## Key facts

- Cited by 79 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2020.

## Built on by (most cited first, 30 of 79)

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- [Randomized Stochastic Variance-Reduced Methods for Multi-Task Stochastic Bilevel Optimization](https://mlsanity.com/paper/2105.02266/) (2021)
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- [Averaged Method of Multipliers for Bi-Level Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2302.03407/) (2023)
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- [Tighter Analysis of Alternating Stochastic Gradient Method for Stochastic Nested Problems](https://mlsanity.com/paper/2106.13781/) (2021)
- [Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence](https://mlsanity.com/paper/2202.12183/) (2022)
- [Bilevel Fast Scene Adaptation for Low-Light Image Enhancement](https://mlsanity.com/paper/2306.01343/) (2023)
- [Bilevel Optimization with a Lower-level Contraction: Optimal Sample Complexity without Warm-start](https://mlsanity.com/paper/2202.03397/) (2022)
- [A Conditional Gradient-based Method for Simple Bilevel Optimization with Convex Lower-level Problem](https://mlsanity.com/paper/2206.08868/) (2022)
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- [Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles](https://mlsanity.com/paper/2306.14853/) (2023)

## Abstract (arXiv)

In recent years, a variety of gradient-based first-order methods have been developed to solve bi-level optimization problems for learning applications. However, theoretical guarantees of these existing approaches heavily rely on the simplification that for each fixed upper-level variable, the lower-level solution must be a singleton (a.k.a., Lower-Level Singleton, LLS). In this work, we first design a counter-example to illustrate the invalidation of such LLS condition. Then by formulating BLPs from the view point of optimistic bi-level and aggregating hierarchical objective information, we establish Bi-level Descent Aggregation (BDA), a flexible and modularized algorithmic framework for generic bi-level optimization. Theoretically, we derive a new methodology to prove the convergence of BDA without the LLS condition. Our investigations also demonstrate that BDA is indeed compatible to a verify of particular first-order computation modules. Additionally, as an interesting byproduct, we also improve these conventional first-order bi-level schemes (under the LLS simplification). Particularly, we establish their convergences with weaker assumptions. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed BDA for different tasks, including hyper-parameter optimization and meta learning.