# Beyond Value Functions: Single-Loop Bilevel Optimization under Flatness Conditions

arXiv: https://arxiv.org/abs/2507.20400 · HTML: https://mlsanity.com/paper/2507.20400/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2025.

## Built on by (most cited first, 9 of 9)

- [Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oracles](https://mlsanity.com/paper/2511.19656/) (2025)
- [On the Condition Number Dependency in Bilevel Optimization](https://mlsanity.com/paper/2511.22331/) (2025)
- [Faster Gradient Methods for Highly-Smooth Stochastic Bilevel Optimization](https://mlsanity.com/paper/2509.02937/) (2025)
- [BiRQ: Bi-Level Self-Labeling Random Quantization for Self-Supervised Speech Recognition](https://mlsanity.com/paper/2509.15430/) (2025)
- [On the Convergence of Single-Loop Stochastic Bilevel Optimization with Approximate Implicit Differentiation](https://mlsanity.com/paper/2602.23633/) (2026)
- [One Algorithm, Two Goals: Dual Scoring for Parameter and Data Selection in LLM Fine-Tuning](https://mlsanity.com/paper/2605.06166/) (2026)
- [Diffuse to Detect: Bi-Level Sample Rebalancing with Pseudo-Label Diffusion for Point-Supervised Infrared Small-Target Detection](https://mlsanity.com/paper/2605.20766/) (2026)
- [Bi-Level Collaborative Learning for Few-Shot Scribble-Supervised Medical Image Segmentation](https://mlsanity.com/paper/2607.25432/) (2026)
- [BRIDGE: Bilevel Retrieval-Credit-Aware Agentic Reinforcement Learning](https://mlsanity.com/paper/2609.36505/) (2026)

## Abstract (arXiv)

Bilevel optimization, a hierarchical optimization paradigm, has gained significant attention in a wide range of practical applications, notably in the fine-tuning of generative models. However, due to the nested problem structure, most existing algorithms require either the Hessian vector calculation or the nested loop updates, which are computationally inefficient in large language model (LLM) fine-tuning. In this paper, building upon the fully first-order penalty-based approach, we propose an efficient value function-free (PBGD-Free) algorithm that eliminates the loop of solving the lower-level problem and admits fully single-loop updates. Inspired by the landscape analysis of representation learning-based LLM fine-tuning problem, we propose a relaxed flatness condition for the upper-level function and prove the convergence of the proposed value-function-free algorithm. We test the performance of the proposed algorithm in various applications and demonstrate its superior computational efficiency over the state-of-the-art bilevel methods.