# Unlocking TriLevel Learning with Level-Wise Zeroth Order Constraints: Distributed Algorithms and Provable Non-Asymptotic Convergence

arXiv: https://arxiv.org/abs/2412.07138 · HTML: https://mlsanity.com/paper/2412.07138/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 118 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Builds on (most cited first, 10 of 10)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2006.02032/) (2020)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [ScaleBiO: Scalable Bilevel Optimization for LLM Data Reweighting](https://mlsanity.com/paper/2406.19976/) (2024)
- [Optimal Algorithms for Stochastic Bilevel Optimization under Relaxed Smoothness Conditions](https://mlsanity.com/paper/2306.12067/) (2023)
- [Federated Zeroth-Order Optimization using Trajectory-Informed Surrogate Gradients](https://mlsanity.com/paper/2308.04077/) (2023)
- [Distributed Zeroth-Order Stochastic Optimization in Time-varying Networks](https://mlsanity.com/paper/2105.12597/) (2021)

## Built on by (most cited first, 3 of 3)

- [PR-Attack: Coordinated Prompt-RAG Attacks on Retrieval-Augmented Generation in Large Language Models via Bilevel Optimization](https://mlsanity.com/paper/2504.07717/) (2025)
- [A stochastic gradient method for trilevel optimization](https://mlsanity.com/paper/2505.06805/) (2025)
- [CHAL: Council of Hierarchical Agentic Language](https://mlsanity.com/paper/2605.12718/) (2026)

## Abstract (arXiv)

Trilevel learning (TLL) found diverse applications in numerous machine learning applications, ranging from robust hyperparameter optimization to domain adaptation. However, existing researches primarily focus on scenarios where TLL can be addressed with first order information available at each level, which is inadequate in many situations involving zeroth order constraints, such as when black-box models are employed. Moreover, in trilevel learning, data may be distributed across various nodes, necessitating strategies to address TLL problems without centralizing data on servers to uphold data privacy. To this end, an effective distributed trilevel zeroth order learning framework DTZO is proposed in this work to address the TLL problems with level-wise zeroth order constraints in a distributed manner. The proposed DTZO is versatile and can be adapted to a wide range of (grey-box) TLL problems with partial zeroth order constraints. In DTZO, the cascaded polynomial approximation can be constructed without relying on gradients or sub-gradients, leveraging a novel cut, i.e., zeroth order cut. Furthermore, we theoretically carry out the non-asymptotic convergence rate analysis for the proposed DTZO in achieving the ε-stationary point. Extensive experiments have been conducted to demonstrate and validate the superior performance of the proposed DTZO, e.g., it approximately achieves up to a 40% improvement in performance.