# L3Ms -- Lagrange Large Language Models

arXiv: https://arxiv.org/abs/2410.21533 · HTML: https://mlsanity.com/paper/2410.21533/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in LLM preference optimization methods, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Placed in

- LLM alignment / RLHF / preference optimisation / [LLM preference optimization methods](https://mlsanity.com/alignment/llm-preference-optimization-methods/)

## Builds on (most cited first, 26 of 26)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Scaling Laws for Reward Model Overoptimization](https://mlsanity.com/paper/2210.10760/) (2022)
- [CodeT5: Identifier-aware Unified Pre-trained Encoder-Decoder Models for Code Understanding and Generation](https://mlsanity.com/paper/2109.00859/) (2021)
- [BeaverTails: Towards Improved Safety Alignment of LLM via a Human-Preference Dataset](https://mlsanity.com/paper/2307.04657/) (2023)
- [Enhancing Chat Language Models by Scaling High-quality Instructional Conversations](https://mlsanity.com/paper/2305.14233/) (2023)
- [Open Problems and Fundamental Limitations of Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2307.15217/) (2023)
- [Safe RLHF: Safe Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.12773/) (2023)
- [Fine-Grained Human Feedback Gives Better Rewards for Language Model Training](https://mlsanity.com/paper/2306.01693/) (2023)
- [Unified Pre-training for Program Understanding and Generation](https://mlsanity.com/paper/2103.06333/) (2021)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Rewarded soups: towards Pareto-optimal alignment by interpolating weights fine-tuned on diverse rewards](https://mlsanity.com/paper/2306.04488/) (2023)
- [GatorTron: A Large Clinical Language Model to Unlock Patient Information from Unstructured Electronic Health Records](https://mlsanity.com/paper/2203.03540/) (2022)
- [Fine-tuning language models to find agreement among humans with diverse preferences](https://mlsanity.com/paper/2211.15006/) (2022)
- [Beyond One-Preference-Fits-All Alignment: Multi-Objective Direct Preference Optimization](https://mlsanity.com/paper/2310.03708/) (2023)
- [Controllable Preference Optimization: Toward Controllable Multi-Objective Alignment](https://mlsanity.com/paper/2402.19085/) (2024)
- [Confronting Reward Model Overoptimization with Constrained RLHF](https://mlsanity.com/paper/2310.04373/) (2023)
- [Deep Reinforcement Learning for Multi-objective Optimization](https://mlsanity.com/paper/1906.02386/) (2019)
- [Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions](https://mlsanity.com/paper/1904.04205/) (2019)
- [Transforming and Combining Rewards for Aligning Large Language Models](https://mlsanity.com/paper/2402.00742/) (2024)
- [Relaxed Logarithmic Barrier Function Based Model Predictive Control of Linear Systems](https://mlsanity.com/paper/1503.03314/) (2015)
- [A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems](https://mlsanity.com/paper/2304.14907/) (2023)

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

- [Multi-Objective Preference Optimization: Improving Human Alignment of Generative Models](https://mlsanity.com/paper/2505.10892/) (2025)
- [E-Scores for (In)Correctness Assessment of Generative Model Outputs](https://mlsanity.com/paper/2510.25770/) (2025)
- [Provable Last-Iterate Convergence for Multi-Objective Safe LLM Alignment via Optimistic Primal-Dual](https://mlsanity.com/paper/2602.22146/) (2026)

## Abstract (arXiv)

Supervised fine-tuning (SFT) and alignment of large language models (LLMs) are key steps in providing a good user experience. However, the concept of an appropriate alignment is inherently application-dependent, and current methods often rely on heuristic choices to drive optimization. In this work, we formulate SFT and alignment as a constrained optimization problem: the LLM is fine-tuned on a task while being required to meet application-specific requirements, without resorting to heuristics. To solve this, we propose Lagrange Large Language Models (L3Ms), which employ logarithmic barriers to enforce the constraints. This approach allows for the customization of L3Ms across diverse applications while avoiding heuristic-driven processes. We experimentally demonstrate the versatility and efficacy of L3Ms in achieving tailored alignments for various applications.