# Joint Prompt Optimization of Stacked LLMs using Variational Inference

arXiv: https://arxiv.org/abs/2306.12509 · HTML: https://mlsanity.com/paper/2306.12509/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 56 references that are in the map.
- First posted on arXiv 21 Jun 2023.

## Builds on (most cited first, 30 of 49)

- [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)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Pre-train, Prompt, and Predict: A Systematic Survey of Prompting Methods in Natural Language Processing](https://mlsanity.com/paper/2107.13586/) (2021)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Making Pre-trained Language Models Better Few-shot Learners](https://mlsanity.com/paper/2012.15723/) (2020)
- [What Makes Good In-Context Examples for GPT-3?](https://mlsanity.com/paper/2101.06804/) (2021)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [Inner Monologue: Embodied Reasoning through Planning with Language Models](https://mlsanity.com/paper/2207.05608/) (2022)
- [Measuring and Narrowing the Compositionality Gap in Language Models](https://mlsanity.com/paper/2210.03350/) (2022)
- [Large Language Models Are Human-Level Prompt Engineers](https://mlsanity.com/paper/2211.01910/) (2022)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [Reasoning with Language Model is Planning with World Model](https://mlsanity.com/paper/2305.14992/) (2023)
- [Automatic Chain of Thought Prompting in Large Language Models](https://mlsanity.com/paper/2210.03493/) (2022)
- [Automatic Prompt Optimization with "Gradient Descent" and Beam Search](https://mlsanity.com/paper/2305.03495/) (2023)
- [Learning To Retrieve Prompts for In-Context Learning](https://mlsanity.com/paper/2112.08633/) (2021)

## Built on by (most cited first, 30 of 32)

- [Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models](https://mlsanity.com/paper/2312.06585/) (2023)
- [Optimizing Instructions and Demonstrations for Multi-Stage Language Model Programs](https://mlsanity.com/paper/2406.11695/) (2024)
- [Automatically Correcting Large Language Models: Surveying the landscape of diverse self-correction strategies](https://mlsanity.com/paper/2308.03188/) (2023)
- [Symbolic Learning Enables Self-Evolving Agents](https://mlsanity.com/paper/2406.18532/) (2024)
- [Amortizing intractable inference in large language models](https://mlsanity.com/paper/2310.04363/) (2023)
- [Privileged Information Distillation for Language Models](https://mlsanity.com/paper/2602.04942/) (2026)
- [FedBiOT: LLM Local Fine-tuning in Federated Learning without Full Model](https://mlsanity.com/paper/2406.17706/) (2024)
- [Trace is the Next AutoDiff: Generative Optimization with Rich Feedback, Execution Traces, and LLMs](https://mlsanity.com/paper/2406.16218/) (2024)
- [PAPILLON: Privacy Preservation from Internet-based and Local Language Model Ensembles](https://mlsanity.com/paper/2410.17127/) (2024)
- [Privacy-Preserving Retrieval-Augmented Generation with Differential Privacy](https://mlsanity.com/paper/2412.04697/) (2024)
- [Beyond Verifiable Rewards: Scaling Reinforcement Learning for Language Models to Unverifiable Data](https://mlsanity.com/paper/2503.19618/) (2025)
- [debug-gym: A Text-Based Environment for Interactive Debugging](https://mlsanity.com/paper/2503.21557/) (2025)
- [CriSPO: Multi-Aspect Critique-Suggestion-guided Automatic Prompt Optimization for Text Generation](https://mlsanity.com/paper/2410.02748/) (2024)
- [On the Effect of Sampling Diversity in Scaling LLM Inference](https://mlsanity.com/paper/2502.11027/) (2025)
- [Verbalized Machine Learning: Revisiting Machine Learning with Language Models](https://mlsanity.com/paper/2406.04344/) (2024)
- [Beyond Outlining: Heterogeneous Recursive Planning for Adaptive Long-form Writing with Language Models](https://mlsanity.com/paper/2503.08275/) (2025)
- [Paraphrase and Solve: Exploring and Exploiting the Impact of Surface Form on Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2404.11500/) (2024)
- [No Free Lunch Theorem for Privacy-Preserving LLM Inference](https://mlsanity.com/paper/2405.20681/) (2024)
- [Ask more, know better: Reinforce-Learned Prompt Questions for Decision Making with Large Language Models](https://mlsanity.com/paper/2310.18127/) (2023)
- [Direct Retrieval-augmented Optimization: Synergizing Knowledge Selection and Language Models](https://mlsanity.com/paper/2505.03075/) (2025)
- [When AI Meets the Web: Prompt Injection Risks in Third-Party AI Chatbot Plugins](https://mlsanity.com/paper/2511.05797/) (2025)
- [A Survey of Automatic Prompt Optimization with Instruction-focused Heuristic-based Search Algorithm](https://mlsanity.com/paper/2502.18746/) (2025)
- [Scaling Textual Gradients via Sampling-Based Momentum](https://mlsanity.com/paper/2506.00400/) (2025)
- [Learning, Fast and Slow: Towards LLMs That Adapt Continually](https://mlsanity.com/paper/2605.12484/) (2026)
- [Compound AI Systems Optimization: A Survey of Methods, Challenges, and Future Directions](https://mlsanity.com/paper/2506.08234/) (2025)
- [Test-Time Fairness and Robustness in Large Language Models](https://mlsanity.com/paper/2406.07685/) (2024)
- [DAST: Difficulty-Aware Self-Training on Large Language Models](https://mlsanity.com/paper/2503.09029/) (2025)
- [One Rank at a Time: Cascading Error Dynamics in Sequential Learning](https://mlsanity.com/paper/2505.22602/) (2025)
- [DebiasRAG: A Tuning-Free Path to Fair Generation in Large Language Models through Retrieval-Augmented Generation](https://mlsanity.com/paper/2605.16113/) (2026)
- [Inductive-bias Learning: Generating Code Models with Large Language Model](https://mlsanity.com/paper/2308.09890/) (2023)

## Abstract (arXiv)

Large language models (LLMs) can be seen as atomic units of computation mapping sequences to a distribution over sequences. Thus, they can be seen as stochastic language layers in a language network, where the learnable parameters are the natural language prompts at each layer. By stacking two such layers and feeding the output of one layer to the next, we obtain a Deep Language Network (DLN). We first show how to effectively perform prompt optimization for a 1-Layer language network (DLN-1). Then, we present an extension that applies to 2-layer DLNs (DLN-2), where two prompts must be learned. The key idea is to consider the output of the first layer as a latent variable, which requires inference, and prompts to be learned as the parameters of the generative distribution. We first test the effectiveness of DLN-1 in multiple reasoning and natural language understanding tasks. Then, we show that DLN-2 can reach higher performance than a single layer, showing promise that we might reach comparable performance to GPT-4, even when each LLM in the network is smaller and less powerful.