# Scaling Textual Gradients via Sampling-Based Momentum

arXiv: https://arxiv.org/abs/2506.00400 · HTML: https://mlsanity.com/paper/2506.00400/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 90 references that are in the map.
- First posted on arXiv 31 May 2025.

## Builds on (most cited first, 30 of 57)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [HotpotQA: A Dataset for Diverse, Explainable Multi-hop Question Answering](https://mlsanity.com/paper/1809.09600/) (2018)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [SGLang: Efficient Execution of Structured Language Model Programs](https://mlsanity.com/paper/2312.07104/) (2023)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [Large Language Models Are Human-Level Prompt Engineers](https://mlsanity.com/paper/2211.01910/) (2022)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Large Language Models as Optimizers](https://mlsanity.com/paper/2309.03409/) (2023)
- [Quantifying Language Models' Sensitivity to Spurious Features in Prompt Design or: How I learned to start worrying about prompt formatting](https://mlsanity.com/paper/2310.11324/) (2023)
- [Automatic Prompt Optimization with "Gradient Descent" and Beam Search](https://mlsanity.com/paper/2305.03495/) (2023)
- [Quest: Query-Aware Sparsity for Efficient Long-Context LLM Inference](https://mlsanity.com/paper/2406.10774/) (2024)
- [Optimizing Instructions and Demonstrations for Multi-Stage Language Model Programs](https://mlsanity.com/paper/2406.11695/) (2024)
- [TextGrad: Automatic "Differentiation" via Text](https://mlsanity.com/paper/2406.07496/) (2024)
- [PromptAgent: Strategic Planning with Language Models Enables Expert-level Prompt Optimization](https://mlsanity.com/paper/2310.16427/) (2023)
- [Large Language Model as Attributed Training Data Generator: A Tale of Diversity and Bias](https://mlsanity.com/paper/2306.15895/) (2023)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [BABILong: Testing the Limits of LLMs with Long Context Reasoning-in-a-Haystack](https://mlsanity.com/paper/2406.10149/) (2024)

## Built on by (most cited first, 6 of 6)

- [Test-Time Policy Adaptation for Enhanced Multi-Turn Interactions with LLMs](https://mlsanity.com/paper/2509.23166/) (2025)
- [AgentGrad: Intervention-guided Prompt Optimization for Multi Agent Systems](https://mlsanity.com/paper/2609.08572/) (2026)
- [HPFA: Hypergraph-Based Paired Failure Attribution for LLM Reasoning](https://mlsanity.com/paper/2608.02026/) (2026)
- [The Optimizer Is the Agent: Reasoning-Driven Search across Prompts, Programs, and ML Workflows](https://mlsanity.com/paper/2608.06714/) (2026)
- [Reflective Context Learning: Studying the Optimization Primitives of Context Space](https://mlsanity.com/paper/2604.03189/) (2026)
- [Trading Strategy Optimization via Textual Gradient](https://mlsanity.com/paper/2610.03128/) (2026)

## Abstract (arXiv)

LLM-based prompt optimization, which uses LLM-provided ``textual gradients'' (feedback) to refine prompts, has emerged as an effective method for automatic prompt engineering. However, its scalability and stability are unclear when using more data in training. We systematically investigate the potential and challenges of scaling training data in textual gradient descent. We show that naively scaling training examples is infeasible due to both explicit context-length limits and an implicit context wall, where long-context degradation yields diminishing returns. Inspired by prior wisdom in stochastic gradient descent, we propose Textual Stochastic Gradient Descent with Momentum (TSGD-M), which reweights updates through momentum sampling, using bootstrapped minibatch validation accuracy as importance weights over historical prompts. To stabilize TSGD and enable effective scaling within a limited context window, TSGD-M carries prior prompts information by textitdynamically exploring the past top performing prompts without expanding input context length. TSGD-M integrates seamlessly into existing prompt optimization frameworks, including TextGrad, DSPy-COPRO, and AdalFlow, and achieves consistent gains across 6 benchmarks.