A Theoretical Framework for Prompt Engineering: Approximating Smooth Functions with Transformer Prompts

Prompt engineering has emerged as a powerful technique for guiding large language models (LLMs) toward desired responses, significantly enhancing their performance across diverse tasks. Beyond their role as static predictors, LLMs increasingly function as intelligent agents, capable of reasoning, decision-making, and adapting dynamically to complex environments. However, the theoretical underpinnings of prompt engineering remain largely unexplored. In this paper, we introduce a formal framework demonstrating that transformer models, when provided with carefully designed prompts, can act as a configurable computational system by emulating a ``virtual'' neural network during inference. Specifically, input prompts effectively translate into the corresponding network configuration, enabling LLMs to adjust their internal computations dynamically. Building on this construction, we establish an approximation theory for $\beta$-times differentiable functions, proving that transformers can approximate such functions with arbitrary precision when guided by appropriately structured prompts. Moreover, our framework provides theoretical justification for several empirically successful prompt engineering techniques, including the use of longer, structured prompts, filtering irrelevant information, enhancing prompt token diversity, and leveraging multi-agent interactions. By framing LLMs as adaptable agents rather than static models, our findings underscore their potential for autonomous reasoning and problem-solving, paving the way for more robust and theoretically grounded advancements in prompt engineering and AI agent design.

MPO: An Efficient Post-Processing Framework for Mixing Diverse Preference Alignment

Reinforcement Learning from Human Feedback (RLHF) has shown promise in aligning large language models (LLMs). Yet its reliance on a singular reward model often overlooks the diversity of human preferences. Recent approaches address this limitation by leveraging multi-dimensional feedback to fine-tune corresponding reward models and train LLMs using reinforcement learning. However, the process is costly and unstable, especially given the competing and heterogeneous nature of human preferences. In this paper, we propose Mixing Preference Optimization (MPO), a post-processing framework for aggregating single-objective policies as an alternative to both multi-objective RLHF (MORLHF) and MaxMin-RLHF. MPO avoids alignment from scratch. Instead, it log-linearly combines existing policies into a unified one with the weight of each policy computed via a batch stochastic mirror descent. Empirical results demonstrate that MPO achieves balanced performance across diverse preferences, outperforming or matching existing models with significantly reduced computational costs.

An Overview of Large Language Models for Statisticians

Large Language Models (LLMs) have emerged as transformative tools in artificial intelligence (AI), exhibiting remarkable capabilities across diverse tasks such as text generation, reasoning, and decision-making. While their success has primarily been driven by advances in computational power and deep learning architectures, emerging problems -- in areas such as uncertainty quantification, decision-making, causal inference, and distribution shift -- require a deeper engagement with the field of statistics. This paper explores potential areas where statisticians can make important contributions to the development of LLMs, particularly those that aim to engender trustworthiness and transparency for human users. Thus, we focus on issues such as uncertainty quantification, interpretability, fairness, privacy, watermarking and model adaptation. We also consider possible roles for LLMs in statistical analysis. By bridging AI and statistics, we aim to foster a deeper collaboration that advances both the theoretical foundations and practical applications of LLMs, ultimately shaping their role in addressing complex societal challenges.

A Statistical Hypothesis Testing Framework for Data Misappropriation Detection in Large Language Models

Large Language Models (LLMs) are rapidly gaining enormous popularity in recent years. However, the training of LLMs has raised significant privacy and legal concerns, particularly regarding the inclusion of copyrighted materials in their training data without proper attribution or licensing, which falls under the broader issue of data misappropriation. In this article, we focus on a specific problem of data misappropriation detection, namely, to determine whether a given LLM has incorporated data generated by another LLM. To address this issue, we propose embedding watermarks into the copyrighted training data and formulating the detection of data misappropriation as a hypothesis testing problem. We develop a general statistical testing framework, construct a pivotal statistic, determine the optimal rejection threshold, and explicitly control the type I and type II errors. Furthermore, we establish the asymptotic optimality properties of the proposed tests, and demonstrate its empirical effectiveness through intensive numerical experiments.

S$^{2}$FT: Efficient, Scalable and Generalizable LLM Fine-tuning by Structured Sparsity

Current PEFT methods for LLMs can achieve either high quality, efficient training, or scalable serving, but not all three simultaneously. To address this limitation, we investigate sparse fine-tuning and observe a remarkable improvement in generalization ability. Utilizing this key insight, we propose a family of Structured Sparse Fine-Tuning (S$^{2}$FT) methods for LLMs, which concurrently achieve state-of-the-art fine-tuning performance, training efficiency, and inference scalability. S$^{2}$FT accomplishes this by "selecting sparsely and computing densely". It selects a few heads and channels in the MHA and FFN modules for each Transformer block, respectively. Next, it co-permutes weight matrices on both sides of the coupled structures in LLMs to connect the selected components in each layer into a dense submatrix. Finally, S$^{2}$FT performs in-place gradient updates on all submatrices. Through theoretical analysis and empirical results, our method prevents overfitting and forgetting, delivers SOTA performance on both commonsense and arithmetic reasoning with 4.6% and 1.3% average improvements compared to LoRA, and surpasses full FT by 11.5% when generalizing to various domains after instruction tuning. Using our partial backpropagation algorithm, S$^{2}$FT saves training memory up to 3$\times$ and improves latency by 1.5-2.7$\times$ compared to full FT, while delivering an average 10% improvement over LoRA on both metrics. We further demonstrate that the weight updates in S$^{2}$FT can be decoupled into adapters, enabling effective fusion, fast switch, and efficient parallelism for serving multiple fine-tuned models.

Differentially Private Learning Beyond the Classical Dimensionality Regime

We initiate the study of differentially private learning in the proportional dimensionality regime, in which the number of data samples $n$ and problem dimension $d$ approach infinity at rates proportional to one another, meaning that $d / n \to \delta$ as $n \to \infty$ for an arbitrary, given constant $\delta \in (0, \infty)$. This setting is significantly more challenging than that of all prior theoretical work in high-dimensional differentially private learning, which, despite the name, has assumed that $\delta = 0$ or is sufficiently small for problems of sample complexity $O(d)$, a regime typically considered "low-dimensional" or "classical" by modern standards in high-dimensional statistics. We provide sharp theoretical estimates of the error of several well-studied differentially private algorithms for robust linear regression and logistic regression, including output perturbation, objective perturbation, and noisy stochastic gradient descent, in the proportional dimensionality regime. The $1 + o(1)$ factor precision of our error estimates enables a far more nuanced understanding of the price of privacy of these algorithms than that afforded by existing, coarser analyses, which are essentially vacuous in the regime we consider. We incorporate several probabilistic tools that have not previously been used to analyze differentially private learning algorithms, such as a modern Gaussian comparison inequality and recent universality laws with origins in statistical physics.

FactTest: Factuality Testing in Large Language Models with Statistical Guarantees

The propensity of Large Language Models (LLMs) to generate hallucinations and non-factual content undermines their reliability in high-stakes domains, where rigorous control over Type I errors (the conditional probability of incorrectly classifying hallucinations as truthful content) is essential. Despite its importance, formal verification of LLM factuality with such guarantees remains largely unexplored. In this paper, we introduce FactTest, a novel framework that statistically assesses whether an LLM can confidently provide correct answers to given questions with high-probability correctness guarantees. We formulate factuality testing as hypothesis testing problem to enforce an upper bound of Type I errors at user-specified significance levels. Notably, we prove that our framework also ensures strong Type II error control under mild conditions and can be extended to maintain its effectiveness when covariate shifts exist. %These analyses are amenable to the principled NP framework. Our approach is distribution-free and works for any number of human-annotated samples. It is model-agnostic and applies to any black-box or white-box LM. Extensive experiments on question-answering (QA) and multiple-choice benchmarks demonstrate that \approach effectively detects hallucinations and improves the model's ability to abstain from answering unknown questions, leading to an over 40% accuracy improvement.

Finite-Sample and Distribution-Free Fair Classification: Optimal Trade-off Between Excess Risk and Fairness, and the Cost of Group-Blindness

Algorithmic fairness in machine learning has recently garnered significant attention. However, two pressing challenges remain: (1) The fairness guarantees of existing fair classification methods often rely on specific data distribution assumptions and large sample sizes, which can lead to fairness violations when the sample size is moderate-a common situation in practice. (2) Due to legal and societal considerations, using sensitive group attributes during decision-making (referred to as the group-blind setting) may not always be feasible. In this work, we quantify the impact of enforcing algorithmic fairness and group-blindness in binary classification under group fairness constraints. Specifically, we propose a unified framework for fair classification that provides distribution-free and finite-sample fairness guarantees with controlled excess risk. This framework is applicable to various group fairness notions in both group-aware and group-blind scenarios. Furthermore, we establish a minimax lower bound on the excess risk, showing the minimax optimality of our proposed algorithm up to logarithmic factors. Through extensive simulation studies and real data analysis, we further demonstrate the superior performance of our algorithm compared to existing methods, and provide empirical support for our theoretical findings.

MMed-RAG: Versatile Multimodal RAG System for Medical Vision Language Models

Artificial Intelligence (AI) has demonstrated significant potential in healthcare, particularly in disease diagnosis and treatment planning. Recent progress in Medical Large Vision-Language Models (Med-LVLMs) has opened up new possibilities for interactive diagnostic tools. However, these models often suffer from factual hallucination, which can lead to incorrect diagnoses. Fine-tuning and retrieval-augmented generation (RAG) have emerged as methods to address these issues. However, the amount of high-quality data and distribution shifts between training data and deployment data limit the application of fine-tuning methods. Although RAG is lightweight and effective, existing RAG-based approaches are not sufficiently general to different medical domains and can potentially cause misalignment issues, both between modalities and between the model and the ground truth. In this paper, we propose a versatile multimodal RAG system, MMed-RAG, designed to enhance the factuality of Med-LVLMs. Our approach introduces a domain-aware retrieval mechanism, an adaptive retrieved contexts selection method, and a provable RAG-based preference fine-tuning strategy. These innovations make the RAG process sufficiently general and reliable, significantly improving alignment when introducing retrieved contexts. Experimental results across five medical datasets (involving radiology, ophthalmology, pathology) on medical VQA and report generation demonstrate that MMed-RAG can achieve an average improvement of 43.8% in the factual accuracy of Med-LVLMs. Our data and code are available in https://github.com/richard-peng-xia/MMed-RAG.

NEAT: Nonlinear Parameter-efficient Adaptation of Pre-trained Models

Fine-tuning pre-trained models is crucial for adapting large models to downstream tasks, often delivering state-of-the-art performance. However, fine-tuning all model parameters is resource-intensive and laborious, leading to the emergence of parameter-efficient fine-tuning (PEFT) methods. One widely adopted PEFT technique, Low-Rank Adaptation (LoRA), freezes the pre-trained model weights and introduces two low-rank matrices whose ranks are significantly smaller than the dimensions of the original weight matrices. This enables efficient fine-tuning by adjusting only a small number of parameters. Despite its efficiency, LoRA approximates weight updates using low-rank decomposition, which struggles to capture complex, non-linear components and efficient optimization trajectories. As a result, LoRA-based methods often exhibit a significant performance gap compared to full fine-tuning. Closing this gap requires higher ranks, which increases the number of parameters. To address these limitations, we propose a nonlinear parameter-efficient adaptation method (NEAT). NEAT introduces a lightweight neural network that takes pre-trained weights as input and learns a nonlinear transformation to approximate cumulative weight updates. These updates can be interpreted as functions of the corresponding pre-trained weights. The nonlinear approximation directly models the cumulative updates, effectively capturing complex and non-linear structures in the weight updates. Our theoretical analysis demonstrates taht NEAT can be more efficient than LoRA while having equal or greater expressivity. Extensive evaluations across four benchmarks and over twenty datasets demonstrate that NEAT significantly outperforms baselines in both vision and text tasks.