Ayeong Lee | Searcharxiv

arXiv2026

Variance Reduction Methods for Dirichlet Expectations

Dirichlet distributions are probability measures on the unit simplex. They are often used as prior distributions in modeling categorical data, such as in topic analysis of text data. Motivated by this application, we consider Monte Carlo estimation of expectations $\mathbb{E}[\exp(nH(θ))]$, where $θ$ has a Dirichlet distribution, $H$ is a real-valued function, and $n$ is a parameter. We develop variance reduction techniques particularly designed to work well for large $n$. Our analysis is guided by the Laplace method for approximating integrals, which we extend to fit our problem setting. We develop an importance sampling method that achieves a near-optimal asymptotic relative error. We use related ideas to select a provably effective control variate. We illustrate these results through their application in topic analysis.

stat.ME↗

arXiv2025

How Well do LLMs Compress Their Own Chain-of-Thought? A Token Complexity Approach

Chain-of-thought prompting has emerged as a powerful technique for enabling large language models (LLMs) to solve complex reasoning tasks. However, these reasoning chains can be verbose, raising concerns about efficiency. In response, recent works have sought to decrease response lengths through simple prompting strategies (e.g. 'be concise'). In this work, we conduct the first systematic study of the relationship between reasoning length and model performance across a diverse range of compression instructions (e.g. 'use 10 words or less' or 'remove all punctuation'). In doing so, we discover a universal tradeoff between reasoning length and accuracy that persists across even very distinct reasoning chains. We demonstrate that this tradeoff emerges from a sharp threshold behavior at the question level: each task has an intrinsic 'token complexity' - a minimal number of tokens required for successful problem-solving. We show how token complexity enables us to compute information-theoretic limits on the accuracy-compression tradeoff, and find that prompt-based compression strategies operate far from these theoretical limits. This suggests there may be significant room for improvement and our framework provides a benchmark to help researchers evaluate progress in reasoning efficiency. Our work also highlights the importance of adaptive compression -- giving shorter responses for easier questions -- and we show that token complexity is a useful tool for measuring this capability.

cs.CL↗