Total Variation Distance Estimation in Autoregressive Models
Eric Price, Kevin Tian, Zhiyang Xun, Yusong Zhu
AI 速读 基于摘要
研究在样本、logit 与含噪 logit 三种访问模型下，估计两个长度 n 自回归分布之间全变差距离的查询复杂度，并给出紧致性与实证评估。
cs.LG · cs.DS · stat.ME · stat.ML
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Modern LLM deployments use a number of implementation choices and inference optimizations (e.g., batching, custom kernels, and quantization) on top of fixed weights, so two engines serving "the same model" can produce meaningfully different distributions. We study the problem of estimating the total variation (TV) distance between two length-$n$ autoregressive distributions to additive error $\varepsilon$, under three access models. (1) Under sample access, we use $\widetilde{O}(n^2 K/\varepsilon^2)$ queries, where $K$ is the maximum support of the next-token distribution. This improves upon the $\widetilde{O}(n^3 m/\varepsilon^5)$-query estimator of Meel et al. (2025), where $m \geq K$ is the total size of the token alphabet. (2) Under logit access, we use $O(n/\varepsilon^2)$ queries, and this is tight. (3) Under noisy logit access, we smoothly interpolate between the above two guarantees: if probability values are given to relative error $σ$, we use $\widetilde{O}((n+n^2σ^2)/\varepsilon^2)$ queries. We complement our theoretical results with an empirical evaluation of our algorithms, for example measuring the distance between SGLang and vLLM serving identical weights. Our experiments highlight the robustness and practicality of estimating the total variation distance, which remains estimable where the KL divergence is infinite. Our code is available at https://github.com/XunZhiyang/llm-tv-estimation.
We address the problem of active logistic regression in the realizable setting. It is well known that active learning can require exponentially fewer label queries compared to passive learning, in some cases using $\log \frac{1}{\eps}$ rather than $\poly(1/\eps)$ labels to get error $\eps$ larger than the optimum. We present the first algorithm that is polynomially competitive with the optimal algorithm on every input instance, up to a factor $d^2$ and factors polylogarithmic in the error and domain size. In particular, on a simple family of logistic threshold instances with a uniform distribution, our algorithm uses $\polylog(1/\eps)$ labels, whereas every passive learner requires $Ω(1/\eps^2)$ labels. Our algorithm is based on sampling and can be extended to learn more general class of functions. We further support our theoretical results with experiments demonstrating performance gains for logistic regression compared to existing active learning algorithms.
For some hypothesis classes and input distributions, active agnostic learning needs exponentially fewer samples than passive learning; for other classes and distributions, it offers little to no improvement. The most popular algorithms for agnostic active learning express their performance in terms of a parameter called the disagreement coefficient, but it is known that these algorithms are inefficient on some inputs. We take a different approach to agnostic active learning, getting an algorithm that is competitive with the optimal algorithm for any binary hypothesis class $H$ and distribution $D_X$ over $X$. In particular, if any algorithm can use $m^*$ queries to get $O(η)$ error, then our algorithm uses $O(m^* \log |H|)$ queries to get $O(η)$ error. Our algorithm lies in the vein of the splitting-based approach of Dasgupta [2004], which gets a similar result for the realizable ($η= 0$) setting. We also show that it is NP-hard to do better than our algorithm's $O(\log |H|)$ overhead in general.