Demystifying Group Relative Policy Optimization: Its Policy Gradient is a U-Statistic

Group relative policy optimization (GRPO), a core methodological component of DeepSeekMath and DeepSeek-R1, has emerged as a cornerstone for scaling reasoning capabilities of large language models. Despite its widespread adoption and the proliferation of follow-up works, the theoretical properties of GRPO remain less studied. This paper provides a unified framework to understand GRPO through the lens of classical U-statistics. We demonstrate that the GRPO policy gradient is inherently a U-statistic, allowing us to characterize its mean squared error (MSE), derive the finite-sample error bound and asymptotic distribution of the suboptimality gap for its learned policy. Our findings reveal that GRPO is asymptotically equivalent to an oracle policy gradient algorithm -- one with access to a value function that quantifies the goodness of its learning policy at each training iteration -- and achieves asymptotically optimal performance within a broad class of policy gradient algorithms. Furthermore, we establish a universal scaling law that offers principled guidance for selecting the optimal group size. Empirical experiments further validate our theoretical findings, demonstrating that the optimal group size is universal, and verify the oracle property of GRPO.

A Difference-in-Difference Approach to Detecting AI-Generated Images

Diffusion models are able to produce AI-generated images that are almost indistinguishable from real ones. This raises concerns about their potential misuse and poses substantial challenges for detecting them. Many existing detectors rely on reconstruction error -- the difference between the input image and its reconstructed version -- as the basis for distinguishing real from fake images. However, these detectors become less effective as modern AI-generated images become increasingly similar to real ones. To address this challenge, we propose a novel difference-in-difference method. Instead of directly using the reconstruction error (a first-order difference), we compute the difference in reconstruction error -- a second-order difference -- for variance reduction and improving detection accuracy. Extensive experiments demonstrate that our method achieves strong generalization performance, enabling reliable detection of AI-generated images in the era of generative AI.

Designing Time Series Experiments in A/B Testing with Transformer Reinforcement Learning

A/B testing has become a gold standard for modern technological companies to conduct policy evaluation. Yet, its application to time series experiments, where policies are sequentially assigned over time, remains challenging. Existing designs suffer from two limitations: (i) they do not fully leverage the entire history for treatment allocation; (ii) they rely on strong assumptions to approximate the objective function (e.g., the mean squared error of the estimated treatment effect) for optimizing the design. We first establish an impossibility theorem showing that failure to condition on the full history leads to suboptimal designs, due to the dynamic dependencies in time series experiments. To address both limitations simultaneously, we next propose a transformer reinforcement learning (RL) approach which leverages transformers to condition allocation on the entire history and employs RL to directly optimize the MSE without relying on restrictive assumptions. Empirical evaluations on synthetic data, a publicly available dispatch simulator, and a real-world ridesharing dataset demonstrate that our proposal consistently outperforms existing designs.

Learn-to-Distance: Distance Learning for Detecting LLM-Generated Text

Modern large language models (LLMs) such as GPT, Claude, and Gemini have transformed the way we learn, work, and communicate. Yet, their ability to produce highly human-like text raises serious concerns about misinformation and academic integrity, making it an urgent need for reliable algorithms to detect LLM-generated content. In this paper, we start by presenting a geometric approach to demystify rewrite-based detection algorithms, revealing their underlying rationale and demonstrating their generalization ability. Building on this insight, we introduce a novel rewrite-based detection algorithm that adaptively learns the distance between the original and rewritten text. Theoretically, we demonstrate that employing an adaptively learned distance function is more effective for detection than using a fixed distance. Empirically, we conduct extensive experiments with over 100 settings, and find that our approach demonstrates superior performance over baseline algorithms in the majority of scenarios. In particular, it achieves relative improvements from 57.8\% to 80.6\% over the strongest baseline across different target LLMs (e.g., GPT, Claude, and Gemini).

Double Fairness Policy Learning: Integrating Action Fairness and Outcome Fairness in Decision-making

Fairness is a central pillar of trustworthy machine learning, especially in domains where accuracy- or profit-driven optimization is insufficient. While most fairness research focuses on supervised learning, fairness in policy learning remains less explored. Because policy learning is interventional, it induces two distinct fairness targets: action fairness (equitable action assignments) and outcome fairness (equitable downstream consequences). Crucially, equalizing actions does not generally equalize outcomes when groups face different constraints or respond differently to the same action. We propose a novel double fairness learning (DFL) framework that explicitly manages the trade-off among three objectives: action fairness, outcome fairness, and value maximization. We integrate fairness directly into a multi-objective optimization problem for policy learning and employ a lexicographic weighted Tchebyshev method that recovers Pareto solutions beyond convex settings, with theoretical guarantees on the regret bounds. Our framework is flexible and accommodates various commonly used fairness notions. Extensive simulations demonstrate improved performance relative to competing methods. In applications to a motor third-party liability insurance dataset and an entrepreneurship training dataset, DFL substantially improves both action and outcome fairness while incurring only a modest reduction in overall value.

Detecting LLM-Generated Text with Performance Guarantees

Large language models (LLMs) such as GPT, Claude, Gemini, and Grok have been deeply integrated into our daily life. They now support a wide range of tasks -- from dialogue and email drafting to assisting with teaching and coding, serving as search engines, and much more. However, their ability to produce highly human-like text raises serious concerns, including the spread of fake news, the generation of misleading governmental reports, and academic misconduct. To address this practical problem, we train a classifier to determine whether a piece of text is authored by an LLM or a human. Our detector is deployed on an online CPU-based platform https://huggingface.co/spaces/stats-powered-ai/StatDetectLLM, and contains three novelties over existing detectors: (i) it does not rely on auxiliary information, such as watermarks or knowledge of the specific LLM used to generate the text; (ii) it more effectively distinguishes between human- and LLM-authored text; and (iii) it enables statistical inference, which is largely absent in the current literature. Empirically, our classifier achieves higher classification accuracy compared to existing detectors, while maintaining type-I error control, high statistical power, and computational efficiency.

ReDiF: Reinforced Distillation for Few Step Diffusion

Distillation addresses the slow sampling problem in diffusion models by creating models with smaller size or fewer steps that approximate the behavior of high-step teachers. In this work, we propose a reinforcement learning based distillation framework for diffusion models. Instead of relying on fixed reconstruction or consistency losses, we treat the distillation process as a policy optimization problem, where the student is trained using a reward signal derived from alignment with the teacher's outputs. This RL driven approach dynamically guides the student to explore multiple denoising paths, allowing it to take longer, optimized steps toward high-probability regions of the data distribution, rather than relying on incremental refinements. Our framework utilizes the inherent ability of diffusion models to handle larger steps and effectively manage the generative process. Experimental results show that our method achieves superior performance with significantly fewer inference steps and computational resources compared to existing distillation techniques. Additionally, the framework is model agnostic, applicable to any type of diffusion models with suitable reward functions, providing a general optimization paradigm for efficient diffusion learning.

PyCFRL: A Python library for counterfactually fair offline reinforcement learning via sequential data preprocessing

Reinforcement learning (RL) aims to learn and evaluate a sequential decision rule, often referred to as a "policy", that maximizes the population-level benefit in an environment across possibly infinitely many time steps. However, the sequential decisions made by an RL algorithm, while optimized to maximize overall population benefits, may disadvantage certain individuals who are in minority or socioeconomically disadvantaged groups. To address this problem, we introduce PyCFRL, a Python library for ensuring counterfactual fairness in offline RL. PyCFRL implements a novel data preprocessing algorithm for learning counterfactually fair RL policies from offline datasets and provides tools to evaluate the values and counterfactual unfairness levels of RL policies. We describe the high-level functionalities of PyCFRL and demonstrate one of its major use cases through a data example. The library is publicly available on PyPI and Github (https://github.com/JianhanZhang/PyCFRL), and detailed tutorials can be found in the PyCFRL documentation (https://pycfrl-documentation.netlify.app).

Log-Sum-Exponential Estimator for Off-Policy Evaluation and Learning

Off-policy learning and evaluation leverage logged bandit feedback datasets, which contain context, action, propensity score, and feedback for each data point. These scenarios face significant challenges due to high variance and poor performance with low-quality propensity scores and heavy-tailed reward distributions. We address these issues by introducing a novel estimator based on the log-sum-exponential (LSE) operator, which outperforms traditional inverse propensity score estimators. Our LSE estimator demonstrates variance reduction and robustness under heavy-tailed conditions. For off-policy evaluation, we derive upper bounds on the estimator's bias and variance. In the off-policy learning scenario, we establish bounds on the regret -- the performance gap between our LSE estimator and the optimal policy -- assuming bounded $(1+\epsilon)$-th moment of weighted reward. Notably, we achieve a convergence rate of $O(n^{-\epsilon/(1+ \epsilon)})$ for the regret bounds, where $\epsilon \in [0,1]$ and $n$ is the size of logged bandit feedback dataset. Theoretical analysis is complemented by comprehensive empirical evaluations in both off-policy learning and evaluation scenarios, confirming the practical advantages of our approach. The code for our estimator is available at the following link: https://github.com/armin-behnamnia/lse-offpolicy-learning.

Demystifying the Paradox of Importance Sampling with an Estimated History-Dependent Behavior Policy in Off-Policy Evaluation

This paper studies off-policy evaluation (OPE) in reinforcement learning with a focus on behavior policy estimation for importance sampling. Prior work has shown empirically that estimating a history-dependent behavior policy can lead to lower mean squared error (MSE) even when the true behavior policy is Markovian. However, the question of why the use of history should lower MSE remains open. In this paper, we theoretically demystify this paradox by deriving a bias-variance decomposition of the MSE of ordinary importance sampling (IS) estimators, demonstrating that history-dependent behavior policy estimation decreases their asymptotic variances while increasing their finite-sample biases. Additionally, as the estimated behavior policy conditions on a longer history, we show a consistent decrease in variance. We extend these findings to a range of other OPE estimators, including the sequential IS estimator, the doubly robust estimator and the marginalized IS estimator, with the behavior policy estimated either parametrically or non-parametrically.