Arne Vanhoyweghen

Dominik Baumann, Erfaun Noorani, Arsenii Mustafin et al.

In reinforcement learning, we typically aim to optimize the expected value of the sum of rewards an agent collects over a trajectory. However, if the process generating these rewards is non-ergodic, the expected value, i.e., the average over infinitely many trajectories with a given policy, is uninformative for the average over a single, but infinitely long trajectory. Thus, if we care about how the individual agent performs during deployment, the expected value is not a good optimization objective. In this paper, we discuss the impact of non-ergodic reward processes on reinforcement learning agents through an instructive example, relate the notion of ergodic reward processes to more widely used notions of ergodic Markov chains, and present existing solutions that optimize long-term performance of individual trajectories under non-ergodic reward dynamics.

Arne Vanhoyweghen, Vincent Holst, Melika Mobini et al.

Providing timely and individualised feedback on handwritten student work is highly beneficial for learning but difficult to achieve at scale. This challenge has become more pressing as generative AI undermines the reliability of take-home assessments, shifting emphasis toward supervised, in-class evaluation. We present a scalable, end-to-end workflow for LLM-assisted grading of short, pen-and-paper assessments. The workflow spans (1) constructing solution keys, (2) developing detailed rubric-style grading keys used to guide the LLM, and (3) a grading procedure that combines automated scanning and anonymisation, multi-pass LLM scoring, automated consistency checks, and mandatory human verification. We deploy the system in two undergraduate mathematics courses using six low-stakes in-class tests. Empirically, LLM assistance reduces grading time by approximately 23% while achieving agreement comparable to, and in several cases tighter than, fully manual grading. Occasional model errors occur but are effectively contained by the hybrid design. Overall, our results show that carefully embedded human-in-the-loop LLM grading can substantially reduce workload while maintaining fairness and accuracy.

Bert Verbruggen, Arne Vanhoyweghen, Vincent Ginis

Reinforcement Learning (RL) remains a central optimisation framework in machine learning. Although RL agents can converge to optimal solutions, the definition of ``optimality'' depends on the environment's statistical properties. The Bellman equation, central to most RL algorithms, is formulated in terms of expected values of future rewards. However, when ergodicity is broken, long-term outcomes depend on the specific trajectory rather than on the ensemble average. In such settings, the ensemble average diverges from the time-average growth experienced by individual agents, with expected-value formulations yielding systematically suboptimal policies. Prior studies demonstrated that traditional RL architectures fail to recover the true optimum in non-ergodic environments. We extend this analysis to deep RL implementations and show that these, too, produce suboptimal policies under non-ergodic dynamics. Introducing explicit time dependence into the learning process can correct this limitation. By allowing the network's function approximation to incorporate temporal information, the agent can estimate value functions consistent with the process's intrinsic growth rate. This improvement does not require altering the environmental feedback, such as reward transformations or modified objective functions, but arises naturally from the agent's exposure to temporal trajectories. Our results contribute to the growing body of research on reinforcement learning methods for non-ergodic systems.

Arne Vanhoyweghen, Brecht Verbeken, Andres Algaba et al.

Fine-tuning Large Language Models (LLMs) with reinforcement learning to produce an explicit Chain-of-Thought (CoT) before answering produces models that consistently raise overall performance on code, math, and general-knowledge benchmarks. However, on benchmarks where LLMs currently achieve low accuracy, such as Humanity's Last Exam (HLE), they often report high self-confidence, reflecting poor calibration. Here, we test whether measurable properties of the CoT provide reliable signals of an LLM's internal confidence in its answers. We analyze three feature classes: (i) CoT length, (ii) intra-CoT sentiment volatility, and (iii) lexicographic hints, including hedging words. Using DeepSeek-R1 and Claude 3.7 Sonnet on both Humanity's Last Exam (HLE), a frontier benchmark with very low accuracy, and Omni-MATH, a saturated benchmark of moderate difficulty, we find that lexical markers of uncertainty (e.g., $\textit{guess}$, $\textit{stuck}$, $\textit{hard}$) in the CoT are the strongest indicators of an incorrect response, while shifts in the CoT sentiment provide a weaker but complementary signal. CoT length is informative only on Omni-MATH, where accuracy is already high ($\approx 70\%$), and carries no signal on the harder HLE ($\approx 9\%$), indicating that CoT length predicts correctness only in the intermediate-difficulty benchmarks, i.e., inside the model's demonstrated capability, but still below saturation. Finally, we find that uncertainty indicators in the CoT are consistently more salient than high-confidence markers, making errors easier to predict than correct responses. Our findings support a lightweight post-hoc calibration signal that complements unreliable self-reported probabilities and supports safer deployment of LLMs.