The Free Energy Principle (FEP) states that self-organizing systems must minimize variational free energy to persist (Friston, 2010, 2019), but the path from principle to implementable algorithm has remained unclear. We present a constructive proof that the FEP can be realized through exact local credit assignment. The system decomposes gradient computation hierarchically: spatial credit via feedback alignment, temporal credit via eligibility traces, and structural credit via a Trophic Field Map (TFM) that estimates expected gradient magnitude for each connection block. We prove these mechanisms are exact at their respective levels and validate the central claim empirically: the TFM achieves 0.9693 Pearson correlation with oracle gradients. This exactness produces emergent capabilities including 98.6% retention after task interference, autonomous recovery from 75% structural damage, self-organized criticality (spectral radius ρ 1.0), and sample-efficient reinforcement learning on continuous control tasks without replay buffers. The architecture unifies Pri-gogine's dissipative structures (Prigogine, 1977), Fris-ton's free energy minimization (Friston, 2010), and Hopfield's attractor dynamics (Hopfield, 1982; Amit et al., 1985a,b), demonstrating that exact hierarchical inference over network topology can be implemented with local, biologically plausible rules.

Intelligent reflecting surface (IRS) assisted unmanned aerial vehicle (UAV) systems provide a new paradigm for reconfigurable and flexible wireless communications. To enable more energy efficient and spectrum efficient IRS assisted UAV wireless communications, this paper introduces a novel IRS-assisted UAV enabled spectrum sharing system with orthogonal frequency division multiplexing (OFDM). The goal is to maximize the energy efficiency (EE) of the secondary network by jointly optimizing the beamforming, subcarrier allocation, IRS phase shifts, and the UAV trajectory subject to practical transmit power and passive reflection constraints as well as UAV physical limitations. A physically grounded propulsion-energy model is adopted, with its tight upper bound used to form a tractable EE lower bound for the spectrum sharing system. To handle highly non convex, time coupled optimization problems with a mixed continuous and discrete policy space, we develop a deep reinforcement learning (DRL) approach based on the actor critic framework. Extended experiments show the significant EE improvement of the proposed DRL-based approach compared to several benchmark schemes, thus demonstrating the effectiveness and robustness of the proposed approach with mobility.

Safety is a major concern in reinforcement learning (RL): we aim at developing RL systems that not only perform optimally, but are also safe to deploy by providing formal guarantees about their safety. To this end, we introduce Probabilistic Shielding via Risk Augmentation (ProSh), a model-free algorithm for safe reinforcement learning under cost constraints. ProSh augments the Constrained MDP state space with a risk budget and enforces safety by applying a shield to the agent's policy distribution using a learned cost critic. The shield ensures that all sampled actions remain safe in expectation. We also show that optimality is preserved when the environment is deterministic. Since ProSh is model-free, safety during training depends on the knowledge we have acquired about the environment. We provide a tight upper-bound on the cost in expectation, depending only on the backup-critic accuracy, that is always satisfied during training. Under mild, practically achievable assumptions, ProSh guarantees safety even at training time, as shown in the experiments.

Aligning language models using LLM judge feedback offers a scalable alternative to human annotation, yet is plagued by judgment inconsistencies that destabilize reinforcement learning. While prior work has focused on judge accuracy, the critical issue of logical coherence--particularly preference cycles (A B C A)--has been largely unaddressed. To address this gap, this work introduces an end-to-end framework to systematically detect and resolve these inconsistencies within the reinforcement learning training loop. Our framework features two core contributions: the Conflict Detection Rate (CDR), a novel metric to quantify judgment conflicts, and Deconflicted Graph Rewards (DGR), a signal-purification framework that eliminates cycles before policy optimization. DGR constructs preference graphs from raw judgments, transforms them into conflict-free Directed Acyclic Graphs (DAGs), and generates a logically coherent reward signal compatible with any policy optimizer. Experiments confirm that our framework significantly improves training stability and model performance over strong baselines, establishing logical consistency as a crucial and now-addressable dimension of AI feedback. Aligning large language models (LLMs) with human preferences, traditionally achieved through Reinforcement Learning from Human Feedback (RLHF) (Ouyang et al., 2022), is critical for safe AI deployment. However, the reliance on costly and slow human annotation has created a scalability bottleneck, pushing the field towards Reinforcement Learning from AI Feedback (RLAIF) (Bai et al., 2022; Lee et al., 2023). Within RLAIF, the pairwise comparison paradigm--where an LLM judge selects the better of two responses--has become the de facto standard, prized for its intuitive nature and fine-grained feedback signal that underpins many state-of-the-art alignment techniques (Song et al., 2024; Wang et al., 2024). Recent advances in pairwise methods include the Pairwise-RL framework (Xu et al., 2025), which addresses the fundamental misalignment between generative base models and discriminative reward tasks by unifying reward model training and reinforcement learning application in a consistent pairwise paradigm. This framework combines generative reward modeling with pairwise policy optimization, leveraging generative modeling techniques to improve reward model performance and score calibration. Consequently, our work focuses on the pairwise paradigm, building upon these foundational approaches. However, this scalability comes at a hidden cost: the erosion of logical consistency.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm to improve Large Language Models on reasoning tasks such as coding, math or logic. To assess the reasoning boundary (the fraction of problems a model can solve) researchers often report Pass@k at large sampling budgets. Recent results reveal a crossover phenomenon: while RLVR models outperform the base model at small k values, the base model usually outperforms them when sampling a very large number of completions. This has been interpreted as evidence that base models have a larger reasoning boundary. We argue that on tasks with discrete answer spaces, such as math with numeric outputs, Pass@k at large k reflects the increasingly higher chance of success in the limit of the number of trials rather than genuine reasoning, and can therefore be misleading. We propose Cover@tau, which measures the fraction of problems that a model can solve for which at least a tau proportion of completions are correct. Unlike Pass@k, Cover@tau captures reasoning under an explicit reliability threshold: models that rely on random guessing degrade rapidly as tau increases. We evaluate several RLVR models using Cover@tau-based metrics and illustrate how the relative rankings of popular algorithms change compared to Pass@1, offering a different perspective on reasoning boundaries.

Humanoid table tennis (TT) demands rapid perception, proactive whole-body motion, and agile footwork under strict timing -- capabilities that remain difficult for unified controllers. We propose a reinforcement learning framework that maps ball-position observations directly to whole-body joint commands for both arm striking and leg locomotion, strengthened by predictive signals and dense, physics-guided rewards. A lightweight learned predictor, fed with recent ball positions, estimates future ball states and augments the policy's observations for proactive decision-making. During training, a physics-based predictor supplies precise future states to construct dense, informative rewards that lead to effective exploration. The resulting policy attains strong performance across varied serve ranges (hit rate $\geq$ 96% and success rate $\geq$ 92%) in simulations. Ablation studies confirm that both the learned predictor and the predictive reward design are critical for end-to-end learning. Deployed zero-shot on a physical Booster T1 humanoid with 23 revolute joints, the policy produces coordinated lateral and forward-backward footwork with accurate, fast returns, suggesting a practical path toward versatile, competitive humanoid TT.

This paper provides a self-contained, from-scratch, exposition of key algorithms for instruction tuning of models: SFT, Rejection Sampling, REINFORCE, Trust Region Policy Optimization (TRPO), Proximal Policy Optimization (PPO), Group Relative Policy Optimization (GRPO), and Direct Preference Optimization (DPO). Explanations of these algorithms often assume prior knowledge, lack critical details, and/or are overly generalized and complex. Here, each method is discussed and developed step by step using simplified and explicit notation focused on LLMs, aiming to eliminate ambiguity and provide a clear and intuitive understanding of the concepts. By minimizing detours into the broader RL literature and connecting concepts to LLMs, we eliminate superfluous abstractions and reduce cognitive overhead. Following this exposition, we provide a literature review of new techniques and approaches beyond those detailed. Finally, new ideas for research and exploration in the form of GRAPE (Generalized Relative Advantage Policy Evolution) are presented.

Abstract-- Reliable autonomous navigation across the unstructured terrains of distant planetary surfaces is a critical enabler for future space exploration. However, the deployment of learning-based controllers is hindered by the inherent sim-to-real gap, particularly for the complex dynamics of wheel interactions with granular media. This work presents a complete sim-to-real framework for developing and validating robust control policies for dynamic waypoint tracking on such challenging surfaces. We leverage massively parallel simulation to train reinforcement learning agents across a vast distribution of procedurally generated environments with randomized physics. These policies are then transferred zero-shot to a physical wheeled rover operating in a lunar-analogue facility. Our experiments systematically compare multiple reinforcement learning algorithms and action smoothing filters to identify the most effective combinations for real-world deployment. Crucially, we provide strong empirical evidence that agents trained with procedural diversity achieve superior zero-shot performance compared to those trained on static scenarios. We also analyze the trade-offs of fine-tuning with high-fidelity particle physics, which offers minor gains in low-speed precision at a significant computational cost. T ogether, these contributions establish a validated workflow for creating reliable learning-based navigation systems, marking a substantial step towards deploying autonomous robots in the final frontier .

Numerous heuristics and advanced approaches have been proposed for exploration in different settings for deep reinforcement learning. Noise-based exploration generally fares well with dense-shaped rewards and bonus-based exploration with sparse rewards. However, these methods usually require additional tuning to deal with undesirable reward settings by adjusting hyperparameters and noise distributions. Rewards that actively discourage exploration, i.e., with an action cost and no other dense signal to follow, can pose a major challenge. We propose a novel exploration method, Stable Error-seeking Exploration (SEE), that is robust across dense, sparse, and exploration-adverse reward settings. To this endeavor, we revisit the idea of maximizing the TD-error as a separate objective. Our method introduces three design choices to mitigate instability caused by far-off-policy learning, the conflict of interest of maximizing the cumulative TD-error in an episodic setting, and the non-stationary nature of TD-errors. SEE can be combined with off-policy algorithms without modifying the optimization pipeline of the original objective. In our experimental analysis, we show that a Soft-Actor Critic agent with the addition of SEE performs robustly across three diverse reward settings in a variety of tasks without hyperparameter adjustments.

A key challenge in reinforcement learning (RL) is reward (mis)specification, whereby imprecisely defined reward functions can result in unintended, possibly harmful, behaviours. Indeed, reward functions in RL are typically treated as black-box mappings from state-action pairs to scalar values. While effective in many settings, this approach provides no information about why rewards are given, which can hinder learning and interpretability. Reward Machines address this issue by representing reward functions as finite state automata, enabling the specification of structured, non-Markovian reward functions. However, their ex-pressivity is typically bounded by regular languages, leaving them unable to capture more complex behaviours such as counting or parametrised conditions. In this work, we build on the Runtime Monitoring Language (RML) to develop a novel class of language-based Reward Machines. By leveraging the built-in memory of RML, our approach can specify reward functions for non-regular, non-Markovian tasks. We demonstrate the expressiveness of our approach through experiments, highlighting additional advantages in flexible event-handling and task specification over existing Reward Machine-based methods.