Reward Design for RL

Reinforcement learning with verifiable rewards (RLVR) has become a standard approach for improving reasoning in language models, yet models trained with RLVR often suffer from diversity collapse: while single-sample accuracy improves, multi-sample coverage degrades, sometimes falling below the base model. We provide a structural account of this phenomenon grounded in the properties of binary rewards. Binary rewards create a fundamental degeneracy for policy gradient methods: the set of distributions maximizing expected reward is infinite, with no distinguished element. KL-control resolves this degeneracy by selecting, in the limit $β\to 0$, the filtered model $p_*:=a(\cdot\mid\mathcal{Y}_1)$ -- the base model conditioned on validity -- which is the unique fully valid distribution closest to the base model in KL divergence. This selection operates through a nontrivial asymmetry: the tilted distribution $p_{[β]}\propto a(y)\,e^{v(y)/β}$ converges to $p_*$ in forward KL as $β\to 0$, yet $p_*$ cannot serve as a direct optimization target because $\mathrm{KL}(q\,\|\,p_*)$ is infinite for any full-support policy $q$. We develop explicit formulas relating the hyperparameter $β$ to the more interpretable target validity rate $μ$. Under model misspecification -- the typical practical regime -- the pressure to decrease $β$ drives the optimizer toward highly concentrated distributions over a small number of valid outputs, collapsing toward ever fewer as $β$ decreases, rather than toward the filtered model. We illustrate this mechanism on a toy autoregressive experiment and discuss how alternative divergences that target $p_*$ directly -- as pursued empirically by \citet{kruszewski_whatever_2026} -- avoid this failure mode by rewarding coverage of $p_*$'s support rather than concentration on high-validity outputs.

The Legg--Hutter universal intelligence measure provides a rigorous scalar assessment of general intelligence as expected reward across all computable environments, weighted by simplicity. However, the measure presupposes an externally specified reward function, raising the question of whether the reward primitive is inherently arbitrary or whether a canonical choice exists. We propose a new measure, called intervention complexity, that has five natural properties: environment-derivedness, universality, minimality, sensitivity, and achievement preference. Given a resource function rho encoding an inductive bias (such as program length, execution time, or energy), rho-intervention complexity is a universal reward. The result yields a family of canonical rewards indexed by resource bias, providing a principled completion of the Legg--Hutter framework that does not require external normative input. We further propose a two-dimensional characterisation of intelligence: agent competence (how well the agent performs relative to the oracle optimum) and learning efficiency (how quickly this competence improves with experience). A separation theorem establishes that the choice of resource bias determines the computability of the resulting measure: action-count IC is computable in polynomial time, while program-length IC without oracle access is uncomputable, with the gap between oracle and bare IC precisely quantifying the information-theoretic content of learning. We discuss implications for superintelligence and for pre-training universal agents.

Mathematical reasoning is a key benchmark for large language models. Reinforcement learning is a standard post-training mechanism for improving the reasoning capabilities of large language models, yet performance remains sensitive to the design of the reward function that drives policy optimization. This paper introduces a search-driven framework that treats the reward specification itself as an object of optimization. The setting of interest is one in which the base model is held fixed and the reward specification is the primary remaining design lever. Candidate reward functions are generated by a frontier language model, validated automatically, screened through 500-step Group Relative Policy Optimization (GRPO) training runs on a Llama-3.2-3B-Instruct base model with Low-Rank Adaptation (LoRA), and ranked by F1 on the GSM8K test set. Ranked summaries from prior rounds are then fed back into the next round of generation. Over five rounds, the search produces 50 candidate rewards. The mean F1 rises from 0.596 in Round 1 to 0.632 in Round 5, and the top individual reward reaches F1 = 0.787. Seven ensemble configurations of top-ranked rewards are evaluated. The best ensemble achieves F1 = 0.795 (95% bootstrap CI [0.756, 0.832]) and accuracy 0.660 [0.635, 0.686], a 0.19 absolute F1 gain over a base-rewards-only GRPO baseline (F1 = 0.609). Pairwise McNemar tests with Bonferroni correction show all five-or-more-reward configurations are statistically indistinguishable at α = 0.05/21. A three-seed re-training of the best ensemble yields F1 of 0.785. A randomly drawn 5-reward control collapses to F1 = 0.047, which shows that the ranked-feedback loop, not the additive signal of having more rewards, drives the gain.

A key challenge in multi-agent reinforcement learning (MARL) lies in designing learning signals that effectively promote coordination among agents. Designing such signals requires estimating how one agent's current action affects its teammates over future interaction steps. To address this, we introduce Multi-step Advantage-Gated Interventional Causal MARL (MAGIC), a framework that estimates multi-step action effects between agents and selectively converts them into intrinsic rewards. MAGIC uses counterfactual action interventions to compare teammate futures under factual and counterfactual branches, and introduces a gate based on advantage to direct exploration toward beneficial behaviors aligned with the task goal. Experiments on Multi-Agent Particle Environments (MPE) and StarCraft micromanagement benchmarks (SMAC and SMACv2) show that MAGIC consistently outperforms leading prior methods, with average relative final performance improvements of 26.9% and 10.1%, respectively.

Reinforcement learning (RL) from unit-test feedback has become a standard post-training recipe for improving large language models (LLMs) on code generation. However, the pass-all-tests binary reward can be sparse, yielding no learning signal on challenging problems where none of the sampled solutions passes all tests. A common remedy is to use the test-case pass rate as a surrogate reward. In this work, we study pass-rate rewards in critic-free RL for code generation (e.g., GRPO and RLOO) and report a consistent pattern across base models and algorithms: despite alleviating reward sparsity, pass-rate rewards do not reliably improve final performance over binary rewards in rigorous controlled experiments. To understand this discrepancy, we analyze reward density and the resulting gradient directions. We find that pass-rate rewards are denser, but the induced gradient updates do not consistently move probability mass toward full-pass solutions. This arises because test-case pass rate is a miscalibrated surrogate for progress toward full correctness, and partial-pass solutions within the same group can induce conflicting gradient directions that cancel out. Overall, our results suggest that, in critic-free RL, pass-rate rewards are insufficient to improve code generation and motivate reward designs that better align optimization with the goal of full correctness.

Large language models (LLMs) make reward design in reinforcement learning substantially more scalable, but generated rewards are not automatically reliable training objectives. Existing work has focused primarily on generating, evolving, or selecting reward candidates, while paying less attention to when such candidates can be verified and deployed during policy optimization. We study this deployment-time problem by treating generated rewards as reward hypotheses whose utility depends on the competence of the current policy and the phase of training. We propose \textsc{RHyVE}, a competence-aware verification and phase-aware deployment protocol that compares small sets of reward hypotheses from shared policy checkpoints using short-horizon fork verification. Our experiments show that reward rankings are unreliable at low competence but become informative after task-dependent thresholds. On a sparse manipulation task, phase-aware deployment improves peak and retained performance under a locked protocol. Updated LLM-generated reward-candidate experiments show candidate-family-dependent behavior: generated pools can exhibit phase-dependent winner changes, but no fixed warm-up schedule is universally optimal. Held-out schedule selection, conservative selector baselines, compute-matched controls, and scale controls further show that \textsc{RHyVE} is best understood as a verification-informed deployment protocol rather than a universal scheduler. Dense and all-failure boundary experiments delimit the scope of the method. Together, these results suggest that reward generation and reward deployment should be studied as coupled problems: generated rewards must be verified and deployed under changing policy competence.

Training language models via reinforcement learning often relies on imperfect proxy rewards, since ground truth rewards that precisely define the intended behavior are rarely available. Standard metrics for assessing the quality of proxy rewards, such as ranking accuracy, treat incorrect rewards as strictly harmful. In this work, however, we highlight that not all deviations from the ground truth are equal. By theoretically analyzing which outputs attract probability during policy gradient optimization, we categorize reward errors according to their effect on the increase in ground truth reward. The analysis establishes that reward errors, though conventionally viewed as harmful, can also be benign or even beneficial by preventing the policy from stalling around outputs with mediocre ground truth reward. We then present two practical implications of our theory. First, for reinforcement learning from human feedback (RLHF), we develop reward model evaluation metrics that account for the harmfulness of reward errors. Compared to standard ranking accuracy, these metrics typically correlate better with the performance of a language model after RLHF, yet gaps remain in robustly evaluating reward models. Second, we provide insights for reward design in settings with verifiable rewards. A key theme underlying our results is that the effectiveness of a proxy reward function depends heavily on its interaction with the initial policy and learning algorithm.

Chip placement is a critical step in physical design. While reinforcement learning (RL)-based methods have recently emerged, their training primarily focuses on wirelength optimization, and therefore often fail to achieve expert-quality layouts. We identify the reward design as the primary cause for the performance gap with experts, and instead of formalizing intricate processes, we circumvent this by directly learning from expert layouts to derive a reward model. Our approach starts from the final expert layouts to infer step-by-step expert trajectories. Using these trajectories as demonstrations or preferences, we train a model that captures the latent implicit rewards in expert results. Experiments show that our framework can efficiently learn from even a single design and generalize well to unseen cases.

Many Multi-Agent Reinforcement Learning (MARL) agents fail to adapt properly to cooperating with agents trained with the same objectives but different seeds, algorithms, or other training differences. This is the problem of Zero-Shot Coordination (ZSC), which focuses on training agents to cooperate well with unknown agents. ZSC has been studied for a variety of tabular cases and simple games such as Hanabi, achieving excellent results. However, existing solutions to ZSC only consider identical rewards for your trained agents and all future partners. This is not realistic for the trained agents, as they do not consider the problem of cooperating with agents that have identical sparse objectives but shape the rewards for those objectives in different manner. To address this issue, we show how to train an ensemble of methods using randomized reward shapings chosen using 4 selection algorithms. Experiments done on the Overcooked environment demonstrate consistent improvements of 62.2%-119.2% in sparse reward over baseline ZSC algorithms when playing with agents that have identical sparse rewards but different reward shapings.

The temporal lag between actions and their long-term consequences makes credit assignment a challenge when learning goal-directed behaviors from data. Generative world models capture the distribution of future states an agent may visit, indicating that they have captured temporal information. How can that temporal information be extracted to perform credit assignment? In this paper, we formalize how the temporal information stored in world models encodes the underlying geometry of the world. Leveraging optimal transport, we extract this geometry from a learned model of the occupancy measure into a reward function that captures goal-reaching information. Our resulting method, Occupancy Reward Shaping, largely mitigates the problem of credit assignment in sparse reward settings. ORS provably does not alter the optimal policy, yet empirically improves performance by 2.2x across 13 diverse long-horizon locomotion and manipulation tasks. Moreover, we demonstrate the effectiveness of ORS in the real world for controlling nuclear fusion on 3 Tokamak control tasks. Code: https://github.com/aravindvenu7/occupancy_reward_shaping; Website: https://aravindvenu7.github.io/website/ors/

Recent advancements in Reinforcement Learning with Verifiable Rewards (RLVR) have significantly improved Large Language Model (LLM) reasoning, yet models often struggle to explore novel trajectories beyond their initial policy distribution. While offline teacher guidance and entropy-driven strategies have been proposed to address this, they often lack deep integration or are constrained by the model's inherent capacity. In this paper, we propose OGER (Offline-Guided Exploration Reward), a novel framework that unifies offline teacher guidance and online reinforcement learning through a specialized reward modeling lens. OGER employs multi-teacher collaborative training and constructs an auxiliary exploration reward that leverages both offline trajectories and the model's own entropy to incentivize autonomous exploration. Extensive experiments across mathematical and general reasoning benchmarks demonstrate that OGER consistently outperforms competitive baselines, achieving substantial gains in mathematical reasoning while maintaining robust generalization to out-of-domain tasks. We provide a comprehensive analysis of training dynamics and conduct detailed ablation studies to validate the effectiveness of our entropy-aware reward modulation. Our code is available at https://github.com/ecoli-hit/OGER.git.

State-of-the-art reasoning models utilize long chain-of-thought (CoT) to solve increasingly complex problems using more test-time computation. In this work, we explore a long CoT setting where the model makes up to K successive attempts at solving a problem, in which each attempt is allowed to build on earlier ones after the model receives a hard verifier feedback. This motivates RL methods that can harness per-attempt rewards by carefully weighting individual attempts. We study optimizing the Verification@K reward (the model succeeds by the K-th attempt) and show that naively weighing the attempts by their pass/fail results in biased gradients. We introduce Calibrated Attempt-Level (CAL) GRPO by devising a weighing strategy to obtain unbiased gradients while maintaining small variance. Our theory reveals how incorporating per-attempt rewards influence the training and the eventual Verification@K performance. Experiments, baselines, and ablations on synthetic and real data corroborate our theory and the benefits of CAL-GRPO over vanilla GRPO as well as naive weighting.

Reinforcement learning (RL) often struggles in real-world tasks with high-dimensional state spaces and long horizons, where sparse or fixed rewards severely slow down exploration and cause agents to get trapped in local optima. This paper presents a fuzzy logic based reward shaping method that integrates human intuition into RL reward design. By encoding expert knowledge into adaptive and interpreable terms, fuzzy rules promote stable learning and reduce sensitivity to hyperparameters. The proposed method leverages these properties to adapt reward contributions based on the agent state, enabling smoother transitions between fast motion and precise control in challenging navigation tasks. Extensive simulation results on autonomous drone racing benchmarks show stable learning behavior and consistent task performance across scenarios of increasing difficulty. The proposed method achieves faster convergence and reduced performance variability across training seeds in more challenging environments, with success rates improving by up to approximately 5 percent compared to non fuzzy reward formulations.

Existing robotic foundation policies are trained primarily via large-scale imitation learning. While such models demonstrate strong capabilities, they often struggle with long-horizon tasks due to distribution shift and error accumulation. While reinforcement learning (RL) can finetune these models, it cannot work well across diverse tasks without manual reward engineering. We propose VLLR, a dense reward framework combining (1) an extrinsic reward from Large Language Models (LLMs) and Vision-Language Models (VLMs) for task progress recognition, and (2) an intrinsic reward based on policy self-certainty. VLLR uses LLMs to decompose tasks into verifiable subtasks and then VLMs to estimate progress to initialize the value function for a brief warm-up phase, avoiding prohibitive inference cost during full training; and self-certainty provides per-step intrinsic guidance throughout PPO finetuning. Ablation studies reveal complementary benefits: VLM-based value initialization primarily improves task completion efficiency, while self-certainty primarily enhances success rates, particularly on out-of-distribution tasks. On the CHORES benchmark covering mobile manipulation and navigation, VLLR achieves up to 56% absolute success rate gains over the pretrained policy, up to 5% gains over state-of-the-art RL finetuning methods on in-distribution tasks, and up to $10\%$ gains on out-of-distribution tasks, all without manual reward engineering. Additional visualizations can be found in https://silongyong.github.io/vllr_project_page/

Reinforcement Learning (RL) has shown strong potential for improving robotic manipulation policies, yet its practical use remains bottlenecked by the difficulty of specifying reward functions that are both semantically meaningful and reusable across tasks. In this paper, we propose Large Reward Models (LRMs), a framework that adapts foundation VLMs into frame-level reward generators for robot policy refinement. We specialize a state-of-the-art VLM on a multi-source dataset spanning real-world robot trajectories, human-object interactions, and simulated manipulation environments. Unlike prior approaches that mainly evaluate trajectories post-hoc, LRMs expose multiple reward interfaces from visual observations: progress estimation, task completion, and temporal contrastive comparison. Starting from an imitation-learned policy, we use these VLM-derived rewards to guide PPO refinement on held-out long-horizon manipulation tasks. Our experiments show that LRM progress rewards provide the strongest non-privileged online refinement signal, improving the IL baseline and narrowing the gap to privileged environment rewards. We further deploy progress rewards for progress-weighted behavioral cloning on four real-world manipulation tasks spanning two robot platforms, improving over SFT on all four tasks. These results suggest that modality-specific specialization of foundation VLMs can provide practical visual reward signals for both simulated policy refinement and physical robot self-improvement without hand-coded task rewards.

Reward machines (RMs) inform reinforcement learning agents about the reward structure of the environment, enabling support for non-Markovian tasks and improving sample efficiency. However, learning with RMs is ill-suited for long-horizon problems where subtasks can be completed in any order. In such cases, the amount of information to learn increases exponentially with the number of unordered subtasks. We address this issue by introducing three generalisations of RMs: (1) Numeric RMs allow users to express complex tasks in a compact form. (2) In agenda RMs, states are associated with an agenda that tracks the remaining subtasks to complete. (3) Coupled RMs have coupled states associated with each subtask in the agenda. In addition, we introduce QCoRM, a new task-decomposition Q-learning-based algorithm that leverages coupled RMs and preserves global optimality guarantees in tabular settings. Our experiments across four domains -- featuring both discrete and continuous action and state spaces -- demonstrate that QCoRM scales better than baseline algorithms for long-horizon problems with unordered subtasks.

Existing robot locomotion learning techniques rely heavily on the offline selection of proper reward weighting gains and cannot guarantee constraint satisfaction (i.e., constraint violation) during training. Thus, this work aims to address both issues by proposing Reward-Oriented Gains via Embodied Regulation (ROGER), which adapts reward-weighting gains online based on penalties received throughout the embodied interaction process. The ratio between the positive reward (primary reward) and negative reward (penalty) gains is automatically reduced as the learning approaches the constraint thresholds to avoid violation. Conversely, the ratio is increased when learning is in safe states to prioritize performance. With a 60-kg quadruped robot, ROGER achieved near-zero constraint violation throughout multiple learning trials. It also achieved up to 50% more primary reward than the equivalent state-of-the-art techniques. In MuJoCo continuous locomotion benchmarks, including a single-leg hopper, ROGER exhibited comparable or up to 100% higher performance and 60% less torque usage and orientation deviation compared to those trained with the default reward function. Finally, real-world locomotion learning of a physical quadruped robot was achieved from scratch within one hour without any falls. Therefore, this work contributes to constraint-satisfying real-world continual robot locomotion learning and simplifies reward weighting gain tuning, potentially facilitating the development of physical robots and those that learn in the real world.