Reward Shaping

Reinforcement learning significantly enhances LLM capabilities but suffers from a critical issue: length inflation, where models adopt verbosity or inefficient reasoning to maximize rewards. Prior approaches struggle to address this challenge in a general and lossless manner, primarily because additive penalties introduce a compensatory effect that creates optimization shortcuts, while heuristic gating strategies lack generality beyond binary feedback. To bridge this gap, we present Group Relative Reward Rescaling (GR$^3$), which reframes length control as a multiplicative rescaling paradigm, effectively establishing a generalized, continuous, and reward-dependent gating mechanism. To further ensure lossless optimization, we incorporate group-relative regularization and advantage-aware calibration, which dynamically adapt length budgets to instance difficulty and preserve the advantage signal of high-quality trajectories. Empirically, across both RLHF and RLVR settings, GR$^3$~maintains training dynamics and downstream performance comparable to standard GRPO while significantly mitigating length inflation, outperforming state-of-the-art length-regularized baselines.

We introduce Reward-Zero, a general-purpose implicit reward mechanism that transforms natural-language task descriptions into dense, semantically grounded progress signals for reinforcement learning (RL). Reward-Zero serves as a simple yet sophisticated universal reward function that leverages language embeddings for efficient RL training. By comparing the embedding of a task specification with embeddings derived from an agent's interaction experience, Reward-Zero produces a continuous, semantically aligned sense-of-completion signal. This reward supplements sparse or delayed environmental feedback without requiring task-specific engineering. When integrated into standard RL frameworks, it accelerates exploration, stabilizes training, and enhances generalization across diverse tasks. Empirically, agents trained with Reward-Zero converge faster and achieve higher final success rates than conventional methods such as PPO with common reward-shaping baselines, successfully solving tasks that hand-designed rewards could not in some complex tasks. In addition, we develop a mini benchmark for the evaluation of completion sense during task execution via language embeddings. These results highlight the promise of language-driven implicit reward functions as a practical path toward more sample-efficient, generalizable, and scalable RL for embodied agents. Code will be released after peer review.

We present ProgAgent, a continual reinforcement learning (CRL) agent that unifies progress-aware reward learning with a high-throughput, JAX-native system architecture. Lifelong robotic learning grapples with catastrophic forgetting and the high cost of reward specification. ProgAgent tackles these by deriving dense, shaped rewards from unlabeled expert videos through a perceptual model that estimates task progress across initial, current, and goal observations. We theoretically interpret this as a learned state-potential function, delivering robust guidance in line with expert behaviors. To maintain stability amid online exploration - where novel, out-of-distribution states arise - we incorporate an adversarial push-back refinement that regularizes the reward model, curbing overconfident predictions on non-expert trajectories and countering distribution shift. By embedding this reward mechanism into a JIT-compiled loop, ProgAgent supports massively parallel rollouts and fully differentiable updates, rendering a sophisticated unified objective feasible: it merges PPO with coreset replay and synaptic intelligence for an enhanced stability-plasticity balance. Evaluations on ContinualBench and Meta-World benchmarks highlight ProgAgent's advantages: it markedly reduces forgetting, boosts learning speed, and outperforms key baselines in visual reward learning (e.g., Rank2Reward, TCN) and continual learning (e.g., Coreset, SI) - surpassing even an idealized perfect memory agent. Real-robot trials further validate its ability to acquire complex manipulation skills from noisy, few-shot human demonstrations.

While Large Language Models (LLMs) form the cornerstone of sequential decision-making agent development, they have inherent limitations in high-frequency decision tasks. Existing research mainly focuses on discrete embodied decision scenarios with low-frequency and significant semantic differences in state space (e.g., household planning). These methods suffer from limited performance in high-frequency decision-making tasks, since high-precision numerical state information in such tasks undergoes frequent updates with minimal fluctuations, and exhibiting policy misalignment between the learned sub-tasks and composite tasks. To address these issues, this paper proposes Normalized Action Reward guided Consistency Policy Optimization (NAR-CP). 1) Our method first acquires predefined dense rewards from environmental feedback of candidate actions via reward functions, then completes reward shaping through normalization, and theoretically verifies action reward normalization does not impair optimal policy. 2) To reduce policy misalignment in composite tasks, we use LLMs to infer sub-observation candidate actions and generate joint policies, with consistency loss ensuring precise alignment between global semantic policies and sub-semantic policies. Experiments on UAV pursuit, a typical high-frequency task, show our method delivers superior performance on independent and composite tasks with excellent generalization to unseen tasks.

Reinforcement Learning with Verifiable Rewards (RLVR) has significantly advanced the reasoning capabilities of Large Language Models (LLMs) by optimizing them against factual outcomes. However, this paradigm falters in long-context scenarios, as its reliance on internal parametric knowledge is ill-suited for tasks requiring contextual grounding--the ability to find and reason over externally provided information. We identify a key reason for this failure: a reward based solely on the final answer is too sparse to effectively guide the model for identifying relevant evidence. We formally prove that the outcome-only reward leads to significant vanishing gradients for the context grounding process, rendering learning intractable. To overcome this bottleneck, we introduce LongRLVR to augment the sparse answer reward with a dense and verifiable context reward. This auxiliary signal directly incentivizes the model for selecting the correct grounding information, providing a robust learning gradient that solves the underlying optimization challenge. We validate our method on challenging long-context benchmarks using Qwen and LLaMA models. LongRLVR consistently and significantly outperforms the standard RLVR across all models and benchmarks, e.g., boosting a 14B model's scores on RULER-QA from 73.17 to 88.90 and on LongBench v2 from 39.8 to 46.5. Our work demonstrates that explicitly rewarding the grounding process is a critical and effective strategy for unlocking the full reasoning potential of LLMs in long-context applications. Our code is available at https://github.com/real-absolute-AI/LongRLVR.

Reward design is of great importance for solving complex tasks with reinforcement learning. Recent studies have explored using image-text similarity produced by vision-language models (VLMs) to augment rewards of a task with visual feedback. A common practice linearly adds VLM scores to task or success rewards without explicit shaping, potentially altering the optimal policy. Moreover, such approaches, often relying on single static images, struggle with tasks whose desired behavior involves complex, dynamic motions spanning multiple visually different states. Furthermore, single viewpoints can occlude critical aspects of an agent's behavior. To address these issues, this paper presents Multi-View Video Reward Shaping (MVR), a framework that models the relevance of states regarding the target task using videos captured from multiple viewpoints. MVR leverages video-text similarity from a frozen pre-trained VLM to learn a state relevance function that mitigates the bias towards specific static poses inherent in image-based methods. Additionally, we introduce a state-dependent reward shaping formulation that integrates task-specific rewards and VLM-based guidance, automatically reducing the influence of VLM guidance once the desired motion pattern is achieved. We confirm the efficacy of the proposed framework with extensive experiments on challenging humanoid locomotion tasks from HumanoidBench and manipulation tasks from MetaWorld, verifying the design choices through ablation studies.

Long-horizon precision manipulation in laboratory automation, such as pipette tip attachment and liquid transfer, requires policies that respect strict procedural logic while operating in continuous, high-dimensional state spaces. However, existing approaches struggle with reward sparsity, multi-stage structural constraints, and noisy or imperfect demonstrations, leading to inefficient exploration and unstable convergence. We propose a Keyframe-Guided Reward Generation Framework that automatically extracts kinematics-aware keyframes from demonstrations, generates stage-wise targets via a diffusion-based predictor in latent space, and constructs a geometric progress-based reward to guide online reinforcement learning. The framework integrates multi-view visual encoding, latent similarity-based progress tracking, and human-in-the-loop reinforcement fine-tuning on a Vision-Language-Action backbone to align policy optimization with the intrinsic stepwise logic of biological protocols. Across four real-world laboratory tasks, including high-precision pipette attachment and dynamic liquid transfer, our method achieves an average success rate of 82% after 40--60 minutes of online fine-tuning. Compared with HG-DAgger (42%) and Hil-ConRFT (47%), our approach demonstrates the effectiveness of structured keyframe-guided rewards in overcoming exploration bottlenecks and providing a scalable solution for high-precision, long-horizon robotic laboratory automation.

Real-world user requests to LLM agents are often underspecified. Agents must interact to acquire missing information and make correct downstream decisions. However, current multi-turn GRPO-based methods often rely on trajectory-level reward computation, which leads to credit assignment problems and insufficient advantage signals within rollout groups. A feasible approach is to identify valuable interaction turns at a fine granularity to drive more targeted learning. To address this, we introduce InfoPO (Information-Driven Policy Optimization), which frames multi-turn interaction as a process of active uncertainty reduction and computes an information-gain reward that credits turns whose feedback measurably changes the agent's subsequent action distribution compared to a masked-feedback counterfactual. It then combines this signal with task outcomes via an adaptive variance-gated fusion to identify information importance while maintaining task-oriented goal direction. Across diverse tasks, including intent clarification, collaborative coding, and tool-augmented decision making, InfoPO consistently outperforms prompting and multi-turn RL baselines. It also demonstrates robustness under user simulator shifts and generalizes effectively to environment-interactive tasks. Overall, InfoPO provides a principled and scalable mechanism for optimizing complex agent-user collaboration. Code is available at https://github.com/kfq20/InfoPO.

Retrieval-Augmented Generation (RAG) enhances large language models (LLMs) by incorporating external knowledge, yet traditional single-round retrieval struggles with complex multi-step reasoning. Agentic RAG addresses this by enabling LLMs to dynamically decide when and what to retrieve, but current RL-based training methods suffer from sparse outcome rewards that discard intermediate signals and low sample efficiency where failed samples contribute nothing. We propose Search-P1, a framework that introduces path-centric reward shaping for agentic RAG training, comprising two key components: (1) Path-Centric Reward, which evaluates the structural quality of reasoning trajectories through order-agnostic step coverage and soft scoring that extracts learning signals even from failed samples, and (2) Dual-Track Path Scoring with offline-generated reference planners that assesses paths from both self-consistency and reference-alignment perspectives. Experiments on multiple QA benchmarks demonstrate that Search-P1 achieves significant improvements over Search-R1 and other strong baselines, with an average accuracy gain of 7.7 points.

The reasoning process of Large Language Models (LLMs) is often plagued by hallucinations and missing facts in question-answering tasks. A promising solution is to ground LLMs' answers in verifiable knowledge sources, such as Knowledge Graphs (KGs). Prevailing KG-enhanced methods typically constrained LLM reasoning either by enforcing rules during generation or by imitating paths from a fixed set of demonstrations. However, they naturally confined the reasoning patterns of LLMs within the scope of prior experience or fine-tuning data, limiting their generalizability to out-of-distribution graph reasoning problems. To tackle this problem, in this paper, we propose Explore-on-Graph (EoG), a novel framework that encourages LLMs to autonomously explore a more diverse reasoning space on KGs. To incentivize exploration and discovery of novel reasoning paths, we propose to introduce reinforcement learning during training, whose reward is the correctness of the reasoning paths' final answers. To enhance the efficiency and meaningfulness of the exploration, we propose to incorporate path information as additional reward signals to refine the exploration process and reduce futile efforts. Extensive experiments on five KGQA benchmark datasets demonstrate that, to the best of our knowledge, our method achieves state-of-the-art performance, outperforming not only open-source but also even closed-source LLMs.

Following the success of Group Relative Policy Optimization (GRPO) in foundation LLMs, an increasing number of works have sought to adapt GRPO to Visual Large Language Models (VLLMs) for visual perception tasks (e.g., detection and segmentation). However, much of this line of research rests on a long-standing yet unexamined assumption: training paradigms developed for language reasoning can be transferred seamlessly to visual perception. Our experiments show that this assumption is not valid, revealing intrinsic differences between reasoning-oriented and perception-oriented settings. Using reasoning segmentation as a representative case, we surface two overlooked factors: (i) the need for a broader output space, and (ii) the importance of fine-grained, stable rewards. Building on these observations, we propose Dr.~Seg, a simple, plug-and-play GRPO-based framework consisting of a Look-to-Confirm mechanism and a Distribution-Ranked Reward module, requiring no architectural modifications and integrating seamlessly with existing GRPO-based VLLMs. Extensive experiments demonstrate that Dr.~Seg improves performance in complex visual scenarios while maintaining strong generalization. Code, models, and datasets are available at https://github.com/eVI-group-SCU/Dr-Seg.

Standard vision-language-action (VLA) models rely on fitting statistical data priors, limiting their robust understanding of underlying physical dynamics. Reinforcement learning enhances physical grounding through exploration yet typically relies on external reward signals that remain isolated from the agent's internal states. World action models have emerged as a promising paradigm that integrates imagination and control to enable predictive planning. However, they rely on implicit context modeling, lacking explicit mechanisms for self-improvement. To solve these problems, we propose Self-Correcting VLA (SC-VLA), which achieve self-improvement by intrinsically guiding action refinement through sparse imagination. We first design sparse world imagination by integrating auxiliary predictive heads to forecast current task progress and future trajectory trends, thereby constraining the policy to encode short-term physical evolution. Then we introduce the online action refinement module to reshape progress-dependent dense rewards, adjusting trajectory orientation based on the predicted sparse future states. Evaluations on challenging robot manipulation tasks from simulation benchmarks and real-world settings demonstrate that SC-VLA achieve state-of-the-art performance, yielding the highest task throughput with 16% fewer steps and a 9% higher success rate than the best-performing baselines, alongside a 14% gain in real-world experiments. Code is available at https://github.com/Kisaragi0/SC-VLA.

Large language models (LLMs) are increasingly deployed as multi-step decision-making agents, where effective reward design is essential for guiding learning. Although recent work explores various forms of reward shaping and step-level credit assignment, a key signal remains largely overlooked: the intrinsic uncertainty of LLMs. Uncertainty reflects model confidence, reveals where exploration is needed, and offers valuable learning cues even in failed trajectories. We introduce SELAUR: Self Evolving LLM Agent via Uncertainty-aware Rewards, a reinforcement learning framework that incorporates uncertainty directly into the reward design. SELAUR integrates entropy-, least-confidence-, and margin-based metrics into a combined token-level uncertainty estimate, providing dense confidence-aligned supervision, and employs a failure-aware reward reshaping mechanism that injects these uncertainty signals into step- and trajectory-level rewards to improve exploration efficiency and learning stability. Experiments on two benchmarks, ALFWorld and WebShop, show that our method consistently improves success rates over strong baselines. Ablation studies further demonstrate how uncertainty signals enhance exploration and robustness.

Large Language Models (LLMs) consistently benefit from scaled Chain-of-Thought (CoT) reasoning, but also suffer from heavy computational overhead. To address this issue, efficient reasoning aims to incentivize short yet accurate thinking trajectories, typically through reward shaping with Reinforcement Learning (RL). In this paper, we systematically investigate the mechanics of efficient reasoning for LLMs. For comprehensive evaluation, we advocate for more fine-grained metrics, including length distribution conditioned on correctness and performance across a wide spectrum of token budgets ranging from 2k to 32k. First, we reveal that the training process follows a two-stage paradigm: length adaptation and reasoning refinement. Through extensive experiments (about 0.2 million GPU hours) in a unified protocol, we deconstruct training prompts and rollouts, reward shaping, and optimization strategies. A central finding is to maintain a sufficient density of positive reward signals and avoid the short-is-correct trap. Moreover, the learned length bias generalizes across domains and difficulty levels. We distill these findings into valuable insights and practical guidelines, and validate them across the Qwen3 models ranging from 0.6B to 30B, demonstrating the robustness and generalization. Weights are available at https://wutaiqiang.github.io/project/Art

Vision-Language-Action (VLA) models have demonstrated significant potential for generalist robotic policies; however, they struggle to generalize to long-horizon complex tasks in novel real-world domains due to distribution shifts and the scarcity of high-quality demonstrations. Although reinforcement learning (RL) offers a promising avenue for policy improvement, applying it to real-world VLA fine-tuning faces challenges regarding exploration efficiency, training stability, and sample cost. To address these issues, we propose IG-RFT, a novel Interaction-Guided Reinforced Fine-Tuning system designed for flow-based VLA models. Firstly, to facilitate effective policy optimization, we introduce Interaction-Guided Advantage Weighted Regression (IG-AWR), an RL algorithm that dynamically modulates exploration intensity based on the robot's interaction status. Furthermore, to address the limitations of sparse or task-specific rewards, we design a novel hybrid dense reward function that integrates the trajectory-level reward and the subtask-level reward. Finally, we construct a three-stage RL system comprising SFT, Offline RL, and Human-in-the-Loop RL for fine-tuning VLA models. Extensive real-world experiments on four challenging long-horizon tasks demonstrate that IG-RFT achieves an average success rate of 85.0%, significantly outperforming SFT (18.8%) and standard Offline RL baselines (40.0%). Ablation studies confirm the critical contributions of IG-AWR and hybrid reward shaping. In summary, our work establishes and validates a novel reinforced fine-tuning system for VLA models in real-world robotic manipulation.

This paper introduces a novel, open-source MARL simulation framework for studying implicit cooperation in LEMs, modeled as a decentralized partially observable Markov decision process and implemented as a Gymnasium environment for MARL. Our framework features a modular market platform with plug-and-play clearing mechanisms, physically constrained agent models (including battery storage), a realistic grid network, and a comprehensive analytics suite to evaluate emergent coordination. The main contribution is a novel method to foster implicit cooperation, where agents' observations and rewards are enhanced with system-level key performance indicators to enable them to independently learn strategies that benefit the entire system and aim for collectively beneficial outcomes without explicit communication. Through representative case studies (available in a dedicated GitHub repository in https://github.com/salazarna/marlem, we show the framework's ability to analyze how different market configurations (such as varying storage deployment) impact system performance. This illustrates its potential to facilitate emergent coordination, improve market efficiency, and strengthen grid stability. The proposed simulation framework is a flexible, extensible, and reproducible tool for researchers and practitioners to design, test, and validate strategies for future intelligent, decentralized energy systems.

Autonomous control of double-Ackermann-steering robots is essential in agricultural applications, where robots must execute precise and complex maneuvers within a limited space. Classical methods, such as the Timed Elastic Band (TEB) planner, can address this problem, but they rely on parameter tuning, making them highly sensitive to changes in robot configuration or environment and impractical to deploy without constant recalibration. At the same time, end-to-end deep reinforcement learning (DRL) methods often fail due to unsuitable reward functions for non-holonomic constraints, resulting in sub-optimal policies and poor generalization. To address these challenges, this paper presents ManeuverNet, a DRL framework tailored for double-Ackermann systems, combining Soft Actor-Critic with CrossQ. Furthermore, ManeuverNet introduces four specifically designed reward functions to support maneuver learning. Unlike prior work, ManeuverNet does not depend on expert data or handcrafted guidance. We extensively evaluate ManeuverNet against both state-of-the-art DRL baselines and the TEB planner. Experimental results demonstrate that our framework substantially improves maneuverability and success rates, achieving more than a 40% gain over DRL baselines. Moreover, ManeuverNet effectively mitigates the strong parameter sensitivity observed in the TEB planner. In real-world trials, ManeuverNet achieved up to a 90% increase in maneuvering trajectory efficiency, highlighting its robustness and practical applicability.

Reinforcement Learning (RL) has emerged as a key driver for LLM reasoning. This capability is equally pivotal in long-context scenarios---such as long-dialogue understanding and structured data analysis---where the challenge extends beyond consuming tokens to performing rigorous deduction. While existing efforts focus on data synthesis or architectural changes, recent work points out that relying solely on sparse, outcome-only rewards yields limited gains, as such coarse signals are often insufficient to effectively guide the complex long-context reasoning. To address this, we propose LongR, a unified framework that enhances long-context performance by integrating a dynamic ``Think-and-Read'' mechanism---which interleaves reasoning with document consultation---with a contextual density reward based on relative information gain to quantify the utility of the relevant documents, without evidence annotations. Empirically, LongR achieves a gain of over 9% on LongBench v2 and consistent improvements on RULER and InfiniteBench, demonstrating robust efficiency in navigating extensive contexts. Furthermore, LongR consistently enhances performance across diverse RL algorithms (e.g., DAPO, GSPO). Finally, we conduct in-depth analyses to investigate the impact of reasoning chain length on efficiency and the model's robustness against distractors.

Query-product relevance prediction is fundamental to e-commerce search and has become even more critical in the era of AI-powered shopping, where semantic understanding and complex reasoning directly shape the user experience and business conversion. Large Language Models (LLMs) enable generative, reasoning-based approaches, typically aligned via supervised fine-tuning (SFT) or preference optimization methods like Direct Preference Optimization (DPO). However, the increasing complexity of business rules and user queries exposes the inability of existing methods to endow models with robust reasoning capacity for long-tail and challenging cases. Efforts to address this via reinforcement learning strategies like Group Relative Policy Optimization (GRPO) often suffer from sparse terminal rewards, offering insufficient guidance for multi-step reasoning and slowing convergence. To address these challenges, we propose TaoSR-AGRL, an Adaptive Guided Reinforcement Learning framework for LLM-based relevance prediction in Taobao Search Relevance. TaoSR-AGRL introduces two key innovations: (1) Rule-aware Reward Shaping, which decomposes the final relevance judgment into dense, structured rewards aligned with domain-specific relevance criteria; and (2) Adaptive Guided Replay, which identifies low-accuracy rollouts during training and injects targeted ground-truth guidance to steer the policy away from stagnant, rule-violating reasoning patterns toward compliant trajectories. TaoSR-AGRL was evaluated on large-scale real-world datasets and through online side-by-side human evaluations on Taobao Search. It consistently outperforms DPO and standard GRPO baselines in offline experiments, improving relevance accuracy, rule adherence, and training stability. The model trained with TaoSR-AGRL has been successfully deployed in the main search scenario on Taobao, serving hundreds of millions of users.

Autonomous exploration of cluttered environments requires efficient exploration strategies that guarantee safety against potential collisions with unknown random obstacles. This paper presents a novel approach combining a graph neural network-based exploration greedy policy with a safety shield to ensure safe navigation goal selection. The network is trained using reinforcement learning and the proximal policy optimization algorithm to maximize exploration efficiency while reducing the safety shield interventions. However, if the policy selects an infeasible action, the safety shield intervenes to choose the best feasible alternative, ensuring system consistency. Moreover, this paper proposes a reward function that includes a potential field based on the agent's proximity to unexplored regions and the expected information gain from reaching them. Overall, the approach investigated in this paper merges the benefits of the adaptability of reinforcement learning-driven exploration policies and the guarantee ensured by explicit safety mechanisms. Extensive evaluations in simulated environments demonstrate that the approach enables efficient and safe exploration in cluttered environments.

Monte Carlo tree search is attractive for robotic manipulation because it can improve action selection through simulation without requiring a fully differentiable policy. In uncertain domains, however, sparse terminal rewards and noisy transitions can make shallow search brittle: many candidate branches remain indistinguishable until late rollouts, and small simulation budgets amplify this ambiguity. This paper presents Reward-Centered ReST-MCTS, a decision-making framework that decomposes intermediate feedback into rule, heuristic, optional neural, and value-estimation channels, centers the resulting process signal against matched task contexts, and uses it to bias or repair search while preserving terminal-task evaluation. The primary evidence is intentionally tiered. Local tasks and matched ManiSkill diagnostics isolate reward-center mechanisms and ablations; matched option-level ManiSkill sweeps test robustness under primitive failure, observation noise, and initial-pose shifts while not claiming standard benchmark superiority; and an official same-backbone OpenVLA-OFT/LIBERO bridge tests bounded VLA action repair. The OpenVLA-OFT clean reproduction reaches 10/10 LIBERO-Spatial successes both with and without RCRM-Guard. A single-suite same-backbone action-channel stress artifact over ten paired LIBERO-Spatial action-channel stress episodes records 0/10 unguarded successes and 9/10 guarded successes. Additional observation-noise, language-perturbation, and visual-distractor probes are reported as coverage and negative-result context rather than superiority evidence. The resulting claim is bounded: Reward-Centered ReST-MCTS is an inspectable test-time verifier for same-backbone high-uncertainty manipulation, not a replacement VLA policy or a broad standard-benchmark superiority claim.

Deep neural networks based on reinforcement learning (RL) for solving combinatorial optimization (CO) problems are developing rapidly and have shown a tendency to approach or even outperform traditional solvers. However, existing methods overlook an important distinction: CO problems differ from other traditional problems in that they focus solely on the optimal solution provided by the model within a specific length of time, rather than considering the overall quality of all solutions generated by the model. In this paper, we propose Leader Reward and apply it during two different training phases of the Policy Optimization with Multiple Optima (POMO) model to enhance the model's ability to generate optimal solutions. This approach is applicable to a variety of CO problems, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Flexible Flow Shop Problem (FFSP), but also works well with other POMO-based models or inference phase's strategies. We demonstrate that Leader Reward greatly improves the quality of the optimal solutions generated by the model. Specifically, we reduce the POMO's gap to the optimum by more than 100 times on TSP100 with almost no additional computational overhead.