Lifelong Learning with Behavior Consolidation for Vehicle Routing
Authors: Jiyuan Pei
Yi Mei
Jialin Liu
Mengjie Zhang
Xin Yao
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Combinatorial Optimization
Vechicle Routing
Lifelong Learning for Optimization
Primary area: optimization
TL;DR: We propose LLR-BC, an effective lifelong learning framework for neural VRP solver to learn from sequentailly arising tasks with different scales and distributions.

Abstract: Recent neural solvers have demonstrated promising performance in learning to solve routing problems. However, existing studies are primarily based on one-off training on one or a set of predefined problem distributions and scales, i.e., tasks. 
When a new task arises, they typically rely on either zero-shot generalization, which may be poor due to the discrepancies between the new task and the training task(s), or fine-tuning the pretrained solver on the new task, which possibly leads to catastrophic forgetting of knowledge acquired from previous tasks. This paper explores a novel lifelong learning paradigm for neural VRP solvers, where multiple tasks with diverse distributions and scales arise sequentially over time. Solvers are required to effectively and efficiently learn to solve new tasks while maintaining their performance on previously learned tasks. Consequently, a novel framework called Lifelong Learning Router with Behavior Consolidation (LLR-BC) is proposed. LLR-BC consolidates prior knowledge effectively by aligning behaviors of the solver trained on a new task with the buffered ones in a decision-seeking way. To encourage more focus on crucial experiences, LLR-BC assigns greater consolidated weights to decisions with lower confidence. Extensive experiments on capacitated vehicle routing problems and traveling salesman problems demonstrate LLR-BC’s effectiveness in training high-performance neural solvers in a lifelong learning setting, addressing the catastrophic forgetting issue, maintaining their plasticity, and improving zero-shot generalization ability.

Published as a conference paper at ICLR 2026
LIFELONG LEARNING WITH BEHAVIOR CONSOLIDA- TION FOR VEHICLE ROUTING
Jiyuan Pei', Yi Mei', Jialin Mengjie Zhang', Xin Yao?
Center of Data Science and Artificial Intelligence & School of Engineering and Computer Science, Victoria University of Wellington, Wellington, New Zealand School of Data Science, Lingnan University, Hong Kong SAR, China jiyuan Eayuan-piiee pwwpppw.w....e.... ac nz, nz, {yi, mei, meng aengiiie jie. 0ecs, www. ac. n2, (jialin. liu, xinyao] @1n edu. .kk
ABSTRACT
Recent neural solvers have demonstrated promising performance in learning solve routing problems. However, existing studies primarily based on one-off training on one or a set of predefined problem distributions and scales, i.e., tasks. When new task arises, they typically rely either zero-shot generalization, which may be poor due to the discrepancies between the new task and the train- ing ts, or fine-tuning the pretrained solver on the task, which possibly leads to catastrophic forgetting of knowledge acquired from previous tasks. This paper explores novel lifelong learning paradigm for neural VRP solvers, where multiple tasks with diverse distributions and scales arise sequentially over time. Solvers are required effectively and efficiently learn to solve new tasks while maintaining their performance previously learned tasks. Consequently, novel framework called Lifelong Learning Router with Behavior Consolidation (LLR- BC) is proposed. LLR-BC consolidates prior knowledge effectively by aligning behaviors of the solver trained on new task with the buffered ones in decision- seeking way. To encourage more focus on crucial experiences, LLR-BC assigns greater consolidated weights to decisions with lower confidence. Extensive exper- iments on capacitated vehicle routing problems and traveling salesman problems demonstrate LLR-BC's effectiveness in training high- high-performance neural solvers in lifelong learning setting, addressing the catastrophic forgetting issue, main- taining their plasticity, and improving zero-shot generalization ability.
INTRODUCTION
Neural solvers have gained significant atten- tion for their remarkable performance on ve- hicle routing problems (VRPs) (Vinyals et al.) 2015 Bello et al.] 2017 Bengio et al. 2021 Bogyrbayeva et al. 2024). By training on multiple specific problem distributions and scales, i.e., tasks, simultaneously. neu- ral solvers can learn construct high-quality solutions or improve existing solutions effi- ciently for problems drawn from the same dis- tribution and of the same scale as the training tasks (Kool et al., 2019 Kwon al., 2020 Lu et al. 2020; TTang & Yao 2024). However, in real-world scenarios, unpredictable problems with new distributions
Tg arises S
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Aeg. objective (total disanne)) on
72
Tg
Time
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Figure 1: Conceptual demonstration of catas- trophic forgetting while fine-tuning on sequential, new tasks with different distributions and scales.
scales often arise over time. For example, a daily delivery service provider receives different numbers of orders with new patterns of delivery locations and demands due unpredictable consumer behavior, business expansion, and new shopping trends. In such scenarios, is challenging to develop a universal solver that can perform effectively across all possible problem distributions and scales after one-off

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training (Liu et al., 2023). Some recent studies (Jiang et al. 2022 Bi et al. 2022 Gao et al. 2024
Wang et al.) 2024b; Xiao et al. 2023) address unpredictable new problems by enhancing the gen- eralization ability of trained solvers. However, the generalization would have a boundary, and on new problems that differ substantially from the training ones, these methods still require additional learning to enhance problem-solving performance (Zhou et al., 2023). A common approach is to
fine-tune the trained solver on new task (Bengio et al.] 2021; Zhou et al. [2023); nonetheless, it leads to catastrophic forgetting of knowledge acquired from previous tasks (Khetarpal et al. (022). Solver's performance on problem instances of the previous tasks will decrease significantly (cf. Fig- ure T. as the model parameters will be overwritten during training on new task (Wang et al.) 2024c). These issues highlight the need for neural solvers capable of lifelong learning, enabling them to continuously acquire knowledge from unforeseen tasks while retaining generic knowledge and their performance on previous tasks.
Existing studies on lifelong learning for neural VRP solvers (Li et al,, 2024, Feng et al., 2025) are largely confined to highly specific scenario settings: tasks solely differing in scale or distance metric, the task order is fixed and known, and the generation of new problem instances is controllable. Furthermore, they are mainly dedicated to the solver's 's performance at the end of the lifelong learning process after being trained on all training tasks. As lifelong learning is an ongoing process (possibly without any end), the solver's performance each time point of the lifelong learning process is also important. Their proposed methods rely on actively generating and training on new problem instances from previous tasks, making them inapplicable when the generation of new instances is uncontrollable, as is the case studied in this work.
In contrast, this work targets more realistic and general lifelong learning scenarios, simultane- ously accommodating changes in both scale and distribution across tasks, without assuming any prior knowledge of the task order or any control over problem instance generation. We propose a novel method for lifelong learning of neural VRP solvers, called Lifelong Learning Router with Behavior Consolidation (I.LR-BC). L.LR-C focuses on addressing the catastrophic forgetting issue that arises during training across new tasks arising sequentially, which is commonly overlooked in prior neural VRP solver research. It buffers and revisits experiences from previous tasks to retain acquired knowledge. To efficiently utilize the limited experience memory, LLR-BC weights expe- riences based on decision confidence, emphasizing more crucial ones. Given that small changes in action probability distribution, especially under low confidence, can alter decisions and drastically affect the constructed route, LLR-BC B-C minimizes reverse
divergence to consolidate behavior in decísion-seeking way. Extensive experiments capacitated vehiele routing problems (CVRPs) and traveling salesman problems (TSPs), equipped with a comprehensive metrics set to evaluate the solver during the ongoing lifelong learning process, under various task sequences and base neural solvers show that LLR-BC effectively mitigates forgetting, maintains plasticity, and improves zero-shot generalization by accumulating transferable knowledge over time.
Our major contributions include: i) we introduce LLR-BC, general lifelong learning framework enabling neural VRP solvers learn from different tasks sequentially and address catastrophic for- getting effectively; ii) we propose Confidence-awann Experience Weighting (CaEW) to prioritize crucial experiences, as a module LLR-BC for enhancing the effectiveness of experiences utiliza- tion; iii) we propose Decision-seeking Behavior Consolidation (DsBC), which works in LLR-BC to preserve past behaviors by minimizing the discrepancy between buffered and current solver behav- iors stored states, with an emphasis replicating past decisions; iv) the effectiveness of LLR-BC in is validated through extensive experiments, supported by detailed analysis and discussion.
2 RELATED WORK
2.1 NEURAL VRP SOLVERS
Machine learning, particularly deep reinforcement learning (DRL., enables neural solvers to learn VRP-solving strategies directly from problem-solving experience, eliminating the need for hand- crafted heuristics (Bengio et al. 2021). Existing methods are categorized as neural construction or
neural improvement approaches (Bi et al. 2022 Kong et al. 2024). Neural construction methods
build solutions from scratch by sequentially selecting the next node in an autoregressive fashion.
Notable examples such as POMO (Kwon et al.) 2020) and DualOpt (Zhou et al. (025) generate high-quality solutions within seconds, rivaling strong heuristics like LKH3 (Helsgaun) 2017). In
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contrast, neural improvement methods leam enhance existing solutions, either by configuring (Wu iet al. 2022 Ma et al. 2023) or selecting (Lu et al. 2020 Pei et al. 2025b; Chen et al. 2025 Guo
et al. 2025 among predefined improvement heuristics. Although improvement methods have the
potential to further improve solution quality by increasing the number of iterations, this also leads to higher computational cost. This paper focuses on lifelong learning for neural construction methods, due to their favorable balance between effectiveness and efficiency.
2.2 CROSS-DISTRIBUTTON CROSS-SCALE GENERALIZATION
Early studies of neural VRP solvers typically focus on solving single task, i.e., problems with fixed scale and one given distribution (e.g.. uniform) (Kool et al., 2019: Kwon et al. 2020). re- sulting in poor generalization across distributions and scales (Bi et al. 2022] Xiao al.) 2025).
which is often encountered in real-world applications. To address this issue, recent studies focus on improving doss--iistibtttonn or cross-scale performance. DROP (Jiang et al. 2022) improves cross- distribution performance by training across multiple distributions and optimizing for worst-case task performance. AMDKD (Bi et al. 2022) distills knowledge from multiple task-specific teacher mod- els into student model, achieving improvement in zero-shot generalization. ELG (Gao et al. 2024) separates local and global policies and, despite training only on uniform problems of scale 100, gen- eralizes well across other tasks. Xiao al. (2025) proposes two network modules to handle different scales and distributions, respectively, improving generalization. INViT (Fang et al. 2024) involves multiple encoders and decoders with multiple views to improve generalization. Omni (Zhou et al., 2023) applies meta-learning to learn a solver that can quickly adapt to a new task.
However, existing studies largely rely one-off training on predefined task set. New tasks typically handled via zero-shot generalization or independent fine-tuning, without leveraging knowl- edge accumulated across new tasks, and with limited consideration of catastrophic forgetting. Con- sequently, their effectiveness in sequential task learning remains limited. In realistic settings, tasks arrive continuously and previously learned tasks can inform future adaptation, motivating the devel- opment of lifelong neural VRP solvers that incrementally learn from a stream of diverse tasks.
2.3 LIFELONG LEARNING
Lifelong learning, also referred to as continual learning (Wang et al. 2024c), involves training a model to sequentially learn multiple tasks, progressively improving its capabilities in a manner analogous to human learning throughout life [Thrun] 1998; Khetarpal et al. 2022). Each task is characterized by distinct data distribution (Wang al. 2024c). A key challenge lies in balancing plasticity, i.e., the ability to acquire knowledge of new tasks, with stability, i.e., the capacity of retaining knowledge from past tasks (Parisi al. 2019). Insufficient stability leads to catastrophic forgetting, where the performance on earlier tasks degrades after training on new tasks.
Several strategies have been proposed mitigate catastrophic forgetting. Training task-specific model components is an example (Wu et al. 2021 Ebrahimi et al.) 2020). but this increases storage costs and poses challenges in selecting the correct model when the task of the test data is unknown. Regularization of parameter updating to preserve prior knowledge is also widely used (Kirkpatrick) al. 2017 Zenke et al. 2017). However, this can hinder learning on new tasks. A further widely adopted method is experience replay, where past experiences on previously learned tasks are stored and revisited during training on new task (Isele & Cosgun| 2018 Buzzega et al., 2020). Nonethe- less, identifying and retaining the most crucial experiences to improve memory efficiency and min- imize interference with learning new tasks often requires problem-specific designs
To the best of our knowledge, only two existing works have studied lifelong learning for neural VRP solver (Li et al. 2024| Feng et al. 2025). However, their studies focused on rather restricted scenarios: (i) tasks differ only in scale (Li et al. 2024 Feng et al., 2025) or distance metric (Feng jt al. 2025), (ii) task order is fixed and known under strong assumptions (e.g.. task scale gradually increases), and (iii) problem instance generation is controllable
that new instances of previously learned tasks can always be actively generated, making these methods inapplicable to scenarios with uncontrollable instance streams. Studies of more generic and practical scenarios desired.
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Time
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Figure 2: LLR-BC in the lifelong learning scenario where new tasks with different distributions and scales sequentially arise over time. Tr: the task time t. τạΣ solver with model parameters 0. B: experience buffer. J: problem instances. { problem solving trajectories. &: experiences. wgt weights of experiences. a: action, i.e., node visit. s: state. var. variance. P: behavior.
LIFELONG LEARNING ROUTER WITH BEHAVIOR CONSOLIDATION
We consider the scenarios where problem instance generation is uncontrollable and only new in- stances of the current task are generated for training, while future tasks and their order are unknown, reflecting practical cases. We assume that all the tasks are equally important, and each has a fixed and known training budget. The main aim is train solver capable of generating high-quality solutions across all the learned tasks arising over time. Therefore, after training on each new task, the solver is tested on all tasks learned so far.
We propose JLR-BC, inspired by the experience replay paradigm (Isele & Cosgun 2018 Buzzega al. | It consolidates knowledge learned from previous tasks by revisiting the old experiences obtained from these tasks. To increase the effectiveness of utilizing limited old experiences, we in- troduce two ingredients I.I-BC: CaEW and DsBC. Figure 2 illlstraats L.RR-BCss full workflow. LLR-BC maintains a fixed-size experience memory and trains the solver learned from the previous tasks on each newly arrived task. Following standard practice in neural VRP solvers, LLR-BG BC trains on each task in epochs. In each epoch, LLR-BC iteratively samples and solves batch of problem instances I from the current task to obtain a set experience trajectories {r). )- A DRL algorithm that maximizes the reward gain based on {r} is applied to update the solver model. Simultancously, a subset of experiences E is sampled from the memory for consolidation of previously learned tasks. CaEW assigns higher weights to more crucial sampled experiences, then DsBC guides model up- dates by minimizing weighted behavioral divergence between the current model and the behavior buffer of the sampled experiences. This joint process mitigates catastrophic forgetting while pre- serving adaptability. Notably, LLR-BC is a general and readily integrable framework, independent of specific model architectures or RL. algorithms, and applicable to many existing neural solvers, such as POMO (Kwon et al., 2020) and INVIT (Fang et al. (024). Core ingredients of LLR-BC. are described below. More details about LLR-BC can be found in Appendix
3.1 EXPERIENCE REPLAY FOR ROUTING
LLR-BC employs reservoir sampling (Vitter 1985) to maintain a fixed-size buffer B. New experi- ences added directly if the memory is not full. Otherwise, each incoming experience replaces a randomly selected buffered experience with probability N' where [B] the buffer size and N denotes the total number of experiences tried to add so far. Reservoir sampling ensures that all expe- riences are equally likely to be buffered while the total number of obtained experiences increasing.
While existing methods (Li et al. 2024 Feng et al. 2025) take one whole problem instance of pre- vious tasks as an experience, we consider experiences at finer granularity. We follow the commonly studied Markov decision process (MDP) formulation of constructive neural VRP solver (Kwon et al. 2020 Zhou et al. 2023 Bogyrbayeva et al., 2024), where each state corresponds to a partial solution composed of visited nodes, and an action
select one node to visit next. In LLR-BC each expe- rience e = (s, P) consists of a state and the solver's 's behavior P on the state s. state s represents the current partial solution for solving instance. The behavior P is the probability distribution of
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Decisions made with low confidence could be more susceptible to change during model up- dates (Farahmand 2011 AAhmed et al.] 2019). Therefore, LLR-BC assigns higher consolidation weights such experiences, encouraging the solver to preserve behaviors in erucial decision points. Published as a conference paper at ICLR 2026
selecting each node as the next visit. The detailed solver design. e.g., the encoding of problems and solutions, follows the underlying base neural solver used. We use the probability distribution over all nodes rather than the single selected node to represent the solver's behavior, it captures richer information about the learned routing strategy (Rusu et al.
2016 Buzzega et al. (020).
To consolidate high-quality behaviors, LLR-BC buffers experiences only during the final epoch of each task, when the solver is expected to be well-trained on the task. At each model update step, random set of experiences & is sampled from buffer B for behavior consolidation. A detailed discussion of the additional memory usage from the buffer can be found in Appendix A.5
3.2 CONFIDENCE-AWARI EXPERIENCE WEIGHTING (CAEW)
While experiences sampled uniformly from the buffer to ensure broad coverage of seen states, their importance for addressing forgetting can vary significantly. Constructive VRP solvers select nodes in sequential manner, hence each decision influences future decisions while some crucial ones have a cumulative and amplified impact on the solution quality (Sun et al.) 2024). Therefore, identifying and prioritizing crucial states and behaviors, i.e., crucial experience, is essential for effective learning and consolidation.
Confidence is measured by the variance of the action probability distribution, where lower variance indicates lower confidence and leads to greater emphasis during consolidation (Spielberg & Azaria 2019 Balasuntharam et al. 2002). Confidence-awart weight of an experience
(s, P) is normal- valinus (PP) where var(P) is the variance of and |P| is the size of P, i..., the
ized by w(e) number of actions. varmax (PP)) = IP) TP7 denotes the maximum possible variance for a distribution
varlP)
with the same number of candidates as P. Then, given a set of sampled experiences E, we rescale
w(e) weights such that the sum of weight equals to 1, i.e., wz(e) Ee'ce wlery Ve E E. Appendix A.4 provides further discussion about the design of CaEW.
3.3 DECISION-SFEKING BEHAVIOR CONSOLIDATION (DsBC)
Existing methods (Li et al. 2024 Feng et al. 2025) replay experiences by generating, solving, and learning instances of previous tasks, ignoring the instructive information contained in historical be- haviors. In contrast, by minimizing the weighted sum of the difference between the current model's behaviors and the buffered behavior buffered states, LLR-BC encourages the consolidation of prior behavior patterns, thereby preserving knowledge and mitigating catastrophic forgetting.
The Kullhack-Leite divergence (KLD) D where P and Q are the teacher and leaner probability distribution respectively. is widely adopted for knowledge distillation of neural VRP solvers [Bi et al., 2022]] Li et al. 2024 Zheng et al. 2025). However, recent studies (Kaplanis let al. 2019 Wang et al. 2024a Qi et al. 2025) have shown that using reverse KLD (RKLD) DRKL (PIQQ) = Dr (Q] [I) (detailed explanation is in Appendix ). could lead to better learn- ing performance. While minimizing KLD pushes the learner distribution to spread probability mass across all modes of the given teacher distribution, minimizing RKLD leads to mode-seeking, where the learner tends to concentrate
the teacher's pig actions, while still maintaining relatively good alignment with the overall distribution (Chan et al. 2022 Wu et al., 2025). During problem-solving of constructive neural VRP solvers, the action (node) with the highest probability is typically chosen to extend the route. To address forgetting, that could manifest in significantly longer routes on previously seen tasks, it is crucial to preserve these highest-probabille probability decisions. Therefore, instead of KLD, LLR-BC employs the RKLD to measure the behavioral difference in decision-sed seeiion way. To consolidate sampled behaviors, LLR-BC minimizes the following behav- ior consolidation loss term, considering the RKLD (Wang et al. 2024a Qi et al. 2025
Po(a)
(B, (2,8)=Em - Σ Pa(a) log
(1)
P(a)
eEc
asA
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where E = fe= (s) is the set of sampled experiences, A is the action space (i.e., available node set), P(a) is the probability corresponding to action a in the buffered behavior P and Po(a) = To(als) is the probability output by the current model To on buffered state s.
Finally, during each model updating in training on a new task, LLR-BC jointly maximizes the ex- pected reward on the new task and minimizes the RKLD between the current policy's behavior on the buffered states and the corresponding buffered behavior. The overall loss function is as follows:
[(e, (T),)) = (0, {T}) + a LBc(
(2)
where {r] is the set of newly obtained experience trajectories from the new task, CDRL(B, (r]) is the loss calculated by the adopted DRL algorithm based on {r} to maximize the reward gain, and a a hyperparameter to balance between consolidation of previous experiences and learning of the new task. Notably, LLR-BC is a generic framework and can be applied to different neural solver methods with different DRL, algorithms, in which LDRL Eq. equation can be implemented in different ways. Appendix A.5 detailedly discusses about computational cost of DsBC.
EXPERIMENTS
We conduct series of experiments to answer the following research questions:
Effectiveness on all learned tasks: Can LLR-BC effectively solve learned tasks after training on them sequentially in the lifelong learning setting?
Stability and plasticity: In lifelong learning process, how does LLR-BC perform in terms of stability and plasticity?
Zero-shot generalization ability: Can LLR-BC effectively acquire transferable knowl- edge across sequentially arising tasks to enhance zero-shot performance on an unseen task?
Hyperparameter sensitivity: How sensitive is LLR-BC to key hyperparameters, including the buffer size [B), number of sampled experiences [8). and the weight of the behavior consolidation term in the loss function?
Applicability: Can LLR-BC work effectively on different base neural solvers?
To answer the questions, we simulate multiple lifelong learning scenarios on CVRP and TSP, and compare LLR-BC against widely used and representative lifelong learning baselines across diverse
m Most experiments are conducted based on POMO (Kwon et al. 2020), given its concise design and broad applicability. To verify the applicability of LLR-BC, we further evaluate LLR- BC on two representative and state-of-the-ar constructive neural solvers designed for good cross-
distribution cross-scale performance, i.e., Omni (Zhou al. 2023 and INViT (Fang et al. 2024).
Dataset. Six distributions for sampling node coordinates are used: four proposed by Bossek al. (2019) and widely used in neural VRP solver studies (Jiang et al. 2022; Zhou et al. (023). i.c., Uniform (U), Gaussian Mixture (GM), Explosion (R), Compression (C), and two additionally de- signed ones for greater diversity: Grid (G) and Ring (R). For CVRP, node demand is also required to be sampled. While prior studies used uniformly random demand for all distributions (Kwon al.) 2020 Jiang et al., 2022 Zhou et al. 2023 Fang et al., 2024). we build six distinct demand distri- butions and assign them the above six distributions to better reflect the diversity of real-world problems (Liu et al. 2021). Following common practice (Kwon et al. 2020 Jiang et al., 2022), we consider three problem scales: 20, 50, and 100. Specifically, tasks U and correspond to scale 20, tasks G and E to scale 50, and tasks C and GM to scale 100, arbitrarily. We construct five task orders by randomly permuting the tasks. In addition, two classic and widely used benchmark datasets, TSPLIB (Reinelt 1991) and CVRPLIB (Uchoa et al.) (017)), are used for generalization ability evaluation. Appendix provides the full definition of tasks and orders.
Training and Test Settings. Following Kwon al. (2020), we train the solver each task over 200 epochs, and solve a batch of sampled problems in parallel to leverage GPU parallelism during training. Therefore, both experience buffering and sampling operate at the batch level rather than the single experience level. Hyperparameters of LLR-BC are set as follows: [B] = 1000, [g] = 16,
Our implementation is available in https///gituuhcco//Peiil.....

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and 100, where [B] and |8] are defined in units of experience batches. The buffer constitutes only about 0.01% of the total training experiences obtained across all tasks. For evaluation, each task has test set of 1,000 problem instances, which different from the training ones. For each trial, all methods are evaluated on the same test sets. Additional details are provided in Appendix
Compared Methods. The goal of this work is not to design a totally new solver that outperforms the state-of-the-art solver on each individual task or fixed set of tasks that can be learned simul- taneously, but to improve the solver's s ability to learn unpredictable tasks sequentially. We consider several existing methods for comparison. First, adapt the methods of (i) Li et al. (2024) (denoted Li (intra) and Li (inter)) and (ii) of Feng et al. (2025) (denoted Feng) to our scenarios and include them as baselines. Then, involve several representative and commonly adopted strategies in ral solver studies (Kwon et al. 2020 (Zhou et al. 2023): (iii) Restart, reinitialize the solver and train from scratch on each task; (iv) Fine-tuning, simply sequential training on each task. Additionally, we evaluate (v) EWC (Kirkpatrick et al. (2017). a widely used method in lifelong learning, especially in recent studies of optimization tasks (Manchanda & Ranu 2023 Pei et al., 2025a). We also adapt and compare with (vi) LiBOG (Pei et 2005aa, a recent method originally proposed for black-box optimization, within the VRP tasks. Appendix C.I
provides more details.
Evaluation Metries. Based on standard practice in lifelong learning research (Chaudhry al. 2018 Wang et al., 2024c), we build the following evaluation metrics. Let di denote the test per- formance, I.c., average objective value (tour length) over all the test instances, of the solver on task T, after training on the first i tasks sequentially (including task T.). As different tasks have different
scales for the objective value, the objective values are normalized by dij = ت..-=ب where dž is smallest (best) test performance achieved on task T3 by all the solvers obtained by all the methods in all lifelong learning scenarios. Based on this notation, after learning & tasks, calculate:
dj
Average Performance (AP): the current average performance on tasks learnt so far, i.e.,
AP = Ei_ide, de.s.
Average Forgetting (AF): the average performance decrease on previously learned tasks after learning all of them sequentially, i.e., AF ト Σ max(0, dr.; - -d,.).
Average Max Forgetting (AMF): the average of maximal forgetting of each task during the lifelong learning, i.c., AMF - Ea=l maxt_041 max(0, d,,- dz,:).
Average Plasticity (API): the average performance solver achieves on each new task after training on it, i.c., API = 1 E
Average zero-shot Generalization (AG): the average performance on each newly arising task before training on it, i.e., AG = k Σk=1 d,
Smaller values of the above metrics indicate better performance. Unless otherwise specified, all reported values of the five metrics in tables are scaled by 10
4.1 PERFORMANCE IN SOLVING SEEN TASKS
The experimental results for CVRP and TSP are summarized in Table with k 6 for all metrics. More detailed results with the absolute solution distance and optimality gap can be found in Ap- pendices D.I D.2 Across all task orders, LLR-BC achieves average AP values of 0.0042 (CVRP) and 0.0034 (TSP), whereas all other methods exceed 0.023 (CVRP) and 0.014 (TSP), respectively, indicating that LLR-BC obtains solutions of problems from learned tasks with substantially better quality. Moreover, LLR-BC exhibits greater robustness to task order variation. Compared methods show large fluctuations (larger Std.) in AP across orders, while LLR-BC consistently maintains low AP values with lower standard deviations. An additional comparison of LLR-BC with methods with non-life oonliielong settings, including POMO in multi-task training settings and INVIT its original training setting (Fang et al., 2024), further demonstrates LLR-BC's superior performance (details can be found in Appendix[D.3 D.3). These results demonstrate that LLR- LLR-BC effectively learns from se- quentially arising tasks and retains the ability to solve all encountered tasks with high effectiveness.
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Table 1: The mean (std.) of metrics over the five task orders.
CVRP
TSP
Mathod
AP
AMF
AP
AMF
API
AG
Li (inter)
(6.8
0.0 (0.0)
(..0)
33.6 (7.7)
(10.5)
56.5 (38.3)
0.4 (0.5)
0.5 (0.5)
61.7 (45.
(5.6)
Li (intra)
(2.1)
0.0 (0.0)
0.1 (0.1)
39.3 (1.7)
48.2 (4.8)
(7.2)
0.2 (0.3)
0.5 (0.4)
62.8 (12.5)
(2.47
Feng
22.0)
3.2 (2.0)
4.5 (1.7)
24.2 (0.7)
39.1 (8.6)
24.1 (3.2)
(1.4)
3.8 (1.9)
(1.6)
45.5 (23.2)
Restart
60.5 (29.3)
41.3 (26.
52.0
9-1
(4.7)
31.7 (7.6)
50.5 (40.0)
(32.8)
(-1
72.4
Hine-tuning
(9.2)
19.9 (3.6)
28.1 (5.0)
3.8 (0.8)
42.1 (6.7)
14.8 (2.3)
28.9 (2.8) (2.8)
3.5 (7.4)
(17.7)
EWC
(9.2)
19.5 (5.4)
25.5 (4.3)
6.9 (1.1)
39.8 (4.2)
18.6 (4.4)
(4.5
(1.7)
(18.3
LIBOG
31.3 (11.9)
19.7 (5.8)
25.1 (5.8)
7.2 (1.0)
40.8 (3.9)
19.2 (1.6)
17.2 (4.5)
22.8 (5.5)
5.8 (1.7)
51.7 (14.8)
LLR-BC
(1.0)
0.7 (0.5)
0.8 (0.4)
3.5 (0.8)
26.7 (4.5)
(1.3)
0.8 (0.3)
(0.4)
2.8 (1.6)
41.1 (21.0)
11.00
5:75
F 10.75
5.50
16
6.50
12.0
11.5
5.50
7.0
3.875
3.825
R
6.00
2,000
5.75
1.975
200
400
800
1000
1200
200
400
600
800
1000
1200
Training Epoch
Training Epoch
LLR-BC
(inter)
Restart
EWC
LLR-BC
(inter)
Feng
(intra)
Fine-tuning
LiBOG
Feng
Li (intra)
Fine Fine-tuning
LiBOG
(a) CVRP
(b) TSP
Figure 3: Forgetting curve of task order 1, measured by average solution distance (vertical axis). Epochs 0-200 (first task) omitted as no forgetting occurs. Notably, some methods obtain too large solution distances and exceed the vertical range.
4.2 STABILITY AND PLASTICITY
We analyze the stability by comparing the AF and AMF values over all five task orders, as shown in Table [ Li (inter) and Li (intra) (Li et al. 2024), perform better in mitigating forgetting than LLR-BC. This is because half of their training resources (in terms of epochs)
the new task are used to learn purely from instances of previous tasks. Although they address forgetting well, they do not learn much from new tasks and perform poorly in terms of AP and API. LLR-BC achieves substantially lower AF and AMF values than the rest of the compared methods, demonstrating its strong stability for effectively mitigating catastrophic forgetting. Figure the average solution distance on seen tasks during lifelong learning of task order 1. At the start of each task, most baselines exhibit sharp performance drop on previously learned tasks. Li (inter) and Li (intra) learn new tasks slowly. Feng suffers from low stability in maintaining learned knowledge. In contrast, LLR-BC mitigates forgetting and maintains consistently lower distances with greater stability.
LLR-BC also achieves the best API values among all methods, indicating its best plasticity. Notably, Restart produces a single API value with no std., as is independent task order and is executed only once, rather than once per order. Figure presents the solver's test performance on the current task throughout lifelong learning. As the number of encountered tasks increases, EWC, as well as LiBOG, gradually loses plasticity due to the cumulative regularization contrast, LLR-BC operates at the behavioral level, enabling the model to explore the parameter space more freely while preserving learned behaviors. Model can discover parameter values that yield good decisions on both new and past tasks. With the guidance from previous experiences, LLR-BC demonstrates even better performance on the new task than fine-tuning.
model parameters. In
Additionally, the rankings of methods differ on the two forgetting metrics, i.c., AF and AMF, For example, fine-tuning achieves lower (better) AF than EWC but yields higher (worse) AME.
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Test Performance
Test Performance
0.02
0.02
0.01
0.01
0.00
0.00
200
400
600
800
1000
1200
200
400
600
800
1000
1200
Training Epoch
Training Epoch
-LLR-BC
L (inter) Restart
EWC
LLR-BC
Li (inter) Restart
EEWC
Feng
Li (intra) Fine-tuning
LIBOG
FFeng
Li (intra)
Fine-tuning - LiBOG
(a) CVRP
(b) TSP
Figure 4: Test performance on the current task during lifelong learning on task order 1.
new task can sometimes improve performance on earlier, similar tasks. Backward transfer also evident in Figure This further underscores the importance of LLR-BC's ability to maintain strong plasticity. Similar patterns are observed across other task orders, as shown in Appendix D In summary, LLR- BC consistently outperforms all baselines in both stability and plasticity.
This discrepancy arises because AF also accounts for backward transfer, i.e., training
4.3 ZERO-SHIOT GENERALIZATION
Beyond performance on learned tasks, we also expect neural solvers acquire general knowledge from randomly ordered tasks, thereby improving their performance on unseen tasks. AG values for all methods are reported in Table ! LLR-BC outperforms all baselines in terms of AG, demonstrat- ing its strong ability to zero-shot generalize to a new task. We further test the solver after learning from task order on TSPLIB and CVRPLIB, with the maximum scale as 1001. As shown by the re- sults in Table LLR-BC performs the best on both benchmarks. Appendix presenss the detailed values. Despite the uncontrollable and unpredictable nature of task order, LLR-BC demonstrates a strong generalization ability by capturing general knowledge across tasks.
Table 2: Mcan (Std.) test performance on benchmark instances.
Benchmark
Restart
Fine-tuning
EWC
LiBOG
CVRPLIB TSPLIB
43.68 (27.37)
8.54 (10.71)
11.86 (11.59)
18.05 (10.86)
99.76 (61.06)
38.16 (93.36)
27.80 (59.58)
27.96 (32.29)
Benchmark
Li (inter)
Li (intra)
Feng
LLR-BC
CVRPLIB TSPLIB
39.31 (23.11)
46.80 (26.96)
27.38 (18.53)
7.88 (8.75)
122.60
58.46 (26.55)
75.15 (30.83)
18.08 (16.98)
4.4 ABLATION STUDY
We conduct ablation studies (denoted -nEW and -KLD) on task order to evaluate the contribution of our key components, i.e. CaEW and DsBC. As shown in Table removing either key component leads to performance degradation across multiple metrics on both CVRP and TSP, confirming the effectiveness and contribution of both modules. Furthermore, we conduct additional ablation studies to better investigate the characteristics of LLR- including: (i) variant that buffers experiences at every epoch instead of only the last epoch of each task (denoted -EE), (ii) variants that use entropy and top-2 margin instead of variance quantify decision confidence (denoted -Ent and -T2M), (iii) a variant that rescales the reservoir sampling probabilities so that the buffered experiences are more uniformly distributed across tasks (denoted -Res), and (iv) variant that buffers all steps for solve an instance as one experience, instead of just one step, denoted as -IB. The results show that both buffering only the final epoch and using unscaled reservoir sampling contribute notably to the performance of LLR-BC. Replacing the confidence measure has only a minor impact, indicating that LLR-BO BC not sensitive to the specific choice of confidence measurement. More details about ablation studies can be found in Appendix
4.5 HYPERPARAMETER SENSITIVITY ANALYSIS
We varied the values of key hyperparameters, i.e., [Σ], [B] and and evaluated the performance of LLR-BC across all metrics under each setting, as shown in Table 3 -nEW and -KLD denote the version without CaEW (using equal weights) and the version using KLD, respectively. -n(v)

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Published as a conference paper at ICLR 2026
denotes the version with set Detailed discussion be found in Appendix D.5 In summary, varying the hyperparameters leads to only minor performance fluctuations across evaluation metrics, compared with the performance gap between LLR-BC and the baselines. This suggests that LLR- BC robust and not overly sensitive to its hyperparameter settings.
Table Metric values LLR-BC in different settings, on task order with k 6.
Method
CVRP
TSP
Method
CVRP
TSP
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
(E(4) (8)
5.7 0.8
0.8
4.8
22.7
1.2
1.2
1.7
21.6
-o(10)
13.2
5.1
5.5
4.2
34.7
1.3
0.4
0.6
1.2
20.5
5.8
1.0
1.1
4.2
22.8
2.0 0.9
1.1
1.5
21.6
-0(50)
6.6
1.4
1.6
4.2
24.7
2.0
0.7
0.9
1.2
22.2
B [250]
6.4
1.9
1.9
4.3
25.3
1.6 0.7
1.1
1.1
20.5
-a(500)
5.1
0.2
0.3
4.9
22.5
3.5
1.3
1.3
3.2
21.2
B
(500)
5.8
1.3
1.3
4.2
29.8
2.2 0.6
0.9
1.7
22.2
-a(1000)
5.9
0.3
0.4
5.8
23.6
4.9
1.4
1.4
4.6
22.3
nFW
5.2 0.8
0.8
4.2
24.5
1.8
0.9
0.8
1.5
22.3
-KID
5.5
0.7
0.7
4.3
23.1
1.9
0.9
0.9
1.5
22.3
EE
7.8 3.1
3.1
3.9
24.1
2.6
2.1
2,4
1.6
22.7
-Ent
4.8
0.5
0.7
4.0
22.7
2.1
0.9
1.0
1.6
21.9
Ress
4.9 0.9
0.9
3.7
23.9
2.0
1.0
1.0
1.3
22.3
-T2M
4.8
1.0
1.2
3.4
22.8
1.7
0.9
1.1
1.0
21.6
Default
4.9 0.6
0.7
4.3
23.5
1.7
0.8
0.9
1.3
21.6
-1B
35.4
23.4
27.2
4.4
31.0
2.5
1.8
2.0
1.0
21.9
4.6
APPLICABILITY
We further implement LLR-BC on Omni Zhou al. (2023) and INVIT Fang al. (002)). and compare with fine-tuning on task order 1. As Table shows, LLR-BC outperforms fine-tuning generally, with similar pattern found POMO. More details of experiment settings, results, and discussions are presented in Appendix D.6 In summary, LLR-BC is effective on the focused lifelong leamingg scenario, without relying on specific characteristics of the base neural solver.
Table 4: Metric values with Omni or INVIT the base neural solver, k 6.
CVRP
TSP
Method
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
Fine-tuning (Omni)
34.7
19
22.1
16.9
64
52,9
13.6
18.8
14.4
56.7
LLR-BC (Omni)
16.5
2,9
4.6
19.9
55.2
12.9
0.9
1.4
11.6
34.7
(INViT)
28.6
9.7
9.7
21.6
35.2
14.5
0.6
0.6
17.1
11.1
LLR-BC (INVIT)
23,8
5.8
5.8
19.6
28.8
14.5
4.5
6.3
11.1
10.7
CONCLUSIONS
We propose LLR-BC, lifelong learning framework for neural VRP solvers, considering learn- ing tasks with varying problem scales and distributions that arrive sequentially. We introduce two core components: CaEW, which emphasizes low-confidence behaviors, and DsBC, which preserves learned behavior effectively in decision-secking way. Experiments across diverse orders of tasks with varying distributions and scales demonstrate that LLR-BC consistently outperforms baselines in terms of performance in solving learned tasks, resistance to catastrophic forgetting, learning ef- ficiency on new tasks, and zero-shot generalization. Ablation and sensitivity studies confirm the effectiveness of the core designs of LLR-BC and the framework's 's robustness to hyperparameter set- tings. LLR-BCC is both model- and RL algorithm-agnnsticc and can be integrated with various neural VRP solvers. Experiments on different base neural solvers verify LLR-BC's applicability.
Although studied focus on dross-distribution and cross-scale settings, LLR- -BC principle appli- cable other lifelong learning scenarios for neural VRP solvers, such tasks that differ in their distance matrices. Lifelong learning across different problem variants (c.g., TSP and CVRP) is also practically important research direction. which we leave for future work, although it may require task-specific model components (Drakulic et al.) 2025). LLR-BC could further reduce its reliance on task identity and adapt to scenarios with continuously evolving tasks by applying reservoir sampling during training on each instance, rather than only at the final epoch of each task. Despite its strong performance, LLR-BC has limitations in certain cases. For example, fixed | can be difficult to tune when task scales differ substantially, potentially leading to reduced plasticity on small-scale tasks, where old experiences dominate new ones batch, and reduced stability on large-scale tasks, where new experiences dominate old ones in a batch. Addressing these issues in future work could further broaden the applicability of LLR-BC.
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ACKNOWLEDGEMENT
This work was supported by the internal grants of the Lingnan University, Hong Kong SAR, China. It also supported in part by the New Zealand MBIE Endeavour Smart Ideas Grant under Contract RTVU2305 and MBIE SSIF Fund on Data Science Programme under contract RTVU1914. Xin Yao was supported by an internal grant from Lingnan University, Hong Kong SAR, China
REPRODUCIBILITY STATEMENT
All algorithmic details (ef. Section 3 and Appendix A), training protocols (cf. Section and Ap- pendix []. and evaluation metrics (cf. Section and Appendix C.2) are described in the main paper and further claborated in the Appendix. For empirical studies, we provide a detailed description of the datasets and instance generation pipeline (ef. Appendix B Hyperparameters and implementa-
tion details for all baselines are also reported Section and Appendix[C We also release our code and scripts for reproducing experiments, including instructions for running and data preparation. Together, these resources enable independent researchers to replicate our results and build upon our contributions.
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A FURTHER DETAILS OF LLR-BC
A.1 OVERALL PROCESS OF LLR-BC
For sequence of tasks, LLR-BC conducts lifelong learning with Algorithm L For each new task in the lifelong learning process, LLR-BC trains the solver model To and updates the experience buffer B as Algorithm demonstrates.. Only the model parameter and the buffer B are transferred between tasks. LLR-BC sequentially train any number of tasks, it does not rely any knowledge about the number or order of tasks.
Algorithm LLR-BC Lifelong Learning
1, Bn 2: Parameters: A, C, CDRL
1: Input: {T
3: Output: 0
4: for E (1, (1,.... ... K} do
5: 0,B < --RBBC One Task Learning (0,B, A,C, (DLL)
6: end for
7: return
Algorithm LLR-BC BC One Task Learning
1: Input: 0, B, A, C, CDrL
2: Parameters: ox |E]
3: Output: 0,
4: for E [I,...,,A] A} do
5: I generate a batch of C' problems from T
6: {r] Solve I with Tg
7:
loss + (6, {r})
8:
if not empty then
9:
+ -uniformly sample
{we(e)}osE + {1 (we (e))ee) + (Erre we(er) JeEE
varrur (P) Jez (x,))
van (ar(P)
(e)ee)
11:
12:
loss loss (0,
13:
end if
Optimize(O,, loss)
15: if i = A then
B reservoir sampling BB end if
17:
end for
19:
return 0,
A.2 CONFIDENCE CONSTRUCTIVE SOLVERS
Figure demonstrates a CVRP example where the low-confidence decision changes when training the solver on a new task, For the solver trained on U, tested on a problem of task U, it generates solution with some low-confidence decisions, e.g., the 5th action. Then, after fine-tuning on task E for 10 epochs, we test it again the same problem of task U, getting new solution (the total distance is larger). On the node corresponding to the previous 5th decision, the new solution differs from the previous Many decisions after are also different, but it could be due to the previous differences. It illustrates that low-confidence decisions could be more likely to drift during training on task.
The 1 11th actions also changed, where the new solver has low confidence.
A.3 REVERSE KLD
The Kullback-Leii divergence (KLD) measures the difference from one learner probability dis- tribution Q target/teacher distribution P and is widely adopted for knowledge distillation of
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Qulorroo Depsi
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Quttoeses Depe:
Top2
0.6
kop3
Figure 5: Example of decision drift in node with low confidence. The number on node indicates its order in the solution. Left: the generated solution problem of task Uby given solver. Right: the top three values action probability from the solver corresponding to the generated solution. Upper: solver trained U task. Lower: fine-tuning the solver task E for 10 epochs after training it on task
neural VRP solvers
(Bi al. 2022 Zheng al. 2025), computed as:
DkL(PI1Q) = Σ P(i) log P(i) Q(1)
(3)
where P(i) and Q(i) are the probability of candidate i of the target/teacher distribution P and of another distribution Q. KLD is asymmetric, i.e., D (PIl)) + DKl(Q P)), although both attain their minimum when = Q. By switching Q and KLD convert to RKLD, i...
Q(1) log Q(1) P(1)
DRkl(PIIQ) = DKL EQ()
When the teacher's action probability P(i) is close to zero, the KLD for action becomes very small. a result, using KLD as a loss term places little emphasis on the mismatch between P(i) and the learner's 's probability (2) in such cases. This leads overly mild penalties when the learner assigns high probability to actions that the teacher has effectively ruled out, i.c., learned incorrect ac- tions. Consequently, the learner tends to spread probability mass across all actions to cover possible
modes (Chan et al. 2022 Wu et al. 2025
In contrast, RKLD places greater emphasis penalizing mismatches when the learner's 's proba- bility Q(t) high. This makes RKLD more effective at discouraging the learner from assigning high probability those learned incorrect actions, thereby encouraging focus the most probable teacher actions. As a result, RKLD promotes mode-seeking behavior (Chan et al. 2022 Wu et al. 2025).
In LLR-BC, the teacher P is the buffered behavior, and the learner Q is the behavior of the current learning solver on the corresponding buffered state. By minimizing DRKL (PI())))) Q), LLR-BC better preserves the learned good decisions of previously learned tasks.
(PI
A.4 CONFIDENCE-AWANL WEIGHTING
low-confidence decision of the solver can arise two situations: (i) there is a single best action, but the model has not fully learned the current state and therefore exhibits high uncertainty; or (ii) there are multiple equally good actions. Since we only buffer experiences at the final epoch of each task, when the model is already well trained that task, both types of low-confidence decisions deserve higher emphasis (larger weights): (i) high uncertainty indicates that the state is difficult to learn and thus crucial for learning the task [3,4]; and (ii) multiple equally good actions,
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which mostly lead to similarly good solutions, create plateaus in the solution space and make the optimization problem harder to solve.
CaEW is designed to assign smaller weights to steps where the model selects the estimated best action with higher probability (i.e., higher decision confidence). Besides variance, other probability- based confidence measures with similar properties could also be used. CaEW is not specialized for any particular confidence measure. We choose variance because is simple, straightforward, and effective.
A.5 COMPUTATIONAL. COST
Importantly, compared with fine-tuning the base neural solver, LLR-BC only increases computa- tional cost during training and does not incur extra overhead during inference.
Memory Compared with fine-tuning, the additional space complexity introduced by LLR-BC primarily arises from the experience buffer storing old experiences, with space complexity as O([B] * se). with s, denotes the space for one unit of experience. 8, could vary depending on the formulation of the solving process of the base neural solver. Under our implementation, with [B] = 1000, the experience buffer occupies only about 400 MB of memory when solving CVRP with POMO as the base neural solver. It is acceptable considering the superior performance of LLR-BC.
Run Time Typically, in a training batch of problems with scale N, fine-tuning constraints least N steps of model forward propagation to generate complete solutions. Compared with fine-tuning the base neural solver, LLR-BC introduces additional time cost from computing the new behaviors of the current model on |E] sampled old states, storing and sampling experiences from the buffer, and calculating the RKLD between new and old behaviors as the LBC. The first part requires [E] additional forward propagations, while the latter two involve only lightweight arithmetic operations, whose run cost is negligible. Therefore, the extra time cost introduced by LLR-BC per problem batch is O(E) * tt,) [g]+t,m. with t. denotes the complexity of forward propagation. With | (already outperforming the compared methods) and an additional computational overhead of less than 20%, 8%, and 4%, respectively, compared with fine-tuning. Under our implementation, for all experiments, LLR-BC introduces no more than 8 additional hours based on fine-tuning of training for the whole lifelong learning process. Given its superior performance, this additional training complexity is reasonable and acceptable trade-off.
20, 50, 100, ideally, LLR-BC incurs only
B TASK AND ORDER DETAILS
B. TASK SETTINGS
Four of the six tasks use the node coordinate distribution defined in Bossek et al. (2019), which is widely used in neural VRP solver studies (Zhou al., 2023 Bi et al. 2022). And we built two more distributions, i.c., Ring and Grid. We also assign different demand distributions to different tasks. The details of node coordinate distribution and demand distribution of each task are follows. Vehicle capacity is set to 30, 40. 50 for the tasks with scale 20, scale 50, and scale 100, following the setting in kwonn al. (2020). Notably, for CVRP, the coordinate of the depot is randomly sampled from [0, 1Ja, independent of the sampling of customers (nodes).
Task Uniform (U): Each customer/city node has coordinates (zt, y:) ~ (1, 1). Demand d; ~ 2111, 10), drawn uniformly for CVRP. Figure demonstrates problem examples of task U CVRP and TSP.
Task Gaussian Mixture (GM): 5 cluster centers {c=)5-z are first uniformly sampled from [0, 50 For each c2 center, 19 customer/city nodes are sampled. Each is drawn as (I;, ys) WC 1). Then all node coordinates are linearly mapped into [0. 1]2 with minmax normalization. Demand of each center is drawn uniformly, de, (111 10). The distance dist, of each node i to its center is calculated and minmax normalized to [0.1]. The demand node is set as 10-d däst, then round the demand. Figure 7 demonstrates problem examples of task GM on CVRP and TSP.
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20
24
2.6
5.4
0.2
02
5.0
3.0
0.6
(a) CVRP
(b) TSP
Figure 6: Problem instance sample of task U.
2.0
5.8
5.6
0.4
0.2
2.0 0.0
3.0
(a) CVRP
(b) TSP
Figure 7: Problem instance sample of task GM.
Task Explosion (E): First nodes are sampled with a uniform coordinate distribution. Then, point p (not a node) is uniformly sampled from [0, 1J2. For each node with distance to p smaller than 0.3, we move customer away from p with moving length 0.3 + 8,8 ~ Exz(( 40). All nodes' coordinates are clamped into [0. 1J2, finally. The demand of each node is sampled from N(5, 1) and then rounded and clamped to [0,10]. Figure Saaneeddemmmstattee problem examples of task E on CVRP and TSP.
2.8
2.0
nace
0.1
0.2
2.0
0.0
0.2
0.5
0.0
(a) CVRP
(b) TSP
Figure 8: Problem instance sample of task E.
Task Compression (C): First nodes are sampled with a uniform coordinate distribution. Then 2 points P1 and P2 are sampled uniformly from [0, 1]2, forming a line l. For each node with distance to the smaller than 0.3, new distance is sampled from (0, The node will be along the direction vertical to the line, so that its distance to equals the new distance. All nodes" coordinates clamped into (0, 1)2, finally. The demand of each node is sampled as 10 Ir, x N5,,, 1), and
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then rounded and clamped to [0,10]. Figure demonstrates problem examples of task C CVRP and TSP.
20
6.5
5.8
28
5.5
5.6
0.6
4.3
4.2
3.5
0.0
0.2
0.4
0.8
0.0
0.2
0.4
0.6
0.8
1.0
(a) CVRP
(b) TSP
Figure 9: Problem instance sample of task C.
Task Grid (G): A ratio p is uniformly sampled from [0.2.0.8]. Sample random number uni- formly from [0.1]. If < 0.5, 1. h = p. otherwise, w p, h = 1. Sample center rectangle e (not node) x U( 2:
2). Make a square grid in the rectangle with center as C, width as w and height as h, a - IV50
" ~ u(
grids in x axis direction, b = -[v 5u grids in
axis direction. Put one node in each grid until the number of customers meets 50 (If it cannot be evenly divided, leave the grids with the largest x-values in the row with the largest y-value empty). For each customer, we calculate its distance to the depot, and add a noisy (0, 1) to the distance. Then we map the distance to [1, 10), round it, and assign it as the demand of the customer. Figure I0 demonstrates problem examples of task G on CVRP and TSP.
3.0
0.8
0.5
0.4
0.2
200
5.0
0.0
6.4
(a) CVRP
(b) TSP
Figure 10: Problem instance sample of task G.
Task Ring (R): A ratio p is uniformly sampled from [0.2,0.8]. For each node, we sample an angle an (1, (T) and radius ra = rai + ra2, r0j ~ (0.33, 0.4), of the node is set as (0.5 ra cos(an), 0.5 + ra sin). p S 0.5, p ~ (0, 1).r + x p. otherwise y + * p. For each customer, we calculate its distance to the depot, and add noisy ~ (0, 2) the distance. Then we map the distance to [1, 10). round it, and assign it as the demand of the customer. Figure demonstrates problem examples of task R on CVRP and TSP.
(.44,rog, r02 ~ (000 0.053). O.053.
The coordinate
B.2 TASK ORDERS
We randomly shuffle the six tasks five times, getting 5 different task orders.
Order 1: E -> C- G - U GM.
Order 2: U GM + -U-G-E--R-G--
Order 3: E
G R
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1.2
10
0.5
0.5
0.1
0.2
0.2
0.3
0.3
0.4
5.6
(a) CVRP
(b) TSP
Figure 11: Problem instance sample of task R.
Order 4: G GM - - E C.
Order 5: G - C R - U- GM
E.
C EXPERIMENT SETTING DETAILS
C.I BASELINES
To adapt Li (inter), Li (intra), and Feng to our focused scenarios where the generation of new prob- lem instances is uncontrollable, we maintain one instance buffer for each seen task instead of gener- ating new instances for their experience replay, that each buffer contains the instances generated during learning the corresponding task. Buffers are updated with reservoir sampling. The buffer size of each task is 200, so that there is total of 1000 buffered instances that can be used for learning the 6th tasks, aligning with the buffer size of LLR-BC. We set a = 0.5 arbitrarily for Li (inter), Li (intra) and Feng, as we did not find a suggested value or method for setting the value of a. Feng contains a learnable model parameter B with fixed size n x n (n is the problem scale). We did not find the description about the way to handle B under scenarios where problem scale changes, so we removed B to make it applicable to our scenarios. To adapt LiBOG in VRP tasks, we directly embed its two key modules, i.e., the intra-task and inter-task consolidation terms, into POMO for lifelong learning. We set the weight of the consolidation term to 10 for EWC and LiBOG, and the weight of the intra-task consolidation term to 1e - 4 for LiBOG. For all runs, the solver learns the first task from an identical randomly generated model. As Restart learns each task independently under any task order without being affected by the order, we run Restart once on each task and calculate metrics under different task orders. Therefore, Restart's API owns not std. value. As in most cases, POMO can complete the learning on each single task for 200 epochs (Kwon et al. 2020). we let each method train for 200 epochs on each task. To ensure comparable training time across tasks of differ- ent problem scales, we vary the number of training problems sampled per epoch: 10,000 for tasks of scale 20, 4,000 for scale 50, and 2,000 for scale 100, with batch sizes of 64, 32, and 16, respectively. Following Kwon et al. (2020), during training, we sample batch of problem instances and run N parallel solving processes per instance, each corresponding to a different starting node, where N is the problem scale. This results in N2 parallel trajectories per batch, yielding N2 independent states and behaviors per experience batch. The size of the experience batch varies across tasks with dif- ferent problem scales. We consider this reasonable, larger-scale tasks are typically more complex and thus require larger experience batches to effectively retain learned knowledge. We experimented with buffering and sampling at the level of individual experiences rather than experience batches. However, when using GPU-based parallelization, operating at the batch level significantly reduces computation time without yielding significant performance differences.
C.2 EVALUATION METRICS
In designing our evaluation metrics, we follow common practices in lifelong learning (Chaudhry jet al. 2018 Wang et al. 2024c). However, when measuring forgetting, we use the test performance obtained immediately after training on task the reference, rather than the best performance
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before training on the kthh task. The latter may include effects of backward transfer, which is not the focus of our study. Notably, we report each metric at a given k value rather than averaging across all h, as averaging would overemphasize early tasks in forgetting metries and later tasks in AG, contradicting our assumption that all tasks are equally important.
C.3 ABLATION STUDY DETAILS
In the ablation version of the confidence measure, negative entropy of action probability distribution, and T margin, i.., the absolute difference between top-2 action probability, are used instead of variance. We use the maximal possible entropy to normalize entropy, similar the normalization of variance. Top-2 margin has a value range of [0.1], J, so it does not need normalization. All three measurements, including variance, intend to assign higher confidence values (thus lower weights) when the model selects the estimated best action with higher probability, as the consolidation process requires. Results show that, although T2M performs slightly worse in terms of forgetting (arguably because it focuses only on the top two actions and thus discards information about the others), the overall performance of the three measures very similar. As negative entropy is commonly used as a confidence measurement in recent works, we further calculate the coorelation between varaince and negative entropy. We compute both the variance and the negative entropy for all experiences stored in the buffer after rumning LLR-BC once under task order 1. The Pearson correlation coefficient between the two measurements is 0.972367 (where I indicates perfect positive correlation), showing that, in practice, variance tracks confidence during the lifelong learning process very well. This suggests that, while variance is a simple and effective choice, LLR-BC is not very sensitive to the specific confidence measure. Since LLR-BC is designed to assign higher weights to lower- confidence steps, other confidence measures with similar properties can also be used.
For instance-based buffering, conduct batch-level buffering, set the buffer size to 20, so that the expected buffered steps are more than the buffer size in the step-hased buffer (20 1133>> 1000). And (batch of) instances will be sampled from the buffer for each consolidation, i.c., & = 1. With the similar buffer size (in terms of buffered steps (state- behavior pairs) ). instance- based design reduces the mistaicedivveritty of the buffer, potentially lead- ing to inferior performance. The experimental results indicate that our state- based buffer design outperforms the instance-based buffer design, supporting its advantage.
6
Though reservoir sampling gives each experience equal probability of being kept in the buffer, since the learning processes of different tasks generate different numbers of experiences, the expe- riences in the buffer are not uniformly added from all tasks. For tasks with larger problem scale, more experiences from these tasks are buffered. For one run on task order with buffer size 1000, after learning all tasks, the numbers of experiences in the buffer from each task are: 1:1 C:255, G:141, U:78, R:77, GM:287. The ratio of the numbers of buffered experiences is highly corre- lated with the ratio of the problem scales. We suggest this is reasonable, as tasks with larger scales may be more difficult and need more experiences for consolidating their knowledge and addressing catastrophic forgetting during later learning. After training all tasks with the rescaling version -Res, the numbers of experiences in the buffer from each task are: E: 148, C:148, G:144, U:199, R:208, GM:153. Though the buffer is more balanced, it performs (cf. Table B very similarly (or even slightly worse than) the original LLR-BC, suggesting the effectiveness of our design.
C.4 OTHER DETAILS
Our experiments are conducted on GPU cluster utilizing single Nvidia A100 GPU per run. Each lifelong learning method implemented based on POMO (won et al., 2020), Omn Zhou et al. 2023 and INViT (Fang al. 2024) builds upon the official codes released by the original papers.
Based on POMO, each compared method conducts lifelong learning on a single task order once for performance evaluation, requiring approximately one and half days. For fairness, for each base neural solver, all methods and task orders begin lifelong learning from the same initial model with
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randomly initialized parameters. Training problem instances generated on-the-fly fly the start of each batch during training.
With 200 training epochs, one task contains 200 10000, * 4000, 200 2000 problem instances,
200+10000 200-4000 200+2000 batchs, and at lest 200+ =20 = 625000, 200+4000 50 = 1250000,
04
32
16
64
32
200+2000 * 100 2500000 experience batchs for scale 20, 50, 100 tasks, respectively. Therefore,
16
the total number of generated experiences after learning the six tasks is 8750000. As [B] 1000,
the buffer contains only 8750000 ~ 0.01% of total generated experiences.
1000
D DETAILED EXPERIMENT RESULTS
D.1 MAIN EXPERIMENTS ON CVRP
The detailed metric values in each task order are given in T Figures 12 and 13 demonstrate the forgetting curve and then learning curve on each current task of each method on task orders 3, 4, and 5. Across all orders, LLR-BC demonstrates the best performance. It aligns with the conclusion in the main text. Additionally, across orders, we found specific pattern. For fine-tuning, learning on GM C (both scale 100) will lead to significant forgetting of U R (both scale 20) if they were learned previously, vice versa. It may suggest that the scale difference greatly affects lifelong learning.
The average solution distance and optimality gap (optimal solutions are obtained by HGS) of the solver trained with each method on all tasks follwoing task order are listed in Table Notably, use the solver trained after the corresponding task to evaluate Restart, making it single task method.
D.2 MAIN EXPERIMENTS ON TSP
The detailed metric values in each task order are given Table Figures [4]an]] 15 demonstratee the forgetting curve and then learning curve each current task of each method on task orders 2, 3, 4, and 5. Across all orders, LLR-BC demonstrates the best performance. It aligns with the conclusion in the main text.
The average solution distance and optimality gap (optimal solutions are obtained by Gurobi) of the solver trained with each method all tasks follwoing task order are listed in Table Notably, use the solver trained after the corresponding task to evaluate Restart, making it single task method.
D.3 COMPARISON WITH NON-LIFELONG SETTINGS
Notably, in lifelong learning scenarios, training tasks arise sequentially, and only the current task be learned. Therefore, multi-task solvers, which learn from training tasks simultaneously and encounter no forgetting issue, are not applicable. To further enhance our experimental study, we im- plement straightforward multi-task learning methods (batch-level (BL) and epoch-level (EL) task switching) based on POMO, denoted as POMO-MT-BL and POMO-MT-EL, and report the model's performance after learn from all tasks. Results, as follows, demonstrate that the difference is very small, and LLR-BO BC even outperforms the multi task settings in the last 4 tasks of CVRP and all tasks of TSP. addition, we add comparison with INViT (the recent aned high-performance, arguable state-of-the-ard cross-distribution and cross-scale neural solver) in the original setting without adap- tation to lifelong learning. Specifically, we directly use a model provided by Fang al.] (2024), which is trained with large budget on the uniform distribution scale 100 task and demonstrates good zero-shot generalization ability to unseen tasks. Comparison results indicate that LLR-BC, with lifelong learning, outperforms INVIT that without requiring additional training lifelong learning mechanisms. further verifies the importance of lifelong learning and the effectiveness of our proposed L.R-BC. Detailed results listed in Table 6 andd]
D.4 RESULT ON BENCHMARK INSTANCES
We the identical representative instance set selected and used by Zhou et al. (2023) Table and [0 present the detailed test performance on CVRPLIB (Set-X) and TSPLIB instances, respec- tively. Following the same protocol of the main experiments, the reported values are normalized
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Table 5: Metric values on each order on CVRP.
k=3
k=6
Order
Method
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
Li (inter)
20.7
0.0
0.0
21.2
26.3
24.3
0.1
0.1
24.8
27.7
Li (intra)
33.4
0.0
0.0
39.4
47.8
33.2
0.0
0.0
37.2
41.5
Feng
23.2
1,4
1.4
18.8
27.6
25.7
2.2
2.9
23.9
34.4
1
Restart
72.0
8.2
8.2
11.8
36.3
95.7
27.1
27.7
9.1
40.2
Fine-tuning
32.8
3.2
3,2
7.5
32.8
34.5
19.9
23.1
4.6
31.3
EWC
35.9
3.1
3.1
8.6
32.6
39.6
19.7
21.5
7.5
33.5
LiBOG
38.3
2.7
2.7
8.9
33.4
47.7
18.0
18.6
8.3
34.7
LLR-BC
8.0
0.0
0.0
7.0
27.1
4.7
0.6
0.7
4.1
23.3
Li (inter)
28.6
0.1
0.1
30.4
47.2
33.2
0.0
0.0
34.4
42.2
Li (intra)
33.1
0.5
0.5
41.1
59.9
37.4
0.0
0.2
41.8
51.6
Feng
27.3
13.6
13.6
26.2
64.7
25.3
3.4
5.8
24.5
43.2
2
Restart
44.9
92.5
92.5
8.7
60.7
42.1
37.2
64.4
9.1
52.6
Fine-tuning
21.8
29.9
29.9
3.7
50.0
18.8
14.3
23.3
2.5
38.4
EWC
19.5
33.1
33.1
4.4
48.6
19.7
12.4
23.3
4.9
37.9
LiBOG
22.4
29.1
29.1
4.9
51.1
22.2
12.1
21.2
5.4
39.2
LLR-BC
2.8
0.0
0.0
3.3
37.1
2.2
0.3
0.3
1.8
22.6
Li (inter)
20.2
0.0
0.0
21.3
26.8
24.0
0.0
0.0
24.8
28.2
Li (intra)
30.3
0.0
0.0
36.5
44.8
35.6
0.0
0.0
39.2
44.5
Feng
19.1
2.9
2.9
15.0
21.5
25.6
1.3
2.4
24.2
26.6
3
Restart
106.8
10.7
10.7
7.8
30.9
95.7
22.8
35.7
9.1
51.9
Fine-tuning
35.6
4.9
4.9
6.1
24.1
33.4
21
27.4
3,9
44.4
EWC
40.3
4.1
4.1
6.7
22.3
38.0
18.0
25,4
6.8
40.5
LiBOG
42.6
4.3
4.3
6.8
25.0
43.0
18.4
25.9
6.7
41.7
LLR-BC
6.4
0.0
0.0
5.9
23.0
5.0
0.9
1.0
3.7
23.1
Li (inter)
36.5
0.2
0.2
40.9
62.2
40.1
0.1
0.1
42.8
52.5
Li (intra)
31.7
0.3
0.3
44.8
75.9
32.5
0.0
0.1
40.3
54.8
Feng
22.1
10.6
10.6
20.9
66.3
20.7
2.1
4.3
25.4
52.4
4
Restart
35.7
44.5
44.5
10.3
77.3
42.1
26.5
35.5
9.1
52.1
Fine-tuning
14.6
32.6
32.6
7.0
70.0
20.2
18.9
29.9
4.5
45.8
EWC
19.2
28.4
28.4
8.4
69.6
27.1
18.5
23.6
7.9
46.4
LiBOG
19.4
29.1
29.1
8.7
69.0
26.5
20.0
24.3
7.9
46.7
LLR-BC
7.0
0.3
0.3
6.4
58.2
4.7
0.3
0.6
4.0
32.6
Li (inter)
29.4
0.1
0.1
31.7
45.1
38.6
0.0
0.0
41.2
Li (intra)
28.5
0.1
0.1
36.7
53.3
31.9
0.0
0.1
37.9
48.4
Feng
26.3
1.3
1.3
24.5
47.6
25.9
7.0
23.2
38.9
5
Restart
27.2
19.3
19.3
9.5
58.6
26.7
93.1
96.5
9.1
50.5
Fine-tuning
11.8
11.8
11.8
7.0
50.2
10.5
25.5
36.7
3.7
50.6
EWC
17.8
14.5
14.5
8.2
49.1
17.0
29.0
33.7
7.5
40.9
LiBOG
18.3
13.5
13.5
8.6
49.0
17.3
29.9
35.7
7.7
41.7
LLR-BC
6.5
0.4
0.4
6.7
45.3
4.2
1.6
1.6
3.7
31.8
based on the base solution found in our experiments instead of the best-known/optimal solutions reported in existing studies.
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Table 6: Avg. solution distance (optimality gap) of each method CVRP.
Method
E
C
G
GM
Li (inter)
10.74 (3.5%)
15.42 (6.1%)
11.59 (5.2%)
6.73 (4.1%)
5.83 (19.1%)
13.44 (6.6%)
Li (intra)
10.89 (5.0%)
15.61 (7.3%)
11.71 (6.4%)
6.76 (4.5%)
5.85 (19.4%)
13.53 (7.3%)
Feng
10.81 (4.2%)
15.54 (6.9%)
11.52 (4.5%)
6.72 (3.9%)
5.79 (18.1%)
13.59 (7.7%)
Restart
10.74 (3.5%)
15.28 (5.1%)
11.38 (3.3%)
6.57 (1.6%)
5.71 (14.4%)
13.17 (4.3%)
Fine-tuning
10.93 (5.4%)
15.40 (5.9%)
11.80 (7.1%)
7.04 (8.9%)
5.87 (17.6%)
13.03 (3.3%)
EWC
10.97 (5.7%)
15.41 (6.0%)
11.86 (7.6%)
7.04 (8.9%)
5.97 (19.5%)
13.10 (3.8%)
LiBOG
11.00 (6.0%)
15.43 (6.1%)
11.89 (7.9%)
7.27 (12.3%)
6.01 (20.3%)
13.10 (3.8%)
INVIT
11.18 (7.8%)
15.96 (9.8%)
11.73 (6.5%)
7.03 (8.6%)
5.97 (19.7%)
13.77 (9.2%)
MOMO-MT-EL
10,66 (2.7%)
15.10 (3.9%)
11.38 (3.3%)
6.59 (1.8%)
5.71 (14.5%)
13.04 (3.3%)
POMO-MT-BL
10.66 (2.8%)
15.10 (3.9%)
11.36 (3.1%)
6.58 (1.7%)
5.71 (14.5%)
13.04 (3.4%)
LLR-BC
10.72 (3.3%)
15.17 (4.3%)
11,36 (3.1%)
6.56 (1.5%)
5,70 (14.1%)
13.03 (3.3%)
Table 7: Avg. solution distance (optimality gap) of each method on TSP.
Method
E
G
GM
Li (inter)
5.57 (1.4%)
6.35 (4.8%)
5.59 (2.5%)
3.84 (0.4%)
1.98 (0.2%)
4.47 (8.8%)
(intra)
5.76 (4.8%)
6.59 (8.9%)
5.60 (2.8%)
3.90 (2.0%)
(1.4%)
4.69 (14.0%)
Feng
5.66 (2.9%)
6.40 (5.7%)
5.58 (2.4%)
3.86 (1.0%)
1.98 (0.2%)
4.51 (9.7%)
Restart
5.57 (1.3%)
6.31 (4.3%)
5.49 (0.5%)
3.83 (0.1%)
(0.8%)
4.25 (3.3%%
Fine-tuning
5.67 (3.2%)
6.32 (4.4%)
5.72 (4.8%)
3.88 (1.4%)
1.98 (0.3%)
4.20 (2.1%)
EWC
5.67 (3.3%)
6,36 (5.0%)
5.61 (2.7%)
3.89 (1.6%)
1,98 (0.4%)
4.23 (2.8%)
LiBOG
5.66 (3.1%)
6.33 (4.5%)
5.71 (4.6%)
3.88 (1.5%)
1.98 (0.4%)
4.23 (2.9%)
INVIT
5.75 (4.6%)
6.53 (7.9%)
5.66 (3.8%)
3.93 (2.8%)
2,00 (1.4%)
4.58 (11.4%)
POMO-MT-EL
5.61 (2.1%)
6,25 (3.2%)
5.48 (0.3%)
3.86 (0.8%)
1.98 (0.1%)
4.25 (3.3%)
POMO-MTBLL
5.59 (1.8%)
6.24 (3.0%)
5.47 (0.3%)
3.85 (0.6%)
1.98 (0.1%)
4.24 (3.1%)
LLR-BC
5.56 (1.3%)
6.21 (2.6%)
5.46 (0.1%)
3.83 (0.2%)
1.98 (0.0%)
4.20 (2.1%)
24

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Published as a conference paper at ICLR 2026
Table Metric values on each order on TSP.
k=3
k=6
Order
Method
AP
AF
AMF
API
AG
AP
AF
AMF
API
AG
Li (inter)
17.5
0.0
0.0
14.8
22.5
20.2
1.4
1.4
18.9
22.7
Li (intra)
42.9
1.4
1.4
51.4
45,9
0.1
0.7
50.3
54.5
Feng
25.3
4.8
4.8
15.9
23.5
26.5
0.7
5.5
21.6
25.1
Restart
67.0
11.3
11.3
10.0
24.6
39.9
40.8
7.1
57.4
Fine-tuning
31.2
5.3
5.3
4.2
25.9
18.2
36.5
47.9
2.5
35.6
EWC
26.2
4.5
4.5
5.1
27.4
17.3
22.4
22.6
4.0
29.2
LiBOG
30.2
3.7
3.7
5.2
28.3
19.4
15.2
22.1
4.3
38.2
LLR-BC
2.9
0.3
0.3
2.5
19.1
1.7
0.8
0.9
1.3
21.6
Li (inter)
37.6
0.0
0.0
35.8
53.7
38.0
0.5
0.8
35.8
42.6
Li (intra)
58.1
0.6
0.6
59.7
92.8
55.3
0.7
1.0
56.6
69.7
Feng
29.5
1.1
1.1
25.1
56.4
26.6
2.0
2.0
22.7
35.5
Restart
39.3
10.0
10.0
5.4
68.7
29.7
128.4
128.4
7.1
76.0
Fine-tuning
16.6
7.9
7.9
2.4
61.33
13.8
42.5
42.5
2.7
59.3
EWC
15.4
8.5
8.5
14.5
18.9
19.2
4.7
48.1
LiBOG
18.3
8.0
8.0
4.0
63.0
18.7
16.8
16.8
4.7
45.1
LLR-BC
4.5
0.1
50.6
2,5
0.6
1.6
27.6
Li (inter)
6.1
0.0
0.0
6.9
9.0
19.6
0.3
0.3
19.6
22.8
Li (intra)
22.0
0.0
29.5
27.5
46.3
0.0
51.4
57.2
Feng
9.4
3.7
3.7
1.3
11.3
25.6
3.9
4.3
21.5
23.8
Restart
49.9
56.8
56.8
3.1
70.5
39.9
14.6
42.8
7.1
72.9
Finc-tuning
19.9
29.2
29.2
1.8
17.6
15.4
15.4
30.5
2.2
41.5
EWC
19.8
19.9
19.9
2.6
16.7
17.0
10.8
19.0
3.8
37.4
LiBOG
20.6
15.6
15.6
2.1
16.7
17.4
11.8
20.2
4.3
37.2
LLR-BC
2.2
0.0
0.0
1.9
10.0
3.0
0.6
0.9
1.8
22.9
Li (inter)
98.5
0.0
0.0
104.5
158.1
109.6
0.0
0.0
119.1
146.4
Li (intra)
73.9
0.4
0.4
100.2
203.6
63.7
0.0
0.1
79.1
122.6
Feng
20.6
0.0
0.0
21.1
186.7
17.9
0.0
1.2
18.5
84.6
Restart
40.9
46.0
46.0
7.0
156.2
29.7
37.6
47.5
7.1
85.7
Fine-tuning
25.1
32.2
32.2
164.3
15.5
33.2
4.4
85.2
EWC
34.1
34.1
34.1
8.6
160.8
22.8
23.2
31.2
7.0
77.6
LiBOG
32.4
34.4
34.4
8.7
154.5
22.2
25.5
33.1
7.5
74.5
LLR-BC
8.9
1.1
1.1
7.7
154.6
5.0
1.4
1.4
4.2
68.9
Li (inter)
75.1
0.1
0.1
100.5
161.7
95.2
115.0
153.6
Li (intra)
42.6
2.0
2.0
69.8
127.3
60.0
0.2
0.8
76.6
107.9
Feng
17.3
1.5
1.5
13.1
100.0
23.9
2.6
5.8
22.8
58.5
5
Restart
14.0
11.1
11_1
8.8
99.5
19.5
30.9
41.5
7.1
69.9
Fine-tuning
11.5
9.1
9.1
9.4
99.1
11.3
16.7
28.9
5.7
65.4
EWC
21.5
12.0
12.0
8.9
102.5
20.0
17.7
31.0
8.1
68.3
LiBOG
18.7
9.4
9.4
8.6
92.1
18.2
16.9
22.0
8.2
63.5
LLR-BC
7.5
0.8
0.8
9.1
106.3
4.8
0.5
1.7
5.3
64.5
25

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Published as a conference paper at ICLR 2026
Table 9: Test performance on CVRPLIB instances.
Instance
Li (intra)
(inter)
Feng
Restart
Fine tuning
EWC
LiBOG
LLR. BC
X-n101-k25
38.33
8.16
8.93
X-n153-k22
52.41
12.65
30.32
38.03
21.71
43.18
23.72
X X-3200.k336
26.28
21.48
10.85
19.54
8.13
7.02
13.22
0
X-a251-k28
12.71
12.98
14.37
14.26
14.25
9.23
1.47
X-0303-k21
83.59
44.96
18.77
66.62
10.03
38.12
27.47
0
X-n351-k40
43.88
X-o401-k29
29.12
39.31
30.61
23.33
1.48
16.77
21.85
0
X-a459-k26
X-a502-k39
32.57
87.95
31.46
9.85
4.42
21,88
20.08
X-n548-k:00
25.36
35.99
4.33
X-n599-492
$5.1
25.09
6.62
27.25
17.92
7,76
25.96
X-n655-k131
66.91
60.46
15.64
31.02
26.97
11.66
0.77
X-n701-k44
34.08
14.51
21.06
58.96
30,82
9.88
13.01
X-n749.k98
57.64
46.36
41.03
43.41
20.44
13.76
7.94
X-n8V1-k40
17.03
24.97
20.79
29.82
1,55
14.34
X-0856-k95
21.53
34.56
47.96
50.49
29.5
13.87
35.03
0
X-n895-k37
X n957-kks
57.95
74.26
61.91
58.04
24.37
9.06
27.86
0
X-n1001-k43
13.28
Mean (Shd.)
46 46.86(26.95)
39.37(23.
27.38(18.55)
43.68 (27.37)
8.54 (10.71)
11.86 (1) (11,5
18.05 (10.86)
7.88 (8.75)
Table 10: Test performance on TSPLIB instances.
Instance
Li (intra)
Li (inter)
Feng
Restart
Fine-tuning
EWC
LIOOG
LLR-BC
115.35
42.67
kroA150
85.3
56.6
11.67
29.43
28.06
0
7.58
12.15
16.93
kroB200
90.12
46.27
$5.42
105.78
10.86
34.53
12.47
1s225
145.3
27.3
102.23
152.45
60.52
70.33
24,24
tsp225
76.08
31.3
28.75
97.43
9.61
0
15
2.47
pr226
30.16
42.36
17.84
0
5.95
16.59
16.12
pr264
279.81
164.86
164.17
47.94
36.59
34,88
46.66
a280
119.41
43.49
80.83
72.55
8.98
0
28.86
19.19
127.96
lin318
127.91
61.94
104.45
97. 15
0
16
46.94
32.34
r3400
128.69
$3.32
$5.73
81.68
18.25
25.14
20.14
B417
28.2
58.18
80.35
40.69
3.67
30,88
21.12
pe439
120.37
50.77
61.96
126.81
29,1
3,8
0
0.14
pcb442
157.56
79,02
81.29
1.17
17.31
11.12
d493
40
$2.33
89.3
253.56
443.1
280.94
123.82
127.96
rai575
152.37
51.56
77.56
122.04
D
24.43
20.29
$2.95
150.62
5.04
26.84
d657
126.42
50.54
104.57
267 62
136,78
81.08
101.14
u724
161.09
58.65
83.95
66.06
30.1
2,83
25.18
ral783
155,97
65.08
75.42
135.86
10.48
23.03
34.46
pr1002
147.73
68.96
89.49
65.53
2.33
0.67
29.98
Mean (Std.)
122.60(51.78)
58.46(26.55)
75.15(30.83)
99.76(61.06)
38. 38.16/93.36)
27 27.80199.58)
27, 27.96(32.29)
18.08(16.98)
26

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Published as a conference paper at ICLR 2026
11.00
10.75
14
11.0
6.00
12
120
11.
(G-GM
14
200
400
1000
400
1200
Training Epoch
Trinnig Epoch
LLR-BC
U (inter)
Restart
EWC
LLR-BC
Li (inter)
Restart
EWC
Feng
(intra)
Fine-tuning
LIBG
Feng
Li (intra)
Fnn-tuniing
UBOG
(a) Order
(b) Order 3
12.0
15
11:00
6.00
7.0
7.0
6.00
GM
14
5.75
4 11.00
MA
10.75
200
400
600
1000
1200
200
400
600
800
1000
1200
Training Epoch
Training Epoch
LLR-BC
(inter)
Restart
EWC
LLR-BC
Li (inter)
Restart
- EWC
Feng
(intra)
Fine tuuinu
LIBO
Feng
Li (intra)
Fine tuning LBOG
(c) Order 4
(d) Order 5
Figure 12: Forgetting curve on CVRP task orders 2-5, measured by average solution distance (ver- tical axis). Epochs 0-200 (first task) are omitted as forgetting occurs. Restart is excluded due to significantly higher forgetting than other methods.
D.5 HYPERPARAMETER SENSITIVITY ANALYSIS DETAILS
As Table 3 demonstrated, on CVRP, increasing a reduces AF and AMF while increasing API as expected, as it leads greater emphasis on stability over plasticity, which aligns with expectations since a higher weight in the consolidation loss term will lead to more focus on stability rather than plasticity. contrast, on the TSP, increasing a results in worse performance across evaluation metrics. A potential reason is that TSP tasks are more similar to each other than those in CVRP, as TSP does not involve differences demand distribution. This higher similarity increases the likelihood of beneficial backward transfer when learning new task. In such cases, using lower consolidation weight could more advantageous. Nevertheless, LLR-BC consistently outperforms all baseline methods under each tested setting (cf. Table [I. No significant pattern in performance changing is found by varying |g] or [B. But the impact of changing them is substantially small, verifying the robustness of L.R-BBC.
27

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Published as a conference paper at ICLR 2026
Test Performanc
Test Performanci
0.02
0.01
0.01
0.00
0.00
200
400
600
800
1000
1200
200
400
600
800
1000
1200
Training Epoch
Training Epoch
-LR--BC
Li (inter)
Restart
EWC
LLR-BC
Li (Inter)
Restart
EWC
Feng
(intra)
Fine-tuning
LIBOG
Feng
(Intra)
Fine-tuning
LIBOG
(a) Order 2
(b) Order 3
Test Performanci
Test Performance
0.02
0.02
0.01
0.01
0.00
0.00
200
400
600
800
1000
1200
200
400
600
800
1000
1200
Training Epach
Training Epoch
-LRR-BC
Li (inter)
Restart
-EWC
-LRR-BC
Li (inter)
Restart
EWC
Feng
Li (intra)
Fine-tuning - LIBOG
Feng
Li (intra)
Fine-tuning
LIBOG
(c) Order
(d) Order 5
Figure 13: Test performance on the current task during lifelong learning on CVRP task orders 2-5.
D.6 DETAILS OF APPLICABILITY EXPERIMENTS
We evaluate LLR-BC on Omni Zhouu et al. 2023) and INVIT Fang et al. (0024) to verify its ap- plicability (cf. Section (.6)). Omni designed to produce a strong initial model that can quickly adapt to new tasks, through meta-learning Accordingly, we adopt the meta-learned initial model provided by iZhou et al.) (202) as the starting model for the first task, and run lifelong learning with 10 epochs for each task, following their protocol. INVIT aims at learning from one task to achieve strong zero-shot generalization to other tasks. For INVIT, we sequentially train on tasks for 100 epochs each. As the default setting (Fang et al. 2024), data augmentation with size 8 is used in INVIT. Hyperparameters of LLRBCC are set as follows: [ = 1000, [ = 16, and a = 100, identical the one used on POMO. Hyperparameters of base neural solvers are set identically the original papers. All other settings are consistent with those used with POMO, as above. Since our goal is not to compare performance differences across base neural solvers, normalization for each base neural solver, i.c., for different base neural solvers the value of dł is different. This allows us to demonstrate the applicability of LLR-BC across different base solvers more clearly, without the distraction from performance differences in base neural solvers.
apply independent
For Omni, only on API of CVRP, LLR-BC is slightly higher than fine-tuning. sonable as Omni is designed to quickly adapt and owns outstanding plasticity, which overwrites the benefit from the potential forward transferring of LLR-BC. For INVIT, LLR-BC outperforms fine-tuning on most metrics, except for the forgetting measures CVRP. A potential reason for this weaker performance that TSP tasks are highly similar to one another (relative to the greater diversity among CVRP tasks), and INVIT is known to exhibit strong ability to learn general knowl- edge of relatively similar tasks from one task (Fang et al., 2024). such cases, incorporating prior experience that is not directly relevant to the current learning task may introduce distracting signals, which can outweigh the benefits in mitigating forgetting.
considered rea-
Notably, the performance of LLR-BC can be further improved by tuning its hyperparameters for each individual base neural solver. Overall, LLR-BC outperforms fine- finetuning across most evaluation metrics, regardless of the used base neural solver, thereby confirming its broad applicability.
28

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Published as a conference paper at ICLR 2026
4.5
2,000
5:75
5.50
1.975
TL-R
1:900
1.975
6.25
5:75
Ww
Te=GM
5.0
200
1000
1200
Train
Epoch
Taining Epoch
LLR-BC
(inter)
Restart
EWC
LLR-BC
Li (inter)
Restart
EWC
Feng
(intra)
tuning
Feng
(intra)
Aie-tuniing
LIBG
(a) Order
(b) Order 3
5.75
5.50
5.0
6.50
5.75
Md
5.50
2.975
Tu=U
3.875
3.900
3.850
5.0
1:98
4.5
1.97
6.4
5.75
6.2
200
400
600
1000
1200
200
400
800
1000
1200
Training
Epoch Restart
Training Epoch
LLR-BC
(Inter)
EWC
LLR-C
Li (inter)
Restart
- -EWC
Feng
(Intra)
Aie-ttniigg
LIBGG
Feng
Li (intra)
Fine tuning LiBOG
(c) Order 4
(d) Order 5
Figure 14: Forgetting curve on TSP task orders 2-5, measured by average solution distance (vertical axis). Epochs 0-200 (first task) are omitted forgetting occurs. Restart is excluded due to significantly higher forgetting than other methods.
E LLM USAGE STATEMENT
We used ChatGPT (GPT-5) only assistive tool for grammar checking and language polishing. The model was not involved in research ideation, algorithm design, experiment execution, or result analysis. All scientific content and conclusions entirely the work of the authors.
29

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Published as a conference paper at ICLR 2026
Test Performance
Test Performance
0.02
0.01
0.01
0.00
0.00
200
400
600
800
1000
1200
200
400
600
800
1000
1200
Training Epoch
Training Epoch Restart
-LR--BC
(inter)
Restart
LLR-BC
Li (inter)
EWC
Feng
(intra)
Fine-tuning
LIBOG
Feng
(Intra)
Fine-tuning
LIBOG
(a) Order
(b) Order 3
Test Test Performance
Performanci
0.02
0.01
0.01
0.00
1000
1200
400
600
800
Tranning Epoch
Training Epoch
-LLR-BC
Li (inter)
Restart
- EWC
-LR--BC
Li (inter)
Restart
EWC
Feng
Li (intra)
Fine-tuning
- LIBOG
Feng
(intra)
Fine-tuning
LIBOG
(c) Order
(d) Order
Figure 15: Test performance the current task during lifelong learning TSP task orders 2-5.
30


Reviews:
[{"review_id": "T4aCPKWevX", "reviewer": "Reviewer_A4WR", "summary": "This paper proposes a lifelong learning framework, LLR-BC, for enhancing the generalization performance of existing neural models for vehicle routing problems (VRPs). It trains multiple tasks (including VRPs with different distributions and sizes) sequentially, and employs experience replay strategy and introduces behavior consolidation to preserve knowledge of prior tasks while learning new ones. Experimental results show the effectiveness of the proposed method.", "questions": "Please refer to the weakness.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper employs lifelong learning to solve the generalization issue in VRPs across different distributions and problem sizes.\n2. In experiments, this paper introduces interesting evaluation metrics.\n3. Experimental results show the effectiveness of the proposed method against baseline methods.", "weaknesses": "1. The core idea of this paper, using lifelong learning to address the generalization issue of VRP while alleviating catastrophic forgetting is similar to [1][2]. Although the authors claim that [1][2] focuses on a fixed task order, while the proposed LLR-BC could handle flexible task orders, the core contribution seems to be similar.\n2. This paper aims to enhance the generalization ability of neural models for VRPs, while the comparison of the LLR-BC with other neural models that are specialized for generalizability enhancement is not reported.\n3. The experience replay strategy is a standard strategy in lifelong learning. And the core idea of behavior consolidation seems to be similar to \"inter-task regularization\" and \"intra-task regularization\" in [1], which both learn to preserve knowledge from past ones when training on new tasks. A more clear discrepancy between [1] and this paper should be explained.\n\n[1] Jingwen Li, Zhiguang Cao, Yaoxin Wu, and Tang Liu. Enhancing the cross-size generalization for solving vehicle routing problems via continual learning, 2024. URL https://openreview.net/forum?id=WdvT2UgsTK.\n\n[2] Shaodi Feng, Zhuoyi Lin, Jianan Zhou, Cong Zhang, Jingwen Li, Kuan-Wen Chen, Senthilnath Jayavelu, and Yew-Soon Ong. Lifelong learner: Discovering versatile neural solvers for vehicle routing problems, 2025. URL https://arxiv.org/abs/2508.11679.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nrzSBHE6lz", "reviewer": "Reviewer_ckwk", "summary": "This paper proposes LLR-BC solvers that address the catastrophic forgetting problem when adapting to sequential tasks with different scales and distributions. It further introduces two key modules: CaEW and DsBC. Moreover, Extensive experiments demonstrate the effectiveness of LLR-BC.", "questions": "Clarification on the Logic of the CaEW Mechanism: In Section3.2 , CaEW aims to assign higher weights to “low-confidence” decisions, noting that “lower variance indicates lower confidence.” However, the formula actually assigns higher weights to experiences with low variance (i.e., high confidence). Could you please clarify?\n\nRegarding the limitation of fixed sampling size |ε|: In conclusion that using a fixed |ε| is a limitation when task scales vary significantly. Could you elaborate on why this is problematic? Is it because tasks of different scales require different numbers of experiences for effective consolidation?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe paper introduces two novel modules CaEW and DsBC， with the innovative use of Reverse KL Divergence enabling decision-focused behavior consolidation. It also removes the common “fixed task order” constraint, evaluating on randomized sequential tasks for greater realism and originality.\n2.\tThe model's performance was evaluated across multiple task sequences and VRP scenarios of varying scales and distributions. Furthermore, the paper includes comprehensive ablation experiments and hyperparameter sensitivity analyses, demonstrating the independent effectiveness of the CaEW and DsBC modules as well as the model's robustness to parameter variations.\n3.\tThe presentation is clear and easy to follow.", "weaknesses": "1. Limited Innovation: The paper shows limited novelty based on the existing lifelong VRP models. The comparative experiments include only Fine-tuning, EWC, and LiBOG. However, no comparisons are made with more recent lifelong learning approaches (e.g., Li et al., 2024; Feng et al., 2025) or with cross-problem/multi-task frameworks. Including these would better position the proposed method within the current research landscape.\n2. Potential Bias in Experience Buffering: Experience buffering is conducted only during the final epoch of each task. A further analysis or ablation study on this design choice would enhance the completeness and reliability of the evaluation.\n3. The manuscript contains many typos (e.g., “stuides” → “studies,” “LLB-RC” → “LLR-BC”)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XaWYL2NZT5", "reviewer": "Reviewer_9mCf", "summary": "This paper presents a lifelong learning framework called LLR-BC for learning to solve Vehicle Routing Problems (VRPs) across different distributions and scales. LLR-BC employs a customized experience replay strategy across sequentially arising tasks to prevent catastrophic forgetting. The approach introduces two key components: a confidence-aware experience weighting (CaEW) module and a decision-seeking behavior consolidation (DsBC) loss based on reverse KL divergence. Results on TSP and CVRP validate the effectiveness of the proposed methods against baselines.", "questions": "Other than the questions listed in the weakness, I also have the following questions:\n* Which epoch of the model training is used to update the buffer experience, and what is the rationale for this choice? How sensitive is performance to this decision?\n* How are the hyperparameters for buffer batch size determined?\n* How well does variance actually serve as a proxy for confidence? Are there empirical studies or theoretical analyses that validate this design choice against alternative confidence measures?\n* How diverse is the experience buffer after running the algorithm? Does the buffer maintain sufficient diversity across tasks, or does it become dominated by certain task types or distributions?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The paper studies an interesting and relatively underexplored area for neural VRP solvers: lifelong learning across diverse problem distributions and scales. \n* The presentation and illustrations are clear, making the paper easy to follow. \n* The work presents comprehensive experiments covering multiple task orders, diverse distributions and scales, and multiple backbone VRP neural solvers.\n* The performance appears promising under the employed metrics.", "weaknesses": "* There is some conceptual overlap with existing works, such as Li et al. 2024 and Feng et al. 2025, which address similar lifelong learning setups. It would be useful to directly compare with these existing methods. Also, while the customization of the replay buffer for VRPs is appreciated, the idea of experience replay is a standard technique in lifelong learning literature.\n* The paper proposes using the variance of node selection probabilities as a proxy for model confidence. However, this lacks clear empirical evidence or theoretical justification that probability variance truly represents confidence. My concern is that this may be too strong and fails to distinguish between two scenarios: (1) a genuinely low-confidence model with high uncertainty, and (2) a policy that has learned multi-modal decisions where multiple nodes are equally good selections at certain division points with high confidence. \n* The design and naming regarding “DsBC” seem confusing to me, as the loss functions in Equations (1) and (2) are relatively straightforward applications of reverse KL divergence. It is unclear what new conceptual insights or contributions DsBC provides here.\n* While the experiments are extensive in scope, the absolute performance of the model on each individual task is never reported. In practice, practitioners care not only about whether a lifelong learning model can avoid catastrophic forgetting, but also about the actual solution quality (such as optimality gaps) achieved on each new task. \n* The paper lacks a comparison with other state-of-the-art methods that target generalization across distributions without requiring additional training or lifelong learning mechanisms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TEQUPT0gxE", "reviewer": "Reviewer_iZkt", "summary": "This paper introduces a lifelong learning setting for the VRP domain and proposes the Lifelong Learning Router with Behavior Consolidation (LLR-BC) to address the issue of catastrophic forgetting. Specifically, LLR-BC incorporates Confidence-aware Experience Weighting (CaEW) to prioritize critical experiences and Decision-seeking Behavior Consolidation (DsBC) to emphasize replicating past decisions. Extensive experiments on TSP and CVRP benchmarks demonstrate the effectiveness of LLR-BC.", "questions": "* Is Fig. 1 a conceptual illustration or based on real experimental results?\n* Could LLR-BC be applied to SL-based solvers as well?\n* The experience buffer is state-based, why not consider an instance-based design?\n* Is there any empirical evidence supporting the claims made in lines 236–240?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The paper is overall well-written and easy to follow.\n* The studied topic, i.e., generalization of neural VRP solvers, is important, and the proposed setting is sound.\n* The experiments are generally extensive and solid, demonstrating that the catastrophic forgetting issue is significantly mitigated, as shown in Fig. 3.", "weaknesses": "* This paper applies lifelong learning to the VRP domain, incorporating a reweighting mechanism and reverse KL divergence. The overall technical novelty appears somewhat limited.\n* In line with the no free lunch principle, while the proposed method mitigates catastrophic forgetting on previous tasks, it may incur trade-offs elsewhere. Could the authors clarify what is sacrificed or traded off in their approach or setting?\n* The caption of Fig. 2 should explicitly explain all notations.\n* The writing in two parts could be improved:\n  * In Section 3.1, the authors should first introduce the principle of reservoir sampling.\n  * In Section 3.3, the connection between KLD and RKLD should be highlighted as a change in direction from $KL(\\pi_\\theta | \\mathcal{P}) \\to KL(\\mathcal{P} | \\pi_\\theta)$. The authors should also mathematically explain why this directional change fundamentally alters model behaviour during learning.\n* The experiments primarily compare lifelong learning baselines, but lack a comprehensive evaluation against traditional and neural VRP solvers. Could the authors include direct comparisons with recent generalizable neural VRP solvers, e.g., by multi-task training them on the same task distribution?\n* Minor on line 152: (i.e., )\n\n----\n\nI intend to assign a borderline rejection in the first round of evaluation and would like to see the authors’ rebuttal addressing my main concerns, particularly the experimental comparisons.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)