Instance-wise Adaptive Scheduling via Derivative-Free Meta-Learning
Authors: Hefang Qing
Miao Zhang
Yaoxin Wu
Weinan Huang
Jianhao Yang
Wen Song
Gang Wang
Venue: ICLR 2026 Poster (2026)
Keywords: Scheduling
Neural Combinatorial Optimization
Meta-learning
Primary area: applications to robotics, autonomy, planning

Abstract: Deep Reinforcement Learning has achieved remarkable progress in solving NP-hard scheduling problems. However, existing methods primarily focus on optimizing average performance over training instances, overlooking the core objective of solving each individual instance with high quality. While several instance-wise adaptation mechanisms have been proposed, they are test-time approaches only and cannot share knowledge across different adaptation tasks. Moreover, they largely rely on gradient-based optimization, which could be ineffective in dealing with combinatorial optimization problems. We address the above issues by proposing an instance-wise meta-learning framework. It trains a meta model to acquire a generalizable initialization that effectively guides per-instance adaptation during inference, and overcomes the limitations of gradient-based methods by leveraging a derivative-free optimization scheme that is fully GPU parallelizable. Experimental results on representative scheduling problems demonstrate that our method consistently outperforms existing learning-based scheduling methods and instance-wise adaptation mechanisms under various task sizes and distributions.

Published as a conference paper at ICLR 2026
WNST ADAPTIVE SCHEDULING VIA DERIVATIVE- FREE META-LEARNING
Hefang Qing". Miao Zhang Yaoxin Wu¹, Weinan Huang?,
Jianhao Yang"*. Wen Song 1x Gang Wang'
Shennhen Research Institute of Shandong University Xiiaoi Corporation
Eindhoven University of Technology $Zhongguaneun Academy
qinghefang@emaile sdu. edu. y. wu2@tue. ml baanngeennnn@exx wensong@email i. sdu..du.eeo eata..dd.ccmm cn gangwang@umn. edu
edu. en changmiao20@xiaaaaoo com
com yangj ana.iiinnaiiiiiiiiioooaooo com
ABSTRACT
Deep reinforcement learning has achieved remarkable progress in solving NP-hard scheduling problems. However, existing methods primarily focus on optimizing average performance over training instances, overlooking the core objective solving each individual instance with high quality. While several instance- wise adaptation mechanisms have been proposed, they are test-time approaches only and cannot share knowledge across different adaptation tasks. Moreover, they largely rely on gradient- optimization, which could be ineffective in dealing with combinatorial optimization problems. We address the above issues by proposing an instance- wise meta-learning framework. It trains a meta model to acquire a generalizable initialization that effectively guides per-instance adaptation during inference, and overcomes the limitations of gradient-based methods by leveraging a derivative-free optimization scheme that is fully GPU parallelizable. Experimental results on representative scheduling problems demonstrate that our method consis- tently outperforms existing learning-basee based scheduling methods and instance-wist adaptation mechanisms under various task sizes and distributions.
INTRODUCTION
Scheduling aims to optimize resource allocation for task completion within specified time constraints, playing a pivotal role in a wide range of practical domains such as manufacturing, logistics, and healthcare (Khadivi et al.; 2025). As two fundamental models, Job-shop Scheduling Problem (JSP) and its extension, Flexible Job-shop Scheduling Problem (FJSP) receive much attention. However, solving JSP and FISP optimally remains significant challenge due to their well-known NP-hardness (Michael 1995 Mazyavkina et al.) (02T). Especially for industrial level large-scale instances, exact algorithms such Mixed-Integer Programming (MIP) and Constraint Programming (CP) are often prohibitive due to the excessive computational cost (Da Col & Teppan 2019). Heuristic and metaheuristic methods could strike a balance between solution quality and computational time, but they are typically less accurate due to their reliance on predefined rules and lack of adaptability to specific scenarios (Li et al. 2024a).
Recently, Deep Reinforcement Learning (DRL.). as an emerging alternative method (Feng et al. 2026), has been successfully applied to complex scheduling problems. notable direction is to use DRL. to learn Priority Dispatching Rules (PDRs) (Zhang et al.] 2020) [ong et al. 2023] Wang et al.) 2023). The learned policies are often superior to manually designed PDRs, and can generate solutions within short run time (Mazyavkina et al. 202T). However, the solution quality of existing learning based scheduling methods are still relatively far from optimaliy. One reason is that they focus on training a deep policy model to optimize its average performance training instances. Given that currently DRL can only obtain suboptimal policy, this means well-trained policy can still produce
These authors contributed equally to this work.
" "Corresponding authors.

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Published as a conference paper at ICLR 2026
poor solutions for some testing instances, even when they come from the training distribution (Wang & Li )023).
One way overcome this limitation is test-time adaptation, which is to fine-tune the pre-trained model on each specific instance being solved, Active Search (AS) (Bello et al. (0166) and Efficient Active Search (EAS) (Hottung et al. 2021b) are two representative methods, and the latter shows good performance on JSP. However, this popular paradigm suffers from two limitations. First, works as a pure test-time mechanism that fine-tunes the model on each instance separately, which is inefficient since the adaptation knowledge is discarded and not reusable. Second, existing works use gradient- based methods for adaptation, which performs well in conventional deep (reinforcement) learning tasks but could fall short when dealing with the instance- search task for complex combinatorial optimization problems such as JSP and FJSP, since they could easily fall into local optimum.
In this paper, we address the above issues by proposing meta-learning method driven by derivative- free optimization, for solving complex scheduling problems. Our first contribution is an instance-wist meta-learning framework based on Model-Agnostic Meta-Learning (MAML) (Finn et al.) 2017). which simulates the fine-tuning process during training so as to obtain a meta-model that explicitly considers the needs of fine-tuning, thus providing a well-initialized model for each new instance. This framework is model-agnostic, and is generally applicable to wide range of deep policy model. Next, motivated by the recent success of evolutionary strategies in DRL (Salimans et al. 2017 Song et al. 2020 Kirsch al.. 2022), we design fully derivative- method to train the meta-model, which not only overcomes the limitation of gradient-based -baee methods in instance-wise searching but also bypasses the complicated gradient computation in the original MAML. We design two Monte Carlo (MC) strategies for gradient estimation in the inner loop, which effectively improves the training performance. Finally. we design
ippulatiionparallle framework that shifts the CPU-intensive computational tasks in traditional evolutionary strategies (Salimans et al., 2017) to GPU parallel processing, significantly reducing the training overhead.
We validate the effectiveness of our method mainly on FISP, which is much harder than JSP. Specif- ically, we deploy our method to state-of-the- FISP PDR learning model in (Wang al.) (023). Experimental results demonstrate that the instance-wise fine-tuning strategy significantly improves the model's adaptability to unseen test instances. Moreover, our approach consistently achieves superior performance across benchmark datasets of varying sizes and distributions, outperforming existing instance-level adaptation methods. We also extend our method to JSP in a non-reinforcement learning setting (Corsini al. 2024), showcasing its strong compatibility and adaptability across different learning paradigms.
RELATED WORK
Learning based scheduling. Motivated by the recent success of Neural Combinatorial Optimization (NCO) (Bengio et al. 2021). researchers have begun to utilize deep (reinforcement) learning to tackle JSP and FJSP. The most popular paradigm is PDR learning, which formulates the PDR based schedule construction as Markov Decision Process (MDP), and uses DRL to automatically train the scheduling policy. A common choice in this direction is to represent construction states based on disjunctive graph (e.g., (Zhang et al.) 2020 Park al.) [021 Song al. 2023
ct al. 2023)), and design Graph Neural Network (GNN) based policy network to achieve size- invariance. Besides GNN, other types of neural architecture based Pointer Network (Corsini et al.) 2024) and Attention Mechanism (Wang et al. 2023) Chen et al. 2022, Pirma & Grimm 2024a b) have also been proposed and can achieve even better performance. Another direction is learn
control policies for local search algorithms ęZhang et al., 2024a b). which tend to deliver better solutions than PDR learning methods at the cost of longer run time.
Instance-wise adaptation. Above methods follow the convention in machine learning to optimize the average performance over training instances, which often leads to suboptimal performance on unseen instances. This could be alleviated by Active Search (AS) (Bello et al.) 2016) which dynamically adjusts pre- trained model parameters on each testing instance. Efficient Active Search (EAS) (Hottung et al. 2021b) improves AS by updating only subset of model parameters to reduce computational costs. However, AS and EAS do not change the original training objective of average performance and the adaptation is purely test-time. Meta-models for more efficient adaptation have been explored
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in other combinatorial optimization problems such vehicle routing and graph optimization QQuu et al. 2022 Wang & Li. 2023 Son et al. 2023). but they rely on croblem-specifice techniques and are not directly applicable here. Moreover, these methods use gradient-based optimization for fine-tuming whereas our method avoids strong assumptions about the problem or neural architecture and the first to employ gradient-free optimization for such tasks. Additionally, some research performs instance- wise search in a learned continuous space for high-quality solution distributions (Hottung) al.
2023 2024b), but these methods focus less on adaptation, and models not updated per instance. Our approach is orthogonal to these works and could potentially be combined for better performance.
al. 202
PRELIMINARIES
Job-shop scheduling problem involves a set of jobs J [J,, JJ., Jn to processed on machines M [M1 MM,...,,,, where each job J, consists of sequence of operations O {O11, Oi2..... .. } } to be assigned to machines. In JSP, each operation can only be assigned to one specific machine, whereas FJSP allows multiple machine options. The goal is to minimize the makespan, defined as the maximum completion time across all operations, i.e., Cina = mj (Crr)-
In this paper, The model treats scheduling as a sequential decision task. At each step t, a neural network receives the state 8, (including operations and machines) and outputs an action a, that assigns unscheduled operation to an available machine. This repeats until all operations are scheduled, with rewards based on the final makespan. The policy my (ag (s)) is trained using Proximal Policy Optimization (PPO) (Schulman et al. (017)) to maximize cumulative reward. After training, the policy can be used in greedy or sampling modes. Greedy selects the best action each state, while sampling generates multiple schedules and returns the best solution, though at a higher computational cost.
the PDR learning model for FJSP in (Wang et al.) (023) to evaluate the problem.
Derivative-free optimization (DFO), also known as zero-order optimization, encompasses methods that do not rely on gradients. DFO is effective in optimiza- tion problems where gradients are costly or inacces- sible. Here we use OpenAl's Evolutionary Strategy (ES) (Salimans et al. 2017) for optimization, which be- longs to the class of Natural Evolution Strategies (NES)
Algorithm ES Gradient Estimation
Require:
Neural network parameter set 0, noise standard deviation a, popu- lation size
Sample : , ~N(11)
2: for i I..... do
(Wierstra et al.] 2014). Let F be the objective func-
Compute fitness: F; F(0 + +0,)
tion over parameter vector 0. NES models the popula- tion as a distribution pa (0), parameterized by vh. The optimization process aims to maximize the expected objective Ee-pe [F)] by updating v via stochastic
4: end for
5: return
Σ 1==1 Pe
gradient ascent, with the gradient V..TEo-p,, [F(0)] Ee [F(O) log Pos (0)]. To avoid non- smoothness from the environment or discrete policy actions, we follow OpenAl's implementation use an isotropic multivariate Gaussian distribution for the population, with mean &i and fixed covariance σ (Salimans et al. 2017). This allows the expected objective to be expressed as Egrpi [F() E.,, F(O+ J.)]. We optimize over 0 directly using stochastic gradient ascent: V (o [F(O + oæ)] = 1 Ve [F(O + oe)s], which can be approximated through the Monte Carlo sampling procedure in Algorithm
4 METHODOLOGG
In this section, we propose general instance-wise meta-learning framework for well-adapted parameter initialization in downstream fine-tuning, fully exploiting the potential of derivative-fres optimization in instance-level adaptation tasks. Additionally, we design an efficient population- parallelization strategy that significantly enhances computational efficiency.
4.1 INSTANCE-WISS DERIVATIVE-FRES META-LEARNING FRAMEWORK
We define a FJSP instance class as Ω = (n, m, o, t). where n and m denotes the number of jobs and machines, 0 [Omin; Omax] specifies the range of the number of operations in each job, while
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Ted Instance
Optimintion Pammeter Nimmbuttonn
Pg
0
Gr
Inne--oop Optimization
G3
Intiaaiie
Training Gndien
(O(
ES Gnaliens
Dinection
Ouice-luup Optinistioon
Go
0
Meta- Meta-Training
Fine-Tuning
Figure 1: overall framework of our method
t t= [iii fmax defines the processing time range for operations executed on different machines. We denote G, E Ω specific scheduling instance sampled from the defined instance class.
Unlike traditional methods that aim to learn a single model with optimal average performance training set, goal is learn initialization of model parameters that enables efficient adaptation to unseen instances at test time. This initialization allows the model to rapidly converge to high-quality, instance-specifee solutions through fine-tuning during inference, which offers more potential than test-time only methods such as AS (Bello et al. 2016) and EAS (Hottung et al. 2021b). Accordingly, our training problem is:
0 arg min EG, ~ F (Ock) Go
G.)
where du denotes the moda- that provides starting parameter for each instance, and 0 (k) is the fine-tuned model obtained after K steps of gradient updates from 0g on instance Gr, which is then used to generate the final solution (i.e., schedule) for G,-
We implement this learning task using the MAML framework (Finn al. 2017). The meta-training process consists of two stages: inner-loop optimization and outer-loop optimization. The inner loop performs simulated fine-tuning on each individual instance to capture instance-specific features and enable rapid adaptation. The outer loop aggregates feedback from multiple instances to update the meta-model parameters, thereby enhancing its ability to quickly adapt to new, unseen instances. The pseudocode for the meta-tres meta-training process is presented in Algorithm] 2
Algorithm 2 Instance-wise Derivative-Free Meta- Training
Require:
Training instance set Ω mini-batch size B, number of inner-loop updates K, adaptation step size a, meta step size 3, noise standard deviation o, number of epochs T';
1: Randomly initialize the meta-model 6u
2: for I.....7 do
3: for each randomly sampled instance G, E == do
4: Initialize instance-specific model: g On
5: for k = 1...., K do
6: 0.h) - --1 a g(=-1) EEMNTO.)
FF (0 (oft-1) + oe G+r)
7: end for
8: end for
9: On f- On DB Vo, E BEWN(OL
[ (oL + ( G.)
end for
key issue in Algorithm 2 is how to compute the inner-loop and outer-loop gradients Line (computing gradients of the instance-wise model) and Line 9 (computing gradients of the meta-model defined in Eq. T. Unlike the mainstream gradient-bee based methods (e.g. (Manchanda et al. 2022 Zhouu et al. 2023 Qiu et al.) 2022) Wang & Li (223)), propose full derivative-free ffree method to
estimate these gradients. Leveraging the black- nature of DFO, our framework relies solely on the evaluability of the objective function, and makes assumptions about the underlying MDP and the
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environment's 's structural priors, therefore is easy to implement and potentially applicable to various scheduling problems. Moreover, using DFO in the inner loop is beneficial for finding high-quality solution for each instance due to its strong global search ability. The overall architecture is illustrated in Figure
4.1.1 INNER-LOof OPTIMIZATION
In our approach, inner-loop optimization updates instance-specific parameters. For a training instance Gr. its model is initialized to the meta-model, i.e., 0 <- On- Then, K steps of ES gradient updates applied to obtain a model o,*), At each step k, a population of μ individuals is sampled from the parameter distribution (o. and their fitness F; evaluated. The ES gradient computed according to Algorithm|] 1 and used to update the instance-specific model 0,.
In standard NES (Wierstra al., 2014, Salimans et al., 2017), fitness F; is computed by a single policy rollout. However, this single-sample strategy is unstable in complex scheduling problems due to environmental stochasticity. To address this, we propose a parallel sampling mechanism, generating L solutions per individual by sampling its policy network 8 + dee, for more accurate gradient estimation.
MC averaging estimation. Our first method computes the mean of the L objective values to obtain the individual's s fitness, meaning that we use the following equation for fitness computation in Line in Algorithm] 1
1 L Σ p (ee,s))
where r2!0( (0 + oe;) is the objective value (i.e., makespan) of the I-th solution. This simple strategy effectively reduces variance in gradient estimation and improves training performance. However, when the policy is used in the sampling mode, MC averaging is less effective since the mean value cannot reflect the sampling result. To further enhance performance under sampling, design another ES gradient estimator that incorporates sampling information into the inner-loop optimization process.
MC best-sample estimation. To better align with the sampling mode commonly used for NCO policies, we replace the averaging of objective values across multiple feasible scheduling solutions with best-sample-basee strategy. Specifically, during each inner-loop update, we exclusively utilize the best objective value among the L solutions of each individual its fitness estimate for ES gradient computation. The fitness F; of each individual in Algorithm iis then computed as follows:
F = F(O + +Ee)) min
(=1...,L {
F (1) + +E.)
(3)
4.1.2 OUTER-LOOP OPTIMIZATION
In Algorithm 2 the aim of the outer-loop optimization is to update the meta-pant pettaparameters that govern the model's 's ability to rapidly adapt. Un- like conventional dist-bistritution- based meta-learning approaches that primarily focus on task-level gener- alization, our method shifts the op- timization emphasis toward improv- ing the model's 's adaptability to individ- ual instances. The outer loop aggre- gates adaptation outcomes from the inner loop across multiple training in- stances to optimize the shared meta- parameters, aiming to maximize the expected performance of the model af- ter instance-lewe fine-tume min-tunnng. Specifi- cally, for each training instance (treated pseudo-test instance) we optimize the neta-model by
Algorithm Meta-Level ES Gradient Estimation
Require: Population size μ., adaptation steps K, adaptation step size a, noise standard deviations d, Sample . ~ (0, I). .1.............................. .. (O,
2:
for j 1, 1,...,,44 do
3:
for 1: = =
4:
for i = 1, 1....., do F(g.)) = F(0 + ngg + oE4)
5:
6:
end for
7:
0k + n19 + 0--1 + n9; - a pa Σi F(g)
8:
end for
9: F, F(Ox n99))
10: end for
11: return Mn プド F5 F5.
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maximizing the performance after K steps of ES gradient-based adaptation. The computation of the meta ES gradient on a single instance is detailed in Algorithm 3 During training, the meta- veeta-arameters updated once after simulating the K-step adaptation process a mini-batch of B instances {Gr) as in Line 9 of Algorithm
First-order Approximations. In our framework, DFO removes the need for second-order derivatives, but both optimization loops remain population-based. introducing perturbations to form a distribution of candidate solutions. This differentiation, increasing the variance of ES gradient estimates. Moreover, given that derivative-free methods are intrinsically slower converge. the resulting computational overhead renders the training process practically infeasible in real-world applications.
mechanism resembles higher-order
We address this challenge by utilizing the first-order approximation scheme in FOMAML (Finn et 2017), which explicitly discards second-order terms and updates the meta-parseeters using only first-order gradients. Although originally developed for gradient- methods, it can be analogously applied to our derivative-free setting. Specifically, the first-order approximation of meta-model update can be expressed as:
0o 0 0o
B B
Σ
Vo, Be
[F' (o!*) oe c.)]
We use the above equation to replace the meta-model update in Line 9 of Algorithm 2 The only notation difference is that replace the full meta-gradient Vo, in Line 9 of Algorithm 2 with the instance-specific gradient Ve,, which is the essence of first-order approximation and significantly reduces the computational overhead.
4.2 GPU-BASED PARALLELIZATION
Population-based DFO methods such as ES are naturally parallelizable. However, traditional imple- mentations (Salimans al., 2017 Song et al. 2020) typically rely on CPU clusters and distributed schedulers such as Dask or Ray for fitness evaluation and population evolution, limiting their ability to fully utilize the parallization power of GPU. The associated communication and synchronization overheads also become major performance bottlenecks.
In this paper, we develop a GPU-based framework for population-level parallel fitness evaluation, as illustrated in Figure 2 which significantly improves the computational efficiency of ES by parallelizing Line 2-4 in Algorithm [ on GPÚ. The core idea to replace the conventional per- individual evaluation scheme with batch inference mechanism applied the entire population. We adapt the neural network architecture to process all individuals' input states in a single forward pass, thereby achieving model-level full parallelism. Speficially, we construct a perturbation matrix [ Ey] E R where each column E, _N(0, I) is the Gaussian noise added to the mean parameter vector 0 (of dimension d) generate the parameter vector of the i-th individual, 0,. We then construct a population network, represented in matrix form as:
(5) The population network performs parallel forward propagation to produce action policies for all individuals, denoted πI F(),S), where S $,,-..; sn] is the collection of input states for each individual, and 11 is the resulting policy output matrix. The network is used solely for efficient inference and does involve backpropagation.
θ 0 +
Furthermore, we vectorize the FJSP environment to support population-Ievel parallel interactions. We construct vector of environments & Er, ..., Eu), which receives the full batch of policy outputs II and synchronously executes the interaction process for each individual in the environment and collect the corresponding individual fitness as R (II), where R [ri,..,, rp] E Rº denotes the fitness vector of the population. Then we can compute the ES gradient in Algorithm as:
1
V Ee [F(( o=))
- Re
(6)
μo
which is subsequently used update the model parameters.

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Synchronized Update in Paraliel Environment
Stile Mattix (lob Machhnno
Parullelism
Layer'
Action,
Layer
Action Mauix
2
GPU-paralki
Actiong
Environment
Perturbation Matrix
Population Network
Policy Nerwork
Layar
Action
Fitness Values
Gradient Estimation
Figure 2: Overview the GPU-based parallel ES framework
With this mechanism, the entire fitness evaluation process is offloaded to GPU, substantially reducing CPU dependence and data transfer overhead, while improving overall efficiency and system through- put. We will demonstrate the advantage of this implementation in the experiments. Remarkably, approach even outperforms the speed-fooc focused EAS algorithm on most instances.
EXPERIMENTAL RESULTS
In this section, we perform evaluation on FISP which harder than JSP. We apply our method to the state-of-the-art FJSP PDR learning model, Dual-Attention Network based Reinforcement Learning (DANIEL) (Wang et al. 2023). In Appendix D we also provide an evaluation using the self-supervised training JSP model, Self labeling Pointer Network (SPN) (Corsini et al. 2024). to
demonstrate the versatility of our method. Our code available at: https: http://ggttuu..... com/
1m/DFF-META
5.1 EXPERIMENTAL SETUP
Dataset. Following Wang et al. 2023). we generate six group of synthetic FJSP instance of sizes
10 5, x 10, 5. 20 10, 30 10. and 40 10 for training and evaluation. Two types of datasets are generated: SD1 (following (Song et al. 2023)) allows each job to contain variable number of operations, increasing structural diversity; while SD2 (adopted from (Wang et al. 2023) fixes the number of operations per job but significantly broadens the processing time range across alternative machines for each operation, thereby increasing scheduling complexity. Model training is conducted on the four smaller sizes, while the two larger sizes reserved to assess generalization capability. For testing, besides synthetic datasets, we also use four public benchmarks that differ substantially in size (ranging from 10 x 5 to 30 x 10) and distribution, to assess -ross-iistibuttion generalization ability, including the ten classic mk instances (mk01 to 10) from (Brandimarte) 1993), and three groups of la instances from (Hurink al. 1994), namely rdata, edata, and vdata, each containing 40 instances. More details can be found in Appendix A
Training setup. For our method, set the population size μ 100, noise standard deviation = 0.2. Step size of the inner and outer loops are set to a = 3 = 5 x 10-2 Each epoch generates B = 20 training instances, and the inner loop executed for K 3 steps per instance. The training runs for a total of 200 epochs. During the inner-loop adaptation phase, for the two proposed gradient estimation methods, the number of parallel samples per population member is
20 for the MC averaging and T. = 100 for the MC best-sample. All hyperparameters are tuned on the smallest instance size (10
and kept fixed across instance sizes. All experiments are conducted on workstation with Intel Core 19-9900K CPU and single NVIDIA RTX 4090 GPU.
Baselines. We use four types of baselines for comparison: 1) Google OR-Tools (Da Col Teppan 2019), high-performing exact constraint optimization solver with 3600 seconds run time limit; 2)
The best-performing manual PDR Most Work Remaining (MWKR) (Brandimarte 1993 Montazeri
& Van Wassenhovel 1990) reported in (Song al. 2023 Wang al. 2023): ) 3) the original
DANIEL model (Wang et al.) 2023); and 4) two test-time fine-tuning strategies, Active Search
(AS) (Bello et al. 2016) and Efficient Active Search (EAS) (Hottung et al. 2021b). For the latter,
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Table 1: Performance evaluation on synthetic test sets. Sizes marked with R were unseen in training.
10x 5
20 x Obi (Gap)
15 10
20 10
30x 10"
40%<
ORj, (Gup)
Time
Time
ORj, (Capp)
Timee
Obj. (Gap)
Time
Obj. (Gup)
Time
Obj.. (Guup)
Time
Citnels
3 coutee)
0.91
158.3
(0006))
1h
145.0 (0,00%)
995
196.2
(0.00%5
1h
(0-00963 225.8
1h
367 (0.00%)
1h
MWKR
(17.88))
0.16c
206.
(11 W6B)
0. 97x
171.1
1999)
0.50c
216.1
(10.145)
0.71+
(18.489)
1.09c
(12.992)
1.50
Samplig Greedy
DDNNELL
50%)
0.4%
197.
96361
0.94s
161.3
(10,88)))
1:35s
198.5
(1.16)
1.85
280.5 (2.0016)
2.766
371.5(0.155
3.77
A8
104. (399))
0.27m
1999
(2.97%)
0.47m
(7.99)))77))
0.85m
194.2
(-0.66%)
1.1mm
378.4 (0-9418)
117m
(0) 1993
2:65m
SDI
FAS
(7.66))))))
0.27m
1040 (3028))
0.42m
4 1758)
194.5
(40.876)
1:10m
578.5 (0-088))
178m
168
2.55m
103.1 (7.06%)
0.14m
190.2
(1.01
0.31m
153.6 (5.11%)
0.55m
192.1 (.2.0%%
0.79ts
275.3 (4.188)
1.02m
364.6
2.55m
(-1
129.3 2293)
12.97%
170.5
(4.36%
0.31m
191.5 (1.20%)
(3.779))
1.2m
192.5
277.9 (0.765)
3.42m
368.5 (0.2515)
5.12m
EAS
(4.15))
0.25m
191.9
(3.70
1.10m
19251
1.996)
277.7 (0.0916)
2:45m
368.5 (0.3515)
4.7lm
(3.32%)
0.15m
185.6
(8.16%)
149,0 (3,05%)
0.52m
189.1
(-362%)
273.4 (0887%)
(1.14%)
4.55m
Ohtocla
(0.020)
0.51m
507.7
(0.005)
1h
(2.99))
0.77
451.9
(0.00%)
1h
660.2 (0.005)
1h
938. 3 (0.00%)
tb
MWKR
549.
(68.856)
0.166
1026.3
(71.706)
0.335
830.1
(120.2716)
0,52
1041.1
(125.445)
0UTES
1540. (130.29%)
1,0%5
2006.5
(117.44)
1.50s
Sampling Greedy
DANIEL
AS
408 408
(220120.175)
0.25m
0.4
671
(7 (7.85))
(12.29))
0.47m
0.00s
981 (56 865))
S52.2 (43 (95))
0.86m
1:36s
610.1 (32.09%))
571.5
1.22m
1.20
(15758))
397.6
(10:225))
2.17m
2.55
927.6 (-) 14%)
962.6 2.55
2.6mm
3.44c
502
EAS
(86.5(16.655)
0.22m
640.8 (7.22%)
0.45m
542.7
(65.32%)
0.72m
598.
(252219)
102m
739.5 (10.118)
1.75m
922.9
(.645)
2.5m
Dors
369.2 (1.18%)
0.13m
(4.33%)
0.28m
551.4
(41.00%)
0.55mm
$59.9
(21.22)
081m
732.6
(9.87%)
1.55m
926.1
(1.94%)
2.55m
DANTEL
(12.45))
0.88
1.84
(S8.459)
3.83
552.6 64%)
5.97
(8 58%)
12.17
914.0 99%)
21.0%
AS
356.1 (9.1716)
0.23m
620.7
(3.85%)
5026 (33.3516)
1.17m
557.1
(16 (8%)
1,78m
712.8 (6.52%)
900.8
5.1lm
154.9 (8.906)
0.25m
619.4
500.
540.8
1,62
714.9 (6.836)
900. (:3/225)
4.0mm
547.7 (6.5%%
0.17m
0.8mm
$29.9 (14.72%)
(5.10%)
(.8.26))
Table 2: Generalization performance evaluation on public benchmark datasets.
(rdata)
(edata)
(ydata)
Obj. (Gap)
Time
Obj. (Gop)
Time
Obj. (Gap)
Time
Obj. (Gap)
Time
Ortools
173.9 (0.00%)
0,51h
933.4 (0.00%)
0.71h
1026.9 (0.00%)
5.41m
920.6 (0.00%)
0.75h
MWKR
202.2 (16.27%)
0.49s
1052.8 (12.79%)
0.52I
1218.8 (18.69%)
0.52s
952.0 (3.41%)
0.52s
DANIEL
185.7 (6.79%)
1,29s
1031.6 (10.52%)
1,37
1194.9 (16.366)
1.36s
944.9 (2.64))
1.37s
SD1 10 5 model
Greedy
AS
182.6 (5.00%
0.85m
1008.2 (8.01%)
0.90m
1159.7 (12.93%)
0.91m
936.7 (1.75%)
0.91m
EAS
182,7 (5.06%)
0.73m
992.0 (6.28%)
0.80m
1139.9
(11.00%)
0.79m
930.3 (1.05%)
0.79m
Ours
182.3 (4.83%)
0.61m
982.8 (5.29%) )
0.67m
1120.8 (9.14%)
0.68m
928.1 (0.81 %)
0.67m
Sampling
DANIEL
(3.97%)
4.13s
978.3 (4.97%)
4.71s
1122.6 (9.25%)
4.73s
925.4 (0.53%)
4.77s
AS
(3.15%)))
1 1,30m
970.5 (4.00%)
1.42m
1107.6 (8.60%)
1.40m
923.0 (0.25%
1.4m
EAS
(3.15%)))
1,20m
969.5 (4.1 1%
1,29m
1101.8 (7.32%%
1.38m
922.7 (0.23%)
1.32m
Ours
(2.25%%)
0.95m
963.9 (3.28%) %)
1.13m
1086.0 (5)
1.13m
921.9 (0.14%)
1.07m
SD1 15 x model
Greedy
DANIEL
(5.02%)
(6.066)))
0.85m
1.30s
1040.0 (1 193%)
1014.5 (8.63%)
0.89m
1.36s
1152.9 (12.30%)
1175.5 (14.88%)
0.93m
1.38s
934.8 (2.29%)
948.7 (3.05%)
0.91m
1.37s
EAS
18.22 %)
0.73m
1004, (7.57%%
0.81m
1 1140.0 (1 .005%)
0.79m
932.0 (1 1.4%))
0.79m
Ours
181.5 (4.38%)
0.61m
984.9 (5.54%)
0.67m
1131.0 (10.18%)
0.67m
928.7 (0.88%)
0.67m
Sampling
DANIEL
180.9(3.9992)
4.08s
983.3 (5.3%)
4.73
11 (8.73%
4,70s
925.7 (0.55%)
4.75s
AS
(2.9992..11))
1.31m
971.2 (4.25%)
1.42m
1109.1 (8.97%)
1.42m
922.9 (0.24%)
1.40m
EAS
178.7 (2.85%)
1.22m
971.6 (4,29%%
1.29m
1106.5 (8.74%)
1.36m
922.5 (0.22%)
1.32m
Ours
(2.03%%
0.95m
967.3 (3.64%)
1.13m
1094.1 (7.31%)
1.15m
921.7 (0.12%)
1.07m
we implement EAS-EMB for comparison due to its superior performance in scheduling problems as reported in (Hottung et al., (021bb. For fair comparison, all fine-tuning methods (AS, EAS and ours) are performed with a fixed number of K 10 adaptation steps per test instance. We use the solutions generated by OR-Tools as reference to compute the objective gap of each method.
5.2 PERFORMANCE EVALUATION
Results synthetic data. Table D reports the average makespan, relative optimality gap, and average run time for solving an instance across all groups Note that to assess the generalization ability to larger problem sizes, we solve the unseen 30 10 and 40x 10 instances using models trained on the closest scale, 20 x 10. For all neural methods, we report their performance under the greedy and sampling mode (100 solutions as in (Song et al. 2023 Wang et al., (003)). For our method, we use MC averaging and MC best-sample as the inner-loop gradient estimator for the greedy and sampling mode, respectively. Clearly, all neural methods significantly outperform the MWKR rule. AS and EAS can improve the original DANIEL model by test-time adaptation, and the latter is better in most cases especially on SD2. Our method consistently outperforms all baselines across all settings, because the meta-learning scheme explicitly considers the fine-tuning process and the trained meta-model provides a good start point for instance-wing adaptation. Moreover, benefiting from its gradient-free nature, method allows full-parameter adaptation without backpropagation, achieving superior efficiency and surpasses EAS in terms of both solution quality and run time. In Appendix B we provide results much larger 50 20 and 100 x 20 instances,
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Table 3: Effectivvnesss of derivative-free meta-learning.
Table 4: Inference time of GPU and CPU implementation.
10 x 5
20 x 5
15 10
20 x 10
Instance
GPU (Ours)
CPU (Ray)
DANIEL
25.20%
12.27%
56.86%
32.09%
AS
20.17%
7.87%
47.97%
23.73%
5
0.5s
EAS
16.65%
7.22%
45.32%
23.21%
20 x 5
1.1s
29.8s
ES Ours
14.38%
5.02%
44.81%
24.20%
15 X 10
2.4s
62.
13.18%
4.54%
41.00%
21.22%
20 10
3,8s
122.3s
Table 5: Comparison with FOMAML (-G: greedy; -S: sampling).
10 x Obj. (Gapp)
20 X (Ciap)
15 x 10 (Gap)
20
30 x Obj. (Capp)
40x Obj. (Gapp)
Time
Obj.
Time
Obj
Time
Obj. (Grp))
Time
Time
Time
ROMAMI-G 394.4 (20.91%) 0.14m. 646.6 (8.188) 0.30m 557.1 (47.83%) 0.51m 583.8 39%) 0.71m
(13.61) 1:3m 949.3
1.63m
(13.18%) 0.13m
(41.00%)
732.6 (9.47%) 1.5mm
(-194%) 2.55m
FOMAML-S
13%)
0.16m
625.2 (4.61) 0.37m
$14.0 36.366)
0.68m
36%)
1.01m
733.3 (888))
1.83m 928.7 (2%)
2.95m
Oar-S
347.7 (6.59%) 0.17m 607.9 (1.71
%) 0.38m 499.2 (32.45%) 0.83m $29.9 (14.72%)
1.30m. 703.3 (5.10%)
2.67m 889 (-5.20%) 4.52m
Results on public benchmark instances. We further evaluate the cross-distribution generalization ability of method on the widely used public benchmarks. We follow (Wang et al. 2023) and use the meta-models trained on SD1 with sizes 10 x 5 and 15 x 10 for evaluation. As shown in Table method consistently demonstrates strong performance observed the synthetic data. It outperforms all baseline methods across all settings, with particularly notable improvements on the edata task, highlighting its robust generalization ability in handling out-of-distribution instances. Further statistical analysis can be found in Appendix
5.3 ANALYSES
Effectiveness of derivative-free meta-learning. Here perform more fine-grained analysis on the two major parts of our method, i.e. the DFO method and the meta-learning framework. Specifically, on SD2, we train and fine-tune the DANIEL model using the same DFO procedure as in our method, namely ES. As shown in Table 3 this method can already surpass existing gradient-bar pradient-based training and fine-tuning methods, showing the effectiveness of DFO. Building upon this, the integration of meta-learning further enhances the overall fine-tuning performance.
Comparison with standard meta-learning. We present direct comparison between the instance- wise adaptive meta-learning framework proposed in this work and the standard gradient- based meta-learning method, FOMAML, on the challenging SD2 dataset. All experiments were conducted under identical conditions, including the same model architecture, initialization, training epochs (200) and fine-tuning steps (10) to ensure fairness. As shown in Table under the same setting, our method consistently outperforms FOMAML across all instance scales with comparable inference speed. Training cost comparison with FOMAML is presented in Appendix C due to the space limit.
Analysis of inner-loop gradient estimators. Next, verify the effectiveness of our two inner-loop gradient esti- mators proposed in Section I.I.I MC averaging and MC best-sample. The evaluation is conducted on 10 x 5 in- stances from SD2, and the standard NES implementation (the single-sample estimator), AS and EAS are also incor- porated for reference. All methods are evaluated under the sampling mode. In Figure 3 we plot the average objective value for each the K 10 fine-tuning steps. As shown in Figure 3 the single-sample strategy performs only par with AS and EAS. With our two MC strategies, perfor- mance of the meta-model is significantly boosted, and MC best-sample demonstrated the strongest adaptation ability, showing the effectiveness of our novel design.
EAS
Average Dijective
Single-Samph
MC Averaging
370
MC Best-Semple
365
260
355
Figure 3: Comparison of different inner- loop gradient estimators.

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Published as a conference paper at ICLR 2026
Analysis of inference efficiency. We compare the efficiency of our GPU-based implementation with the Ray-based CPU multithreading strategy commonly used in existing DFO methods. Specifically, measure the run time required to complete one forward pass for evaluating the fitmess of 100 individuals. The Ray-based implementation utilizes all 16 available threads on our machine to maximize CPU parallelism. As shown in T our GPU implementation is much faster than the CPU-based method across various scales. Furthermore, as the problem scale increases, the advantage of our implementation becomes even more prominent.
CONCLUSION AND FUTURE WORK
While deep reinforcement learning has been successfully applied in complex scheduling problems, the per-instance solving performance is still far from optimality. In this paper, we propose derivative- free meta-learning framework to enhance the ability of learming-basee scheduling models in adapting individual instances at time. The trained meta-model provides good start point for instance- wise fine-tuning, and the strong empirical performance of our method is validated on both FISP (reinforcement learning) and ISP self-supervised learning). One limitation of our method that derivative-ffre optimization is known to be relatively slower than gradient-based methods. In the future, we will investigate more effective ways to improve training efficiency.
ACKNOWLEDGMENTS
This research was supported by the National Natural Science Foundation of China under Grants 62473233, U23B2059, and the Guangdong Basic and Applied Basic Research Foundation under Grant 2025A1515011700
REFERENCES
Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial optimization with reinforcement learning. arXiv preprint (rXXiv:611.0994// 2016.
Yoshua Bengio, Andrea Lodi, Antoine Prouvost. Machine learning for combinatorial optimization: methodological tour d' horizoo.. European Journal of Operational Research, 290(2):405 2021.
Paolo Brandimarte. Routing and scheduling in flexible job shop by tabu search. Annals Operations Research, 41(3):157-183, 1993.
Ruiqi Chen, Wenxin Li, and Hongbing Yang. A deep reinforcement learning framework based attention mechanism and disjunctive graph embedding for the job-shop scheduling problem. IEEE Transactions on Industrial Informatics, (9(2):1322-133) 2022.
Andrea Corsini, Angelo Porrello, Simone Calderara, and Mauro Dell' Amico. Self-labeling the job shop scheduling problem. Advances in Neural Information Processing Systems, 2024.
Giacomo Da Col and Erich C Teppan. Industrial size job shop scheduling tackled by present day cp solvers. In Principles and Practice of Constraint Programming: 25th International Conference, CP 2019, Stamford, CT, USA, September 30 Octobbrr 2019, Proceedings 25, pp. 144 160. Springer, 2019.
Zhaohan Feng, Ruiqi Xue, Lei Yuan, Yang Yu, Ning Ding, Meiqin Liu, Bingzhao Gao, Jian Sun, Xinhu Zheng, and Gang Wang. Multi-agent embodied Al: Advances and future directions. Science China Information Sciences, 69(5):151202 2026.
Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. International Conference on Machine Learning, pp. 1126-1135. PMLR, 2017.
André Hottung. Bhanu Bhandari, and Kevin Tiemey. Learning latent search space for routing problems using variational autoencoders. In International Conference on Learning Representations, 2021a.
10

---

Published as a conference paper at ICLR 2026
André Hottung, Yeong-Dac Kwon, and Kevin Tiemey. Efficient active search for combinatorial optimization problems. In International Conference on Learning Representations, 2021b.
Johann Hurink, Bernd Jurisch, and Monika Thole. Tabu search for the job-shop scheduling problem with multi-purpose machines. Operations-eesaach---peruuu 15:205-215, 1994.
Maziyar Khadivi, Todd Charter, Marjan Yaghoubi, Masoud Jalayer, Maryam Ahang, Ardeshir Shojaeinasab, and Homayoun Najjaran. Deep reinforcement learning for machine scheduling: Methodology, the state-of-the-art and future directions. Computers & Industrial Engineering, pp- 110856, 2025.
Louis Kirsch, Sebastian Flennerhag, Hado Van Hasselt, Abram Friesen, Junhyuk Oh, and Yutian Chen. Introducing symmetries to black box meta reinforcement learning. In Proceedings of the AAAI Conference Artificial Intelligence, volume 36, pp. 7202-7210, 2022.
Funing Li, Sebastian Lang, Yuan Tian, Bingyuan Hong, Benjamin Rolf, Ruben Noortwyck, Robert Schulz, and Tobias Reggelin. A wransformerbbaee deep reinforcement learning approach for dynamic parallel machine scheduling problem with family setups. Journal of Intelligent Manufac- turing, pp. 1-34, 2024a.
Yang Li, Jinpei Guo, Runzhong Wang, and Junchi Yan. T2t: From distribution learning in training to gradient search in testing for combinatorial optimization. In Advances in Neural Information Processing Systems, 2023.
Yang Li, Jinpei Guo, Runzhong Wang, Hongyuan Zha, and Junchi Yan. Fast 12t: Optimization consistency speeds up diffusion-based caining-to-testing solving for combinatorial optimization. In Advances in Neural Information Processing Systems, 2024b.
Sahil Manchanda, Sofia Michel, Darko Drakulic, and Jean-Mare Andreoli. On the generalization of neural combinatorial optimization heuristics. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 426-442. Springer, 2022.
Nina Mazyavkina, Sergey Sviridov, Sergei Ivanov, and Evgeny Burnaev. Reinforcement learning for combinatorial optimization: A survey. Computers & Operations Research, 134: 2021.
Pinedo Michael. Scheduling, theory, algorithms and systems. ISBNO-13-7067574 1995.
Mr Montazeri and LN Van Wassenhove. Analysis of scheduling rules for an fms. The International Journal of Production Research, 28(4):885- 782- 1990.
Junyoung Park, Jaehyeong Chun, Sang Hun Kim, Youngkook Kim, and Jinkyoo Park. Learning to schedule job-shop problems: representation and policy learning using graph neural network and reinforcement learning. International Journal of Production Research, 59(1 (0311):3300-377 2021.
Jonathan Pirnay and Dominik G Grimm. cel--improvement for neural combinatorial optimization: Sample without replacement, improvement. Transactions on Machine Learning Research, 2024a.
Jonathan Pirnay and Dominik G Grimm. Take step and reconsider: Sequence decoding for self- improved neural combinatorial optimization. In European Conference on Artificial Intelligence (ECAI), 2024b.
Ruizhong Qiu, Zhiqing Sun, and Yiming Yang. Dimes: A differentiable meta solver for combinatorial optimization problems. In Advances in Neural Information Processing Systems, 2022.
Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever. Evolution strategies as scalable alternative to reinforcement learning. arXiv preprint arXiv:/703.03380 038644 2017.
John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint 1707001707.06347, 2017.
Jiwoo Son, Minsu Kim, Hyeonah Kim, and Jinkyoo Park. Meta-sage: Scale meta-learning scheduled adaptation with guided exploration for mitigating scale shift on combinatorial optimization. In International Conference Machine Learning, pp. 32114-11222222 32210. PMLR, 2023.

---

Published as a conference paper at ICLR 2026
Wen Song, Xinyang Chen, Qiqiang Li, and Zhiguang Cao. Flexible job-shop scheduling via graph neural network and deep reinforcement learning. IEEE Transactions on Industrial Informatics, 19 (2):1600-1610, 2023.
Xingyou Song, Wenbo Gao, Yuxiang Yang, Krzysztof Choromanski, Aldo Pacchiano, and Yunhao Tang. Es-maml: Simple hessian-free meta learning. In International Conference on Learning Representations, 2020.
Eric Taillard. Benchmarks for basic scheduling problems. European Journal of Operational Research, 64(2)277-285, 1993.
Florent Teichteil- Beichteil-Königsuuchh Königsbuch, Guillaume Povéda, Guillermo González de Garibay Barba, Tim Luchterhand, and Sylvie Thiébaux. Fast and robust resource-constrained scheduling with graph neural networks. In Proceedings of the International Conference Automated Planning and Scheduling, volume 33, pp. 623-633, 2023.
Haoyu Peter Wang and Pan Li. Unsupervised learning for combinatorial optimization needs meta learning. International Conference on Learning Representations, 2023.
Runqing Wang, Gang Wang, Jian Sun, Fang Deng, and Jie Chen. Flexible job shop scheduling via dual attention network-based reinforcement learning. IEEE Transactions on Neural Networks and Learning Systems, 35(3):3091-3100 2023.
Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, and Jürgen Schmidhuber. Natural evolution strategies. The Journal of Machine Learning Research, 15(1 15(1):949-99)) 2014.
Cong Zhang, Wen Song, Zhiguang Cao. Jie Zhang, Puay Siew Tan, and Xu Chi. Learning to dispatch for job shop scheduling via deep reinforcement learning. In Advances in Neural Information Processing Systems, 2020.
Cong Zhang, Zhiguang Cao, Wen Song, Yaoxin Wu, and Jie Zhang. Deep reinforcement learning guided improvement heuristic for job shop scheduling. In International Conference on Learning Representations, 2024a.
Cong Zhang, Zhiguang Cao, Yaoxin Wu, Wen Song, and Jing Sun. Learning topological representa- tions with bidirectional graph attention network for solving job shop scheduling problem. In The 40th Conference on Uncertainty in Artificial Intelligence, 2024b.
Jianan Zhou, Yaoxin Wu, Wen Song, Zhiguang Cao, and Jie Zhang. Towards omni reni-generalizzale neural methods for vehicle routing problems. International Conference on Machine Learning, pp. 42769-42789 PMLR, 2023.
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Table 6: SDI instance generation distributions.
Table 7: SD2 instance generation distributions.
Size(n m)
[0:]
MMal
Pij
Size(n m)
|0:1
|Maa]
Piyk
10 x
U(4, 6)
U(1,5) 5)
U(1 20)
10 x 5
5
U(1,5 5)
U(1, 99)
20 x
U(4,
U(1.5)
U(1
20 x
5
U(1.5)
U(1, 99)
15 10
U(8. 12)
U(1, 10)
U(1.
15 10
U(1, 10)
U(1, 99)
20 x 10
U(8. 12)
U(1, 10)
U(1. 20)
20 x 10
10
U(1, 10)
U(1, 99)
30 x 10
U(8, 12)
U(1, 10)
101,200
30 x 10
10
U(1, 10)
U(1, 99)
40 10
U(8, 12)
U(1, 10)
((1,20)
40 10
10
U(1, 10)
U(1, 99)
Table 8: Generalization to very large FJSP instances. For neural methods, values outside (inside) parenthesis are greedy (sampling) results.
50 20
100 20
Ortools DANIEL
972.8
1737.1
1013.9 (959.6)
1649.5 (1612.9)
AS EAS
960.4 (930.7)
OOM
970.1 (942.3)
OOM
Ours (Zero-shot)
949.2 (904.9)
1609.1 (1562.2)
Ours (Fine-tune)
886.5 (875.4)
1510.5 (1492.2)
A FJSP DATASETS
In this study, we consider six synthetic instance sizes for training and testing the FJSP. For the SD1 dataset, the instance generation process follows the classical method proposed in (Brandimarte 1993). Specifically, for each instance size, the number of operations |O,| in each job J,, the number of compatible machines |Ma for each operation Oij. and the average processing time pij of each operation across its compatible machines are all independently sampled from uniform distributions defined in Table Then, the actual processing time Pijk of operation Oty on specific compatible machine M₁ E Mi is sampled from bounded uniform distribution centered around the average processing time, i.e., Pijk (0.8p 1.2pt)). contrast, for the SD2 dataset, the number of operations in each job is set equal to the total number of machines in the shop. The processing time Pijk each operation on each compatible machine is directly sampled from a wider uniform distribution, resulting in greater variability in operation durations. The detailed parameter settings provided in Table
B GENERALIZATION TO LARGE PROBLEMS
To further examine scalability at inference time, we conducted an experiment on very large FISP instances of sizes 50 x 20 and 100 x 20 using SD2 distribution. For our method and DANIEL, we use the model trained on 20 10 instances. As shown in Table 8 our meta-model shows strong generalization performance. It significantly outperforms DANIEL trained on the same size in both greedy and sampling modes. Its zero-shot performance already exceeds the 1-hour results Ortools, and fine-tuning further boosts the performance. Notably, AS/EAS failed the largest 100x20 instances (out-of-memony since their gradient-based fine-tuning requires large GPU memory to store gradients. In contrast, gradient-free approach is much more memory friendly on these very large problems.
C TRAINING COST COMPARISON
To assess the training efficiency of our method, we provide a detailed comparison of the training cost against the FOMAML baseline, shown in Table The comparison considers training time (in GPU hours) and GPU memory usage over 200 epochs. Our method incurs approximately 1.5-2 times higher training time and GPU memory usage compared to FOMAML. This increase is attributed to the nature of ES, which performs a population based search and evaluates multiple perturbations during each update. Nevertheless, we believe this additional training cost is modest and reasonable,
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Table 9: Training Cost Comparison with FOMAML on SD2.
10 x 5
20 x 5
15 x 10
20 10
Training Time
2.1h
4.5h
7.2h
11.2h
Ours
GPU Memory Usage
1.3G
1.6G
2.0G
2.2G
Training Time
1.5h
2.9h
3.7h
5.5h
FOMAML
GPU Memory Usage
0.9G
1.0G
1.0G
1.2G
Table 10: JSP Experiments Taillard's benchmarks. Instance sizes marked with were not seen
during training.
15
15
20 x 15
20 x 20
30 x
15*
30 x 20"
Gap
Time
Gap
Time
Gap
Time
Gap
Time
Gap
Time
SPN
16.86%
0.47s
16.13%
0.67s
19.01% % 0.87s
21.16%
1.06s
22.03%
1.43s
Greedy
AS
11.87%
0.21m
14.00%
0.28m
14.00%
0.39m
16.76%
0.43m
17.87%
0.58m
EAS
11.94%
0.16m
13.26%
0.23m
13.98%
0.28m
16.10%
0.32m
17.76%
0.43m
Ours
10.83%
0.13m
12.73%
0.19m
11.78%
0.26m
15.08% %
0.32m
17.08%
0.42m
Sampling
SPN
8,52%
0.53s
10.31%
0.71s
11.51%
0.96s
13.57%
1.20s
15.91%
1.57s
AS
7.54%
0.23m
9.24%
0.29m
9,43%
0.38m
12.38%
0.43m
14.52%
0.57m
EAS
7.06%
0.17m
9.28%
0.24m
9.84%
0.30m
12.21%
0.33m
14.33%
0.45m
Ours
6.31% %
0.15m
8.76%
0.21m
9.24%
0.28m
11.80%
0.33m
14.07% %
0.43m
since training is offline and the final solution quality during inference time is significantly better than the baselines.
D JSP EXPERIMENTS
In this section, we apply our method to a self-supervised learning model for JSP, the Self-labeling Pointer Network (SPN) (Corsini et al. 2024), to evaluate its effectiveness under different learning paradigm and model to further validate its generality and robustness. SPN is self-supervised neural scheduling framework tailored for JSP. A key advantage of SPN that it eliminates the need for external supervision Pointer Network to construct scheduling solutions by sampling multiple candidate sequences for each instance. It then selects the one with the lowest makespan as a pseudo-label and optimizes the model via cross-entropy loss. This iterative process progressively improves the scheduling quality without relying on ground-truth labels.
reinforcement learning signals. During training. SPN utilizes
Dataset. To ensure a fair comparison, we train both SPN and our meta-model on the same dataset. Following the protocol in (Corsini al. 2024), adopt a fixed training dataset composed of six instance sizes (n x m): 10. 15 x 10. 15 x 15, 20 x 10, 20 x 15, 20 x 20), with 500 randomly generated instances per size, totaling 3,000 training instances per epoch. For evaluation, we assess the generalization ability of the models using five instance sizes selected from Taillard's benchmark
set (Taillard] 1993). ranging from 15 15
x 20.
Training Setup. For our method, the population size is set to μ4 100 and the noise standard deviation to 0.04. The step sizes for both the inner and outer loops are set = f = 10-3 The batch size is fixed to B = 10, consistent with the SPN training configuration. For each instance, we perform K = 3 inner-loop adaptation steps. Given the fast convergence behavior of our method, train for total of 10 epochs, while the SPN baseline is trained for 20 epochs as in the original paper. During the inner-loop adaptation phase, we set the number of parallel samples L for the two proposed gradient estimators follows: L 20 for the MC averaging, and L = 128 for the MC best-sample, which aligns with the number of sampled solutions 3 used in SPN training.
Performance Evaluation. We compare method against the original SPN model well as the two test-time fine-tuning strategies, AS and EAS. All fine-tuning methods, including AS, EAS, and ours, perform fixed number of K 10 adaptation steps each test instance. To evaluate
14

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Published as a conference paper at ICLR 2026
Table 11: Average percentage gaps and standard deviations (mean + std) on synthetic test sets. Instance sizes marked with were not seen during training.
10 5
20 x
15 10
20 x 10
30 x 10"
40x 10"
Gready
DANIEL
10.80% + 6.59%
1.94% + +0.0..100 1.90%
10.48% + 3.91%
1.17%+2..00 2.0%%
2.07% 1.5-1%
1.17% 1.45%
AS
8.83% = 3.1%
2.97%
+ 0.91%
7.47% 3.38%
-0.76% 2: 1.27%
0.91% 1: 1.27%
0.38% #: 1.28%
EAS
7.68%
2.38%
3.03%
1.00%
7.12%
2.98%
-0.87%
1.37%
0.98%
1.30%
0.33%
1.21%
Ours
7.06%
1.88%
1.01%
+ 0.62%
5.21%
+ 2.55%
-2.99%
+ 1.12%
-0.18%
+: 1.19%
-0.71% + 1.22%
Sampling
DANIEL
5.61%
2.01%
2.30%
0.74%
4.93%
1.92%
-.17%
1.00%
1.23%
1.25%
0.90%
1.21%
AS
4.36%
1.27%
1.70%
0.59%
3.77%
1.76%
1.80%
0.94%
0.76%
1.22%
0.35%
1.29%
EAS
4.15%
1.26%
1.91% +: 0.58%%
3.70%
1.73%
-189%%
0.92%
0.61%%
1.201%
0.35%
1.38%
Ours
3.32%
0.93%
0.16% + 0.11%
2.05%
1.93%
3.62%
+ 0.91%
0.87%
1.18%
11.14% + 1.25%
DANIEL
25.20% + 9.09F
12.27%
1.19%
56 86%
63%
32. 099%
8.06%
15% + 5.51%
2 2.59%% 4.25%
Groedy
AS
20.17% 7.07%
1.87%
3.00%
47 47.97%
8.30%
23.73%
4.77%%
20.2%
3.69%
-1.14%
EAS
16.65% 1. 5.80%
7.22%
1. 2.73%
45.
8. 9.15%
23.21%
4.49%
10.51% 1. 3.01%
-1.64%
2.94%
SD2
+
21.22%
9.47% 2.87%
-1.94% 2.86%
Sumpling
DANIEL
12.12% = 4.19%
5.99% + 2.06%
38. 45%
= 6. 6.31%%
19.641% +: 3.34%
8.38% 1: 2.98%
-2.99%
2 2.84%
9.17% 3.12%
3.85%
1.26%
33.35%
5.68%
28%
2.60%
6.32% + 2.51%
-3.78%
2.60%
EAS
8.50%
3.62%
3.64%
+ 1.32%
32.79%
5.30%
17.06%
+ 2.86%
6.83%
2.36%
-3.72%
-2.77%
Ours
6.59% 2.01%
1.71%
0.99%
32.45%
+ 5.08%
14.72% 3.08%
5.10%
2.01%
-5.20%
+ 2.57%
Table 12: Average percentage gaps and standard deviations (mean + std) on public benchmarks.
mk Gap
la (rdata) Gap
la (edata) Gap
la (vdata)
Gap
DANIEL
6.79% = 3.49%
10.52% = 8.20%
16.36% + 5.62%
2.64% + 3.02%
SDI 10 x 5 model
Greedy
AS
5.00% 2.84%
8.01% + 5.93%
12.93% + 5.65%
1.75% 2.63%
EAS
5.06% = 2.26%
6.28% 5.85%
11.00% + 5.48%
1.05%
1.01%
Ours
4.83% + 1.52%
5.29% + 4.67%
9.14% 5.09%
0.81% + 0.84%
Sampling
DANIEL
3.97% = 1.34%
4.97%
4.56%
9.25% 4.51%
0.53% 0.56%
AS
3.15% = 0.30%
4.12% 4.00%
4.49% 8.60%
0.25% 0.49%
EAS
3.15% = 0.44%
4.11% 3.93%
7.32% 4.26%
0.23% 0.38%
Ours
2.25% = 0.67%
3.28% 3.65%
5.75% + 3.96%
0.14% - 0.37%
SDI 15 x 10 model
DANIEL
6.06% = 2.43%
11.39% = 7.72%
14.88% + 5.41%
3.05% 3.17%
Greedy
5.02% = 1.85%
8.63% 7.10%
12.30% + 5.24%
2.29% 2.07%
EAS
5.01% = 1.83%
7.57% + 5.54%
11.02% + 4.97%
1.24% 1.93%
Ours
4.38% + 1.46%
5.54% + 4.87%
10.18% + 5.20%
0.88% + 1.09%
Sampling
DANIEL
3.99% = 1.62%
5.35% 4.12%
8.73% 4.48%
0.55% 0.61%
2.91% = 0.59%
4.25% + 3.64%
8.97% 4.49%
0.24% 0.50%
EAS
2.85% = 1.19%
4.29% 1 3.68%
8.74% 4.30%
0.22% 0.45%
Ours
2.03% = 0.66%
3.64% + 3.63%
7.31% + 4.90%
0.12% + 0.44%
Table 13:
Average percentage gaps and standard deviations (mean + std) on Taillard's benchmarks. Instance sizes marked with were not seen during training.
15 x 15
20 x 15
20 20
30 x 15*
30 20
SPN
16.86%
2.75%
16.12% 3.55%
19.01%
3.26%
21.15% + 1.84%
22.03% + 2.89%
Greedy
AS
11.86% + 1.94%
13.99% 2.54%
14.00%
2.17%
16.76% + 3.46%
17.87% + 2.64%
EAS
11.94% +0.%%++1. 1.34%
13.20%
+ 2.43%
1.96% 11.78% 2.00%
13.97%
16.10%
2.74%
17.76% + 2.13%
Ours
10.83% + 2.42%
12.73% + 1.50%
15.08% + 3.56%
17.08% + 1.51%
Samphing
SPN
8.52% + 1.99%
10.31% + 2.13%
11.50% + 1.16%
13.57% + 2.90%
15.91% + 1.62%
AS
7.53% +:2 1.44%
9.23% = 1.73%
9.43% + 1.18%
12.38% + 2.92%
14.52% + 1.81%
EAS
7.06% 1 1.77%
9.28% = 1.67%
9.84% 1.05%
12.21% + 2.92%
14.33% 1 1.50%
Ours
6.31% + 2.01%
8.75% + 1.10%
9.23% 1.40%
11.80% + 2.87%
14.07% + 1.56%
performance, we use the best solution value of each instance as a reference and report the objective gap accordingly. Table 10 summarizes the results of all methods under both greedy and sampling modes, where the sampling mode generates 128 candidate solutions per instance following Corsini et al. (2024). As shown in the table, our method consistently outperforms all baselines across both evaluation settings, even when applied in this non reinforrrree learning paradigm. These results highlight the effectiveness, generality, and model-agnostic nature of approach.
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Published as a conference paper at ICLR 2026
Machine
Nochine
Machine
100
200
300
400
300
Figure 4: FJSP Gantt Chart with Makespan of 536 (Before Fine- (ine-Tuning)
Machine
Machine
Mochine 2-
100
200
400
500
Figure 5: FJSP Gantt Chart with Makespan of 459 (After Fine-Tuning)
E STATISTICAL RESULTS
Tables 11 and 13 present the average optimality gaps and standard deviations of all neural-based methods across various instance sizes on the synthetic and public benchmark datasets, respectively. These results confirm that our conclusions remain valid even when accounting for variance. Our meta-learning framework explicitly incorporates the fine-tuning process during training, enabling to provide well- vell-initialized parameters for each test instance. Morcover, by leveraging the gradient-free nature of our optimization scheme, the method facilitates high-quality instance- wise adaptation. Across all settings, approach consistently outperforms all baseline methods.
F VISUALIZATION OF DECISION ADAPTATION IN FJSP
To illustrate the adaptation process of scheduling decisions in the FISP, selected representative 10x5 FJSP instance and used Gantt charts to depict the changes in decision becision-making before and after model fine-tuning. In the Gantt charts, blocks of the same color represent the operations of the same workpiece, with the horizontal axis representing processing time and the vertical axis representing different machines. Figure shows the scheduling decisions before model fine-tuning. Since the model was not optimized for specific instance, it resulted in extended idle times for machines, and the completion times of different machines varied significantly. After fine-tuning the model (as shown in Figure 5). idle times were eliminated, and the completion times across different machines became more consistent, leading to significant reduction makespan.
16


Reviews:
[{"review_id": "ETPLjMwJab", "reviewer": "Reviewer_3BUB", "summary": "This paper proposes a derivative-free meta-learning framework for solving NP-hard scheduling problems (JSP and FJSP). The approach aims to address two main limitations of existing learning-based scheduling methods: (1) they optimize average performance rather than instance-specific quality, and (2) existing test-time adaptation methods like Active Search (AS) and Efficient Active Search (EAS) use gradient-based optimization and don't share knowledge across instances.\n\nThe proposed approach builds on MAML (an efficient DRL-based appraoch) and on Natural Evolution Strategies (NES) for derivative-free optimization", "questions": "I invite the authors to address the weaknesses mentioned above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) Novelty: while neither component (MAML and NES) is novel in itself, they are used in a way that is novel for the domain.\n\n2) Comprehensive evaluation: the evaluation covers multiple environment sizes and consists of several public benchmarks.\n\n3) Empirical improvement: the appraoch achieves consistent improvement across multiple experiments.", "weaknesses": "1) Lack of statistical significance tests: while the appraoch shows improvement in most cases, in some cases the improvement is small. The standard deviation is not presented (except for one table in the appendix), so it is impossible to determine whether the results are significant. Statistical significance tests would strengthen the authors' claims regarding the superior performance of their approach.\n\n2) Training cost no sufficiently emphasized and discussed: while parallelization is a plus, the training cost is significant. In some cases, training can take more than twice the time of competitors. Moreover, the gap (i.e. the difference in percentages between the proposed approach and FOMAML) seems to grow as the problem grows more complex. This raises significant questions regarding the feasibility of the approach for large-scale problems.\n\n3) No analysis of hyperparameters or sensitivity analysis: No analysis is provided on sensitivity to key parameters like population size (μ=100), noise standard deviation (σ=0.2), or number of inner-loop steps (K=3). How robust is the method to these choices?", "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": "FuBWmLfmZl", "reviewer": "Reviewer_3mss", "summary": "The paper addresses the problem that existing DRL approaches for job-shop and flexible job-shop scheduling optimize the average performance across training instances, which often leads to poor performance on specific individual instances. Existing test-time adaptation methods improve individual instances through fine-tuning, but they operate only at inference time, do not transfer adaptation knowledge across instances, and rely on gradient-based optimization, which can be ineffective in combinatorial search settings.\n\nThe authors propose a derivative-free instance-wise meta-learning framework. Instead of learning a single general policy, the method meta-learns an initialization of the policy parameters that is explicitly optimized to be adapted efficiently to each new scheduling instance. The approach uses NES to perform both inner-loop adaptation and outer-loop meta-updates, avoiding gradients entirely. Additionally, the paper introduces a GPU-parallelized population evaluation mechanism, enabling efficient large-scale derivative-free optimization.\n\nExperiments on FJSSP (and additionally JSSP with a non-RL model) show that the meta-learned initialization enables faster and better per-instance adaptation than prior approaches, and generalizes to larger unseen scheduling instances.", "questions": "- Could you provide evidence or analysis showing why gradient-based adaptation fails and identify cases where gradient-based adaptation performs similarly or better? Additionally, what properties of the problem or model determine when NES has an advantage over gradient-based optimization?\n\n- Could you provide a more detailed efficiency analysis such as wall-clock runtime breakdown, GPU utilization, and the contributions of population size and sampling to compute cost? Additionally, is the training overhead representative for inference as well, or does inference scale differently?\n\n- Since the computational cost of NES scales roughly linearly with population size, could you provide an ablation study showing how different values of the population size affect solution quality, runtime, and adaptation stability?\n\n- Do you have an idea, which characteristics a problem must have for your method to be effective?\n\n- The empirical section shows strong quantitative improvements. However, you do not provide qualitative insights into how the adapted decisions change. Could you include a qualitative or visual analysis showing how adaptation alters decisions and why it leads to better performance?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The authors explicitly shift the learning objective from maximizing average policy performance to optimizing instance-wise adaptability, which departs from the dominant paradigm in DRL-based scheduling. In addition, the paper contributes a system-level innovation, parallelizing population evaluations on GPUs. Hence, the paper offers a novel combination of ideas, which to my knowledge has not been applied in NCO. While individual components (MAML, NES, DRL scheduling) exist, the integration and the shift in optimization objective represent a meaningful contribution.\n\n- The authors claim that their meta-learned initialization enables more effective per-instance adaptation than gradient-based approaches. The experiments include multiple baselines (heuristic, OR-Tools, DRL, AS, EAS) and include ablation studies that isolate the effect of meta-learning and derivative-free optimization independently.\n\n- The experimental setup is thorough and follows best practice: standard benchmark datasets and larger unseen instances are evaluated. Further, hyperparameters and architectures are reported in sufficient detail for reproducibility. The ablations and generalization tests strengthen the claim that improvements arise from the proposed components rather than from tuning or dataset-specific effects.", "weaknesses": "- The paper combines multiple complex components at once (meta-learning, NES, scheduling, PPO), but the authors do not clearly modularize the explanation. Heavy notation is introduced early, creating a high cognitive load before the conceptual flow is fully established. Although figures exist, they are not tightly integrated with the algorithmic description, which makes it difficult to follow how the components interact.\n\n- The authors state that NES is superior to gradient-based adaptation, but the paper does not provide a deeper narrative or analytical justification beyond empirical observation. There is no analysis of failure modes or why gradient-free adaptation is particularly effective in this setting.\n\n- The experimental section includes many numerical comparisons, but offers very limited interpretation of the observed behavior. The focus is on performance differences rather than understanding the mechanism behind them.\n\n- The authors claim the approach is architecture- and domain-agnostic. The experiments do demonstrate architecture-agnostic behavior to a certain degree, but domain-agnostic generality is not evidenced, as all evaluations are restricted to JSS/FJSS scheduling problems.", "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": "zi9ZS48jUe", "reviewer": "Reviewer_2y5p", "summary": "This paper addresses a key limitation of learning-based approaches for NP-hard scheduling problems: models are often trained to optimize average performance, rather than the solution quality for each specific instance. Existing instance-level adaptation methods are typically used only at test time and rely on gradient-based optimization, which can be ineffective in combinatorial search spaces.\n\nInspired by MAML, the authors propose an instance-level meta-learning framework to learn a model initialization that can quickly adapt to new instances. Its main novelty lies in the use of a *fully derivative-free optimization (DFO)* method, specifically Natural Evolution Strategies (NES), for both the inner-loop adaptation and the outer-loop meta-optimization. This avoids the need for complex gradient calculations and is better suited for instance-level search tasks, which can trap gradient-based methods in local optima.\n\nTo stabilize the DFO process, the paper introduces two Monte Carlo estimation strategies (MC averaging and MC best-sample). Furthermore, they develop an efficient GPU-based parallel framework to manage the computational cost of the population-based NES.\n\nThe effectiveness of the method is demonstrated through extensive experiments on Flexible Job Shop Scheduling (FJSP) and Job Shop Scheduling (JSP), applying it to two base models: Reinforcement Learning (DANIEL) and Self-supervised (SPN). The results show consistent and significant improvements over the base models and sota adaptation baselines, such as Active Search (AS) and Efficient Active Search (EAS).", "questions": "1. **Training Time vs. Baselines:** Appendix C provides an excellent comparison of training costs against FOMAML. However, how does the total meta-training time (e.g., 11.2 hours for $20 \\times 10$) compare to the *original pre-training time* of the DANIEL baseline? Understanding the *total* training budget (pre-training for DANIEL vs. meta-training for the proposed method) would provide a clearer picture of the overall computational trade-off. Training a model that requires 11 hours ($20 \\times 10$) suggests that addressing the conflict between \"inference-time memory advantage\" and \"training-time catastrophic cost\" might be a necessary component.\n2. **Rationale for MC Estimators in Ablation:** In Section 5.2, the paper states \"MC averaging\" is used for greedy mode and \"MC best-sample\" for sampling mode. This is intuitive. However, in the ablation study (Figure 3), all estimators (including MC averaging) are evaluated in *sampling mode*. Can you clarify this? Was MC averaging also evaluated in greedy mode? Was the MC best-sample strategy *also* shown to be suboptimal for greedy mode, or is MC averaging indeed the best choice for greedy mode?\n3. **\"Perturbation-on-Perturbation\" Variance:** The paper makes an interesting point in 4.1.2 that the \"perturbation-on-perturbation\" mechanism increases the variance of the natural gradient estimate. Does this imply that the \"NES\" baseline in Table 3 (i.e., DFO fine-tuning *without* meta-learning) is more stable or converges faster (per-instance) during its test-time adaptation, even if its final solution quality is worse?\n4. **Sensitivity to Population Size:** The population size $\\mu=100$ seems to be a critical hyperparameter. What was the rationale for choosing this specific value? How much does performance degrade if a smaller, computationally \"cheaper\" population is used (e.g., $\\mu=20$ or $\\mu=50$)? A sensitivity analysis on key NES hyperparameters (especially population size $\\mu$ and noise $\\sigma$) may be needed. This is a core issue for DFO methods. Without this analysis, reviewers cannot assess whether your method is robust or requires extreme tuning to achieve the claimed performance.\n5. Given that the authors' method (Appendix C) requires up to 2x the training overhead, while the performance gain over standard gradient meta-learning FOMAML (Table 5) is small, strong evidence may be needed to prove that this complex, computationally expensive DFO framework truly offers a substantial advantage over simple FOMAML. An elaboration on this would strengthen the paper.\n6. **Justification for MC Best-Sample:** Using MC best-sample (Eq. 3) to estimate fitness in the inner loop seems to \"overfit\" to the \"sampling mode\" during the meta-training phase. Does this estimator harm the model's performance in \"greedy mode\"? Did the authors compare whether the greedy mode performance would be better if MC averaging was used uniformly?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper addresses a critical and practical problem in Neural Combinatorial Optimization (NCO).\n2. The core contribution—a fully derivative-free meta-learning framework for instance-level NCO. While MAML and NES are existing methods, their combination to solve the specific limitations of gradient-based instance adaptation in scheduling is a sufficient methodological contribution.\n3. The comprehensive experiments are a strength.\n   - The method is tested on two different scheduling problems (FJSP, JSP), two different learning paradigms (RL and self-supervised), multiple synthetic datasets (SD1, SD2), and standard public benchmarks (mk, la, Taillard).\n   - Comparison against strong and relevant baselines, including exact solvers (OR-Tools), strong heuristics (MWKR), base models (DANIEL, SPN), and state-of-the-art gradient-based adaptation methods (AS, EAS).\n   - Sufficient ablation studies: DFO vs. Gradient Adaptation, DFO-Meta vs. DFO-only, DFO-Meta vs. Gradient-Meta, the proposed MC estimators, and GPU parallelization.\n4. The method performs well on both the RL-based DANIEL and the self-supervised SPN. Furthermore, the scalability experiments on larger $50 \\times 20$ and $100 \\times 20$ instances (Table 8) are impressive, highlighting a benefit of the DFO approach: much higher memory efficiency than gradient-based adaptation (AS/EAS).\n5. The motivation is clear, the logical flow is sound, and it effectively uses figures (Figs 1, 2) and algorithms (Algs 2, 3) to explain the complex framework.", "weaknesses": "1. **Training Cost:** The main drawback is the computational cost of DFO, which the authors acknowledge. Appendix C confirms the method requires approximately 1.5-2x the training time and GPU memory of the gradient-based FOMAML baseline. While the authors argue this is a reasonable trade-off for offline training, the high cost of population-based methods remains a significant barrier. A population size of $\\mu=100$ and 200 epochs represent a substantial computational investment.\n2. **Hyperparameter Sensitivity:** NES and other DFO methods are often highly sensitive to hyperparameters (e.g., population size $\\mu$, noise standard deviation $\\sigma$, step sizes $\\alpha, \\beta$). The paper states these were tuned on the smallest instance size and fixed, which is a reasonable protocol. However, the paper lacks a sensitivity analysis. Given that DFO methods can be \"notoriously\" difficult to tune, including a brief analysis (e.g., in the appendix) showing how performance varies with different $\\mu$ or $\\sigma$ values would significantly enhance the paper's practical value and demonstrate the method's robustness. The sensitivity of DFO methods like NES to hyperparameters such as population size $\\mu$, noise standard deviation $\\sigma$, and learning rates $\\alpha, \\beta$ is well-known. The paper fails to provide any sensitivity analysis for these critical hyperparameters. This calls the reproducibility and robustness of the experimental results into serious question. We cannot judge whether the current SOTA results are the product of immense hyperparameter-babysitting or if the method is genuinely robust.\n3. The paper's core contribution is an effective combination of two well-known techniques—MAML (meta-learning) and NES (derivative-free optimization)—applied to a known problem (instance-level adaptation). Neither meta-learning for NCO nor instance-level adaptation (as shown by AS/EAS) are entirely new concepts.\n4. The choice of NES (a DFO method) as the core optimization strategy might be a \"brute-force\" choice. It essentially uses massive computational resources and sample complexity (population-based search) to bypass the (researchable) challenges faced by gradient-based adaptation. One of the key comparisons in the paper (Table 5) shows that the authors' proposed DFO method offers a relatively weak performance improvement over standard gradient meta-learning (FOMAML) (e.g., on $20 \\times 10$ SD2, greedy mode only improves from 21.22% to 23.21%, and sampling mode only from 14.72% to 19.36%). To achieve this slight improvement, Appendix C (Table 9) shows that the authors' method requires **1.5 to 2 times** the training time and GPU memory. Exchanging 2x the computational resources for a 2-4% performance gain makes the method's adoption in other domains difficult.\n5. DFO not needing to store gradients at inference time, thus having low memory usage, is its inherent characteristic, but its training-stage cost (Table 9) is large (e.g., $11.2$ hours for $20 \\times 10$).", "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": "GGUiasnwnl", "reviewer": "Reviewer_kVaC", "summary": "This paper proposes Instance-wise Adaptive Scheduling via Derivative-Free Meta-Learning, a framework that improves deep reinforcement learning for complex scheduling problems such as the Flexible Job-Shop Scheduling Problem (FJSP). It uses meta-learning to learn a generalizable initialization for fast per-instance adaptation and replaces gradient-based optimization with a derivative-free method based on Evolution Strategies, enhanced by GPU parallelization for efficiency. Experiments show that the approach consistently outperforms existing baselines like Active Search and Efficient Active Search in solution quality, adaptability, and generalization.", "questions": "1. How does the proposed framework meaningfully advance beyond standard MAML or existing derivative-free meta-learning methods, and what concrete evidence supports the need to replace gradient-based approaches?\n\n2. Given the high computational cost of two nested evolutionary loops, can the authors justify why such a design is necessary, and provide analysis demonstrating its scalability and efficiency in practical settings?\n\n3. Since the algorithm appears to follow the OpenAI-ES formulation rather than true NES, and GPU offloading may be problem-dependent, how do the authors substantiate their claims of “natural gradient” estimation and general computational efficiency?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper’s key strengths lie in its novel integration of meta-learning with derivative-free optimization to improve instance-wise adaptability in complex scheduling tasks. By leveraging Evolution Strategies and a GPU-parallelized implementation, the approach achieves efficient, gradient-free training and fine-tuning that reportedly scales well to large problem instances.", "weaknesses": "While the paper demonstrates solid implementation and empirical results, its conceptual novelty is limited. The proposed framework largely repurposes the MAML paradigm with minor modifications, and the use of derivative-free optimization appears incremental rather than innovative. The motivation for replacing gradient-based methods is not well justified, with very limited theoretical or empirical evidence. Consequently, the work feels more like an engineering improvement than a substantial methodological advance, lacking deeper analysis or insight into why the proposed approach is fundamentally superior.\n\nA key weakness appears to be the use of two nested evolutionary processes, one for instance-level adaptation and another for meta-level optimization. They make the method highly computationally expensive and sample inefficient. Since evolutionary algorithms already demand large populations and many evaluations, nesting them greatly amplifies the cost. Although GPU parallelization helps with runtime, it does not address this fundamental inefficiency. The paper also lacks analysis showing that such a two-level structure is necessary, raising concerns about scalability and practical applicability.\n\nAs acknowledged by the authors, a potential advantage of the proposed nested design is that it avoids computing second-order derivatives, which can be costly in traditional MAML frameworks. However, it remains unclear why ES is the preferred or necessary choice for achieving this goal. There exist several other first-order or gradient-approximation approaches that could eliminate second-order terms with far lower computational cost. The paper does not provide justification or comparative analysis to show that ES offers a clear advantage over these alternatives, making the design choice appear arbitrary and potentially inefficient.\n\nA significant flaw lies in the paper’s misuse of the term “natural gradient.” Although the authors claim to adopt Natural Evolution Strategies (NES), their algorithm actually follows the OpenAI-ES formulation, which estimates a standard Monte Carlo gradient without the Fisher preconditioning required for true NES. In particular, Equation (4) is mathematically identical to the OpenAI-ES update rule, showing no theoretical modification or geometric correction. This indicates a misunderstanding of NES and renders the claim of “natural gradient estimation” technically inaccurate, undermining both the paper’s methodological novelty and its conceptual soundness.\n\nThe claim that “the entire fitness evaluation process is offloaded to GPU” in the paper appears overstated. In practice, the feasibility of full GPU offloading strongly depends on the problem structure and the simulator used for fitness evaluation. Many scheduling environments, particularly those involving discrete event simulation or complex resource constraints, are not inherently GPU-friendly and may still require significant CPU-side computation or data transfer. Without demonstrating that the underlying environment is fully GPU-parallelizable, this claim lacks general validity. As a result, the reported efficiency gains may not generalize beyond the specific experimental setup used in the paper.\n\nWhile the experimental results in the paper show consistent improvements over baselines such as Active Search and Efficient Active Search, the gains are generally incremental rather than dramatic. The improvements in optimality gap are often modest (a few percentage points) and achieved under carefully controlled synthetic settings, which may not fully reflect real-world complexity. Moreover, the experiments primarily benchmark against older or relatively simple baselines, without comparison to stronger contemporary meta-learning or hybrid optimization approaches. Given the method’s high computational cost and limited novelty, the reported results, though positive, do not convincingly demonstrate a major performance breakthrough or justify the added algorithmic complexity.", "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)