Efficient Training of Multi-task Neural Solver for Combinatorial Optimization
Authors: Chenguang Wang
Zhang-Hua Fu
Pinyan Lu
Tianshu Yu
Venue: Accepted by TMLR ()

Abstract: Efficiently training a multi-task neural solver for various combinatorial optimization problems (COPs) has been less studied so far. Naive application of conventional multi-task learning approaches often falls short in delivering a high-quality, unified neural solver. This deficiency primarily stems from the significant computational demands and a lack of adequate consideration for the complexities inherent in COPs. In this paper, we propose a general and efficient training paradigm to deliver a unified combinarotial multi-task neural solver. To this end, we resort to the theoretical loss decomposition for multiple tasks under an encoder-decoder framework, which enables more efficient training via proper bandit task-sampling algorithms through an intra-task influence matrix. 
By employing theoretically grounded approximations, our method significantly enhances overall performance, regardless of whether it is within constrained training budgets, across equivalent training epochs, or in terms of generalization capabilities, when compared to conventional training schedules. 
On the real-world datasets of TSPLib and CVRPLib, our method also achieved the best results compared to single task learning and multi-task learning approaches. 
Additionally, the influence matrix provides empirical evidence supporting common practices in the field of learning to optimize, further substantiating the effectiveness of our approach.
Our code is open-sourced and available at \url{https://github.com/LOGO-CUHKSZ/MTL-COP}.

Published in Transactions on Machine Learning Research March/2022
Efficient Training of Multi-task Neural Solver for Combinatorial Optimization
Chenguang Wang
chenggangguanggiiik.ccckk.d
School of Data Science, The Chinese University of Hong Kong, Shenahen Shenchen Research Institule of Big Data
Zhang- Hua Fu
Bechaanghuaccchk..cm edu.cr
School Science and Engineering, The Chinese University of Hong Kong, Shenzhen Shenzhen Institute of Artificial Intelligence and Robotics for Society
Pinyan Lu
Shanghai University of Finance and Economics
Tianshu Yu!
yudiansheuBchkk.cdu.cco
School of Data Science, The Chinese University of Hong Kong, Shenzhen
Reviewed on OpenReview: https: // optp//veee.vvveeeee net/ forum? id= HJbccubMcuuQ
Abstract
Efficiently training a multi-task neural solver for various combinatorial optimization prob- lems (COPs) has been less studied so far. Naive application conventional multi-task learning approaches often falls short in delivering a high-quality, unified neural solver. This deficiency primarily stems from the significant computational demands and lack of ad- equate consideration for the complexities inherent in COPs. In this paper, propose general and efficient training paradigm to deliver unified combinatorial multi-task neural solver. To this end, we resort to the theoretical loss decomposition for multiple tasks un- der an encoder-decoder framework, which enables more efficient training via proper bandit task-sampling algorithmms through an intra-task influence matrix. By employing theoretically grounded approximations, our method significantly enhances overall performance, regardless of hhether it is within constrained training budgets, across equivalent training epochs, or in terms of gencralization capabilitios, when compared to conventional training schedules. On the real-world datasets of TSPLib and CVRPLib, our method also achieved the best results compared to single task learning and multi-task learning approaches. Additionally, the influence matrix provides empirical evidence supporting common practicos in the field of learning to optimize, further substantiating the effectiveness of our approach. Our code open-sourced and available at
1 Introduction
Combinatorial optimization problems (COPs) Korte ct al., 2011 are fundamental in varions fields, including logistics (Caltaruzza al.] 2017 Bao el al.) 2018), finance (Tatsumura et al. 2023), telecommunications (Resende & Pardalos] 22008). and computer science, where they are essential for making optimal decisions over discrete structures. Traditional methods for solving COPs, such exact algorithms (Wolsey 2020 ) ap-
proximation methods (Vazirani 2001) and heuristics (Boussaid et al. 2013), often face significant challenges due to their computational complexity and inefficiency, especially in large-scale and dynamic scenarios. The emergence of deep learning has introduced a transformative approach by leveraging powerful modeling ca- pabilitics to learn from data and generalize solutions across similar instances. Building on this idea, many
"Corresponding author

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Published in Transactions on Machine Learning Research (March/2022
studies have attempted to use machine learning-based methods, such supervised learning
Vinyals et al,
2015
Fu et. al. 2021 ). reinforcement learning (Bello et ad.22017 Kool et al. 2019: Kwon et al.
2020 Lu et al.
2020
Wu et al. 2021b), or unsupervised learning (Hibat- Allah et al. 2021 Schuetz et al.
2022 Sanokowski
ctt al. 2023 ).24). to solve COPs by proposing ncural solvers. However, these approaches typically focus on
a single type of combinatorial optimiiaation problem, or even a specific scale of that problem, significantly limiting the feasibility of neural solvers in practical applications. Therefore, training a neural solver capable of handling different types and sizes of COPs simultaneously is an urgent issue that needs to be addressed. Recently, some research has begun to address this issue, proposing the use of unified neural solver to tackle different problems such as Vehicle Routing Problems (VRPs) (Liu et al. 2024 Lin et al. 2024 Berto et al. 2024 Zhou et al. 2024) and graph-based COPs (Boisvert et al.) 2024). However, these solutions are problems. For instance, they may leverage specialist. knowledge often restricted to a specific calegory of
to construct models based on the combinatorial characteristics of various VRPs (such as VRPTW, OVRP, VRPB) or focus solely on graph-based COPs. As a result, they fail to develop a ncural solver that can simultaneously solve multiple lypes of COPs.
Although generic neural solver for multiple COPs is appealing, training such a neural solver can be prohibitively expensive, especially in the cra of large models and this problem is less studied in the literature. To relieve the training burden and better balance the resource allocation, in this paper, we propose a novel training paradigm from a multi-task learning (MTL) perspective, which can efficiently train a multi-task combinatorial neural solver to handle different types of COPs under limited training budgets.
To this end, we treat each COP with a specific problem scale as task and manage to deliver a generic solver handling set of tasks simultancously. Different from standard joint training in MTL, we employ multi-armed bandits (MAB) algorithms to select/sample one task in each training round, hence avoiding the complex balancing of losses from multiple tasks. To better guide the MTL training, we employ reasonable reward design derived from the theoretical loss decomposition to derive the rewards with theoretical guaran- toes in MAB algorithms for the widely adopted encodor-docoder architocture. This loss decomposition also brings about an influence mairix revealing the mutual impacts between lasks, which provides rich evidence to explain some common practices in the scope of COPs.
To emphasize, our method is the first-of-its- kind to consider training generic neural solver for different kinds of COPs. This greatly differs from existing works focusing on either solution construction (Vinyals et al.
2015 Bello et al. 2017: Kool et al.] 2019 Kwon et al.
2020) or heuristic improvement (Lu et al.
2020
Wu et al.) 2021b Agostinelli et al. 2021 Fu et al.]
2021
Kool et al.
2022). Some recent works seek
to generalize neural solvers to different scales Hou et al.
Li et al. 2021 Cheng et al. 2023 Wang et al.
2024) or varying distributions (Wang et al., 2021 Bi et al. handle multiple types of COPs simultaneously. Compared to methods aimed at building a universal neural
2022: Geisler et al.] 2022), but with no ability to
solver (Liu et al.) 2024 Lin et al. 2024 Berto et al. 2024 Zhou et al. 2024 Boisvert et al., 2024), our focus is a generalized training framework that does not require expert knowledge during the training phase, nor does it necossitate specifying the type of problems. This framework is suitable for any neural solver and is adaptable to a broad range of COPs
Experiments are conducted for 12 tasks: Four types of COPs, the Travelling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), the Orientocring Problem (OP), and the Knapsack Problem (KP), and each of them with three problem scales. We compare our approach with single-task training (STL).
extensive MTL baselines (Mao et al. 2021 Yu al. 2020 Navon et al. 2022 Kendall et al., 2018 Liu et al. 202 b) and the SOTA task grouping method, TAG FFfty et al. 2021 under the cases of the same training budgets and same training epochs. Compared with STL, our approach needs no prior knowledge about tasks and can automatically focus on harder tasks to maximally utilize the training budget. Additionally, when comparing with STL under the same training epoch, our approach not only enjoys the cheaper training cost which is strictly smaller than that of the most expensive task, but also shows the generalization ability by providing a universal model to cover different types of COPs. Compared with the MTL methods, our method only picks the most impacting task to train at each time. This mechanism improves the training efficiency without explicitly balancing the losses. Furthermore, we also compare approach with STL and MTL methods on real-world datasets, TSPLib and CVRPLib, achieving the best experimental performance.
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In summary, our contributions can be concluded as follows: (1) We propose novel framework for efficiently training a combinatorial neural solver for multiple COPs via MTI., which achieves prominent performance against standard training paradigms with limited training resources and can further advise efficient training of other large models; (2) We study the intrinsic loss decomposition for the well-adopted encoder-decoder architecture and propose a theoretically guaranleed approximation for it, leading to the influence matrix reflecting the inherent task relations and reasonable reward guiding the update of MAB algorithms; (3)
We verify several empirical observations for neural solvers from previous works (Kool al.) 2019 Joshi
et al. 2021) by the influence matrix, demonstrating the validity and reasonableness of our approach: (4) We
compare our method to a wide range of baselines on synthetic and real-world datasets, demonstrating that it outperforms STL and MTL approaches in efficiency and results.
2 Related Work
2015) pioncered the application of deep neural networks for solving combinatorial optimization problems. Subsequently, numerous neural solvers have been
Neural solvers for COPs Pointer Networks (Vinyals et al.
developed to address various COPs, such routing problems (Bello et al.] 2017; Kool et al. 2019 Lu et al.
2020
Wu et al. 2021b b), knapsack problem (Bello et al.
2017 Kwon et al.
2020), job shop scheduling
problem (Zhang et al. 2020). and others. There are two prevalent approaches to constructing ncural solvers:
solution construction Vinyals et al. 2015 Bello et al.] 2017 Kool et al.] 2019: Kwon el. al.] 2020 which
sequentially constructs a feasible solution, and heuristic improvement (Lu et al. 2020 Wu et al. 2021b;
Agostinelli et al.] 2021 Fu ct al. 2021 Kool et al.; 2022), which provides meaningful information to guide
downstream classical hcuristic methods. In addition to developing novel techniques, several works (Wang
et al. 2021 Geisler et al. 2022: Bi et al. 2022 Wang et al. 2024) have been proposed to address generalization
issues inherent in COPs. For comprehensive review of the existing challenges in this area, we refer to the
survey Bengio et al. 22020).
Multi-task learning Multi-Task Learning (MTL) aims to enhance the performance of multiple tasks by jointly training a single model to extract shared knowledge among them. Numerous works have emerged to address MTL from various perspectives, such as exploring the balance on the losses from different tasks
(Mao et al. 2021 Yu et al.] 2020 Navon et al.) 2022 Kendall et al.] 2018 Liu et al. 2021a b) designing
modulc-sharing mechanisms Misra et al. 2016 Sun et al. [020 Hu& & Singh 2021 improving MTL through
multi-objective optimization (Sener & Koltun 2018 Lin et al. 2019 Momma et al
2022 and meta-learning
(Song et al. 2022). To optimize MTL efficiency and mitigate the impact of negative transfer, some research
focuses on task grouping (Kumar & 111 2012 Zamir et al.) 2018 Standley et al. 2020 Fifty et al. 2021), with the goal of identifying task relationships and learning within groups to alleviate negative transfer effects in conflicting lasks. On the application level, MTL has been extensively employed in various domains, including
Luong et al. 2016). computer vision (Zamir et al. 2018 Scong et al. 2019), bioinformatics (Xu et al. 2017 and many others.
natural language processing (Collobert & Weston 2008
Foundation models for COPs Recently, research has increasingly focused on foundation models for com- binatorial optimization problems. For instance, studies such as Liu et al. (0224) Lin et al. (2024); Berto et al. 2024); Zhou et al. (2024) tackle cross-problem generalization in vehicle routing problems (VRPs) using unified neural solvers that leverage attribute composition. However, these approaches rely on the com- binatorial nature of VRP structures and necessitate specific expert knowledge for problem decomposition, limiting their applicability to other types of COPs. Meanwhile, Boisvert et al. (2024) proposes a generic representation for COPs via graph-based approaches, facilitating efficient learning with novel graph neural network architecture. Additionally, Drakulic et al. (2024) presents generalist model that efficiently ad- dresses multiple COPs using mixed-attention backbonc and multi-type transformer architecture, achieving lask-specific adaptability through light adapters and showcasing strong transfer learning capabilities. Furthermore, Jiang et al. (2024) introduces a model that solves diverse combinatorial optimization prob- lems using natural an encoder-decoder architecture and trained with conflict gradients erasing reinforcement learning, thereby improving performance across tasks.
formulated instances and a large language model for embedding, enhanced by
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Loss Decomposition (Section 3.1) &
Reward Construction (Section 3.2)
TSP Decoder
OVRP Decoder
OP Decoder
KP Decoder
MAB Algorithm
Section 3.3
Reward
Encoder
TSP Header
CVRP Header
OP Header
KP Header
Pomba
cuteria
Crmeria
Select one to train
Figure 1: Pipeline of MAB for Solving COPs in view of MTL. We consider four types of COPs: TSP. CVRP, OP and KP, each with corresponding header and decoder. The encoder, which is common to all COPs, is also included. For each time step, we utilize the MAB algorithm to select a specific task for training, such as CVRP-100 depicted in the figure. We then obtain the loss for the selected task, perform loss decomposition as detailed Section 3.11 and construct a reward using the methodology outlined in Section 3.2 Finally, we utilize the reward update the MAB algorithm.
Multi-armed bandits Multi-armed bandit (MAB) is classical problem in decision theory and machine learning that addresses the exploration-explottatio trade-off. Several algorithms and strategies have been suggested solve the MAB problem, such as the e-greody, Upper Confidence Bound (UCB) family of
algorithms (Lai et al. 1985 Auer et al. 20022), the Exp3 family (Littlestone & Warmuth, 1994 Auer al.,
1995
Gur al.) )014), and the Thompson sampling (Thompson, 1933 Agrawal & Goyal 2012: Chapelle
& Li 2011p. These methods differ in their balance of exploration and exploitation, and their resilience
under distinct types of uncertainty. The MAB has been extensively studied in both thcoretical and practical
contexts, and comprehensive details can be found in Slivkins el. al. (2019) Lattimore & Szepesvári (0200)
3
Method
We consider K types of COPs, denoted as 7 (i 1, 1,,, ....,KK) K), with n, different problem scales for each COP. Thus, the overall task set is T U T {T}lú = 1,2, n4, = 1,,,..,,K; K}. For each type of COP T we consider neural solver Sg: (Z)) Dj. where O' the parameters for COP 7 Ij and yj are the input instance the output space for COP 7" with the problem scale of n, (termed as task T in the sequel). The parameter vector O (6chare 8*) contains the shared and task-specific parameters for the COP T and the complete set of parameters is denoted by θ UKK O'. This parameter notation corresponds to the commonly used Encoder- Decoder framework in multi-task learning in Figure 1 where (share represents the encoder shared across all tasks, and 0' represents the decoder task-specific for each task. Given the task loss functions (+)) for COP Tii with the problem scale of nj, we investigate the widely used objective function:
K n,
min L( θ
Σ
(0)).
(1)
We propose a general Multi-task Learning (MTL) framowork based on Multi-Armed Bandits (MAB) to dynamically select tasks during training rounds and reasonable reward constructed to guide the selection process. In particular, our approach establishes a comprehensive task relation by the obtained influence
According to the Recoderr framework, encoder commonly refers shared models, whereas decoder specific modules. In this study, the decoder component comprises modules: 'Header' and 'Decoder' illustrated Figure
task-
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matrix, which has the potential to empirically validate several common deep learning practices while solving COPs.
Overview We aim to solve equation | using the MAB approach. Given the set of tasks T = {Ti 1.2, n,.2 1.2....,KK } we select an arm (i.e., task being trained) a E T following an MAB algorithm, which yields a random reward signal r, that reflects the effect of the selection. The approximated expected reward is updated based on the rewards. Essentially, our proposed method is applicable to any MAB algorithm. The general framework of MAB algorithm for solving COPs within the context of MTL is outlined in Figure
3.1 Loss Decomposition
In the framework of solving COPs in view of MTL described in Figure | it crucial design reasonable reward for MAB algorithm to guide its update. In this part, analytically drive a reasonable reward by decomposing the loss function for the Encoder-Decoder framework. Note θ = U O = [Osbare U(0,; =
1., } are all trainable parameters.
We suppose that a meaningful reward should satisfy the following two properties: (1) It can benefit our objective and reveal the intrinsic training signal; (2) When a task is selected, there always positive effect it in expectation.
The decrease loss function is reasonable choice and previous work has used it to measure the task relationship (Fifty et al. 2021). However, such measurement is invalid in our context because there are significant differences among tasks (scc Appendix A.8 What's more, the computational cost of the "lookahead loss' * in TAG (Fifty et al. 2021] is considerably expensive when frequent reward signals are needed. We instead propose a more fundamental way based on gradients to measure the impacts of training one task upon the others.
To simplify the analysis, in Proposition we assume the standard gradient deecent (GD) is used to optimize equation [ by training one task at each step t, and then derive the loss decomposition under the encoder- decoder framework. Any other optimization methods, e.g., Adam (Kingına & Ba 2015), can also be used here with small modifications. Detailed proofs for GD and Adam are in Appendix A.2
Proposition 1 (Loss decomposition for GD) Using encoder-decoder framework with paramelers θ
U O' = =Oshare U(0,, i = 1, 1,2....,K] and updating parameters with standard gradient descent: O(t + 1] = O(t) (o(())) where n7 is the step size. Then the difference of the loss of task Tÿ from training step t to l2 (11 = (t)(() (t)() can be decomposed to:
(L2 t2)
= m (a: "
m ma, (1
(a) effeets of training task TT: oj (f2
effects of training task (T * f): (c((tt
(2)
1p 11
1a, T
goham (r))) (2())
PF1
(c) effects of training task (Tp
where (L(O) means taking gradient w.r.t. θ and ( means taking gradient w.r.t. 0 C O, T (t) is some vector lying between (²(t) and (t) and ((a; = T3) is the indicator function.
Proof 1 (Proof of proposition 1 Observe that.
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1211 t2) (t,
R np
1 (a: = =T;) ]
m-1
Σ
ΣΣ
Σ 1( -T)AA((1)))
(3)
1-
K Ap
((, T()
ΣΣΣ>
1(0, ((==17)AL((1+1)
where l(a =ī)) is the indicator function which is introduced here because we only select one task at each time step, taking 1 if selecting task I at time step t, 0 otherwise. Then based on mean value theorem, we take the first order Taylor expansion for
V (E
ip=
l(a, = (a==1))ii(t
(4)
Tr V gahane C (1(((())) ( (L)))(()
Otherwise
where (1) is some vector lying between (1) and 0 O' (1+1).
Suppose lask Tÿ is selected for es limes between we oblain
time step 1 and l2, Afler plugging equalion 4 in equation 3
-
12 Σ (a==T)nVL((00)
offects of traininn task T3: o3(11
(5)
Th
t2
to
((L)(((()))
1aa -m Vonn (VV ΣΣ Σ 1(a
TT
V gatore L
(M) efeerta of troining took (T. # i): (c1(((1a)))))
(e) efferts of trainiing taok (TT P
The idea behind equation means the improvement on the loss for task T3 from t1 to t2 can be decomposed into three parts: (a) effects of training Tý itself w.r.t. 0i; (b) effects of training same kind of COP {Ty a + j} w.r.t. O; and (c) effects of training other COPs (T", w.r.t. gshare Indeed, we quantify the impact of different tasks Tÿ through this decomposition, which provides the intrinsic training signals for designing reasonable rewards.
3.2 Reward Design and Influence Matrix Construction
In this part, we design the reward and construct the intra-task relations based on the loss decomposition introduced in Section 3.1. Though equation 2 reveals the signal during training, the inner products of gradients from different tasks can significantly differ at scale (see Appendix [A.)). This may greatly mislead the bandit's 's update since improvements may come from large gradient values even when they are almost orthogonal. To address this, we propose to use cosine metric to measure the influence between task pairs. Formally, for task Tj from t to 12, the influence from training the same type of COP ти to T is:
12-1 TT==T)) (T
(L)
(6)
bー Σ 1(a, =Ta=TT) (V0)
1=t
and the influence from training other types of COPs T to Tj is:
q
1 (a TT)
Prrr L (T) geharn (P(()(())
(177;411) t= ビー 1 (at = TT!)
(7)
Mraaar 13 (V)(I V gehare (OP(()))
Given equation 6 and equation
we denote the influence vector to Tÿ as:
Influence from Htseff
(1a, (2) (..., mi (Ti: l1, 12). mi (T): (1,
my (TH 12),
m (T,
yr
(8)
infueence from the some kind of COP
inftuence from other kinds of COPs

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Based on equation an influence matrix (2), = (..., (2)) E m x nz) can be constructed to reveal the relationship between tasks from time step t to 12. There are several properties about influence matrix (t1,, t2): (1) ((1,tt2 has blocks (t, E R in the diagonal position which is the sub-influence matrix of same kind of COP with different problem scales; (2) M(1,, asymmetry which consistent with the general understanding in multi-task learning; (3) The row-sum of M (41,42)
n2)
are the total influences obtained from all tasks to one lask; (4) The column-sum of M(I, (2) are the total influences from one task to all tasks.
According to the implication of the clements in ((1,, f2), the column-sum of (t).
t21 = 1 M(1,, E RRI TM
(9)
provides meaningful reward signal for selecting lasks which we can use to updale the bandit algorithm. Moreover, denote the update frequency of computing the influence matrix as AT and the overall train- ing time is nAT, then an average influence matrix W can be constructed based on influence matrices (MAT, (k + 1)AT), A = 0,2, n - 1} collected during the training process:
1
W = nAT Σ M (KAT, (k + 1)AT),
(10)
revealing the overall task relations across the training process.
3.3 Theoretical Analysis and Implementation
When computing the bandit rewards, an issue arises with approximating 4 ()) in Eqs. equation 6 and equation 7 A practical and justifiable approach involves using the approximation ('L)((( (t))()((t) based on a Taylor expansion, as the parameter changes during neural network updates are typically minimal. Nonetheless, obtaining this gradient information is challenging because only one task is selected per training slot. To address this, we adopt the approximation
V"H (T,,(((), & ('(
where (t) represents the most recent training slot for Tj. This method is straightforward to implement and offers the following theoretical guarantee.
For each lask, suppose there exists M > 0 such that || < M. Moreover, we suppose the gradients have equal norm across the training period. For any task Tp. we denote mT(TT; (2) and m² (T7;)1,122) as the approtimations of using (T,,))(( and 4 (3(O in t2)) then
Theorem 1
(1, (211)222
t2. b)) tz)
Σ 1(a (x,()
t=
where mt, t2) and cf(1,, t2) the marimal learning rate and selection times of Th from t to tz.
In practice, this upper bound reasonably negligible due to the small value of M (Sagun et al. 2017
Ghorbani et al. 2019 Yao et al. 2020 Li et al. 2020) and (mas(ttdd)) Ei, 1( = TT) (t), indicating c2()) good approximation during the implementee implementation. We leave the proofs and discussions Appendix A.3
3.4 Computation Complexity of MTL Methods
In this part, we will make a detailed comparison on the computation complexity between our method and other typical MTL methods. We first define some notations for the time complexity: N, D,, F, and B, where Di, F; and B, the dimension of parameters, the computation cost of feed-forward and backward for task i,
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Table 1: Computation complexity of different MTL methods for one training time slot. "Basic" measures the computation for the feed- feew-forward and backward process, "Extra" measures the extra computations used for guiding MTL, and " "All" is the sum of them.
Naive MTL
Bandit- MTL
PCGrad
Nuck MTLb
UW
IMTL OIN(F B))
CAGmnd
Ours
(NIFF B))
ONNIF
ONNI
DINIP Bi)
O B))
ONN(F B)
OF
Extra
(NN)
(NPD)
O (N
0'11
(NN' N
O(ND)
All
O(N(F B) ONNI + 8+ O(N(F + 8 ND)) OIN(F + B- NDW) O(N(F 4 B) OIN(F B V ND))
O + B ND)
GPU Hoxrss
1.00
1.04
6.02
6.87
5.61
5.24
0.07
and we denote D = 1,,, = 1 N), max analyze the computation complexity for Bandit-MTL. PCGrad, Nash-MTI., Sesertainty-Weighhhe (UW) and our method, results are shown in Table] where "Basic" measures the computation for the feed-forward and backward process, 'Extra' measures the extra computations used for guiding MTL, and 'All" is the sum of them.
N). We
We ignore the complexity of sampling from a discrete distribution with elements, e.g. sampling an arm in MAB algorithm. What's more, we also ignore the optimization process in Nash-MIL and UW because they are quite efficient compute. From the results in the Table [ our method has moderate extra computation costs comparing with other methods, however, when considering the overall computation cost, our method achieves the lowest complexity because only need to perform one feedforward-backwrrr process which is the most time-consuming part during training. We provide detailed descriptions of these MTL methods in Appendix A.3
4 Experiments
In this section, we conduct a comparative analysis between the proposed method against both single-task training (STL) and extensive multi-task lcarning (MTL) methods demonstrate the efficacy of our ap- proach in addressing various COPs under different evaluation criteria. Specifically, we examine two distinct scenarios: (1) Under identical training budgets, we aim to showcase the convenience of our method in auto- matically obtaining universal combinatorial neural solver for multiple COPs, circumventing the challenges of balancing loss in MTL and allocating time for each task in STL; (2) Given the approximately same num- ber of training epochs, we seek to illustrate that our method can derive a potent neural solver with excellent generalization capability. Furthermore, we employ the influence matrix to analyze the relationship between different COP types and the same COP type with varying problem scales.
We explore four types of COPs: the Travelling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), the Orientecring Problem (OP), and the Knapsack Problem (KP). Detailed descriptions can be found in Appendix A.1 Three problem scales are considered for each COP: 20, 50, and 100 for TSP, CVRP, and OP; and 50, 100, and 200 for KP. We employ the notation "COP-scale", such as TSP-20, to denote a particular task, resulting in a total of 12 tasks. We emphasize that the dorivation presented in Section 3.1 applies to a wide range of loss functions encompassing both supervised learning based and reinforcement learning-based methods. In this study, we opt for reinforcement learning- primarily because they do not necessitate manual labeling of high-quality solutions. As a representative method in this domain, we utilize the Attention Model (AM) (Kool et al.) 2019) as the backbone and employ POMO (Kwon et al. 2020) to optimize its parameters. Concerning the bandit algorithm, we select Exp3 and the update frequency is set to 12 training batches. We discuss the selection of the MAB algorithms and update frequency in Appendix A.4 with details on the solving logic of neural solvers and training configurations in Appendix A.7
neural solvers,
4.1 Comparison under Same Training Budgets
We consider a practical scenario with limited training resources available for neural solvers for all tasks. Our method addresses this challenge by concurrently training all tasks using an appropriate task sampling strategy. However, establishing schedule for STL is difficult due to the lack of information regarding
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Table 2: Comparison among our proposed method, multi-task learning (MTL), and single task training (STL) utilizing the same training budget. Specifically, STLavg. and STLbal. denote the allocation of resources, with an even distribution and a balanced allocation ratio of 1 : 3, respectively, among tasks with varying scales from small to large. The reported results depict the optimality gap (L) in the main aspects.
Method
TSP20
TSP50
TSP100
CVRP20
CVRP50
CVRP100
OP20
OP50
OP100
KP50
KP100
KP200
Avg. Cap
STLagg
0.009%
0.347%
3.614%
0.466%
2.301%
5.986%
-1.80%
1.2899
5.719%
0.028%
0.014%
0.017%
1.574%_10.302
STTaa.
0.021%
0.341%
3.176%
0.608%
2.312%
4.775%
00.70%%
1.288%
4.799%
0.034%
0.014%
0.016%
1.308%%+4..15
Small Budget
Naive-MTL
0.040%
0.779%
3.263%
0.663%
2.427%
4.364%
- -0.008%
2.380%
5.138%
0.037%
0.014%
0.017%
1.551%6-10.866
UW
0.039%
0. 1.1%
2.353%
0.164%
1.929%
3.712%
-0.134%
2,26858
1.819%
0.035%
0.015%
0.017%
1.312%% 262
Bandit-MTI.
0.057%
1.203%
4. 1.546%
0.947%
3.119%
5.400%
0.534%
2,05895
4.360%
0.043%
0.016%
0.020%
+.850%
POGrad
0.460%
2. 9.920%
7.776%
1.57098
4.800%
8.112%
0.600%
5,26498
9.129%
16. 16.%
0.019%
0.026%
4.7-65%-+222300
CAGrad
0.810%
4.584%
11 11.057%
1.642%
5.047%
8.208%
0.544%
5.074%
9.075%
0.047%
0.020%
0.036%
3.845%
Nash-MTL
0.372%
2.598%
7.047%
4.2552%
7.181%
0.363%
4.681%
8.526%
0,008
0.018%
0.022%
3.035%/410966
TAG
0.085%
0.678%
2.947%
1.0229%
2.2144%
3.916%
- -0.632%
1.530%
3.8503
0.063%
0.020%
0.012%
1.2749::00008 0.006
Random
0.060%
0.577%
3.02M%%
0.608%
2.2333%
4.117%
- -0.9156
1.0619%
3.483%
0.011%
0.014%
0.021%
0.19666:000000
Ours
0.0244
0.262%
1. 1.407%
0. 0.385%
1 1.521%%
2. 798%
1 1.081%
0.447%
1. 1.883%
0.033%
0. 0.015%
0.018%
0.643%:+0011 016
STLasg
0.005%
0.182%
2.240%
0.374%
1.664%
4.122%
1. 1.156%
0.661% 9
3.627%
0.025%
0.013%
0.015%
0.981%
STLea
0.008%
0.1844%
1.634%
0,450%
1.650%
3.476%
-1. -1.112%
0,677%
2 2.555%
0.028%
0.013%
0.013%
0.80096::00022
Malian Budgel
Naive-MTL
0.022%
0.120%
2. 239%
0.510%
1.974%
3.560%
-0.770%
1,16195
3.1797
0.033%
0.013%
0.0144%
1.021%%:00008
UW
0.031%
0.304%
1. 1.796%
0.392%
1.621%
3.180%
0.741%
1.3700%
3.770%
0.030%%
0.013%
0.016%
0.9625% 0. 198
Bandit-MTL.
0.030%
0.723%
3.319%
0.604%%
2.430%
4.374%
0.800%
1.1749
3.000%
0.041%
0.015%%
0.018%
1. 1.247%-+00.44 404
PCGrad
0.19%%
1.583%
5.171%
1.144%
3.641%
6.215%
0,135%
3.654%
7.164%
16.161%
0.017%
0.0200%
3.775%
CAGrad
0.986%
1.47%%
10.33.4%
-
3.805%
6.576%
0.023%
1.071%
7.537%
0.050%
0.019%
0.034%%
3.2
Nesb-MTL
0.177%
1. 1.400%
4.5664%
3. 132%
5.689%
-0.116%
3.626%
6.809%
0.046%
0.016%
0.020%
2. 2.177898++06000 650
TAG
0.023%
0.3311
1. 1.767%
0.372%
1-180%
2.762%
-.811%
0.88156
2 2.80%%
0.03-1%
0.016%
0.022%
0.807% 074
Random
0.044%
0. 0.290%
1.779%
0.4929
7.70%%
3.310%
1050%
2.028% 0.81889
0.030%
0.01 1%
0.014%
0.796%:+00.07
Ours
0.015%
0.192%
0. 0.955%
0.335%
1 1.226%
2. 260%
1.1211
0. 0.165%
1.127%
0. 0.028%
0.011%
0.013%
0.434%
STLoog
0.002%
0.117%
1 2900
0.2841%
1.278%
3.086%
-1. 1222
0.2 0.2199%
2.1015%
0.018%
0.011%
0.0144%
0.602%%:20012
STLeal
0.004%
0.115%
1. 1.020%%
0.367%
1.284%
2.588%
- -1.176%
0,2559
1 1.927%
0.024%%
0.011% 1%
0.012%
0. 0.50677-+00022
Lorge Budget
NTi
0.015%
0.229%
1 1.300%
0.395%
1.611%
2.901%
0.965%
0.447%
1 1.910%
0.027%
0.013%
0.012%
0.658%_10.040
UW
0.028%
0.203%
1.254%
0.341%
1.364%
2.780%
-0.443%
0.6149%
1.00%%
0.029%
0.013%
0.014%
0.642%
Bandit-MTL
0.008%
0.2311%
1.119%
0.489%
1.366%
2.513%
-1.201%%
0.419%
1.975%
0.033%
0.013%
0.018%
0.5.1%-10.112
PCGrad
-0.0065%
0. 0.11%
3. 3.688%
0.625%
2.250%
1.033%
- -0.878%
1.9388%
3.960%
15.176%
0.013%
0.019%
2. 2.66%% 1.0
CAGrad
0.19%
1.971%
6. 6.9477%
1.303%
2.802%
4.622%%
-0.611%
2.4945%
5.848%
0.053%
0.025%
0.024%
2. 1455%
Nach-MTL
0.019%
0.658%
1.9413%
0.502%%
2.005%
3.478%
-0.985%
1.6412%
3.7127%
0.013%
0.018%
0.021%
1 1.08%%
TAG
0.006%
0.058%
1.193%
0.3125%
1.287%
2.253%
- 0..29%
0.594
2.002%
0.029%
0.013%
0.015%
0.677% 000024
Random
0.019%
0.185%
1.177%
0.3715%
1.449%
2.814%
0.9225%
0.4010%
2 0.00%
0.031%
0.013%
0.012%
0.634%:.100088
Ours
0.01/1%
0.173%
0.866%
0. 0.321%
1.182%
2.165%
-1.133%
0.093%
0.987%
0.023%
0.011%
0.011%
0.393%:20.00 056
resource allocation for each task, and MTL methods are hindered by efficiency issues arising from joint task training. In this section, we compare our method with naive STL and MTL methods in terms of the optimality gap: gap% = (1 - obl. training time budget.
gt. x 100, averaged over 10,000 instances for each task under an identical
Baselines The total training time budget is designated as B. For STL framework, we consider three types of sched- ules for the allocation of time across varying problem scales: (1) Average allocation, denoted as STTavg. indicating a uni- form distribution of resources for each task; (2) Balanced allocation with each type of COP receiving resources equi- tably for f. denoted as STLbal: signifying a size-dependent resource assignment with a 1:2:3 ratio from small to large prob- lem scales, caaalescategoriiingg tasks into mada- levels; (3) Allocation determined by the bandit algorithm, denoted as STLbandit; utilizes the selection ratio of each task after training the neural solver by the proposed method. The first schedule is suitable for realistic scenarios where information regarding the tasks is unavailable and the second is advantageous when prior knowledge introduced. Although the third strategy is not typically available in practical applications, it serves to demonstrate the potential benefits of an adaptive training mochanism inherent in our proposed method. What's more, extensive MTL baselines are considered here: Bandit-MTT. (Ma et al. 2021 PC-
Table 3: Training time per epoch, represented in minutes. The COPs are classified into three scales: small, median, and large, which corre- spond to the sizes of 20, 50, and 100, respec- tively (50, 100, and 200 for KP).
COP
Small
Median
Large
TSP
0.19
0.39
0.75
CVRP
0.27
0.50
0.90
OP
0.20
0.41
0.60
KP
0.34
0.61
1.10
9

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Published in Transactions on Machine Learning Research March/2023
Grad Yuu ct al. 2020 Nash-MTL Navon et al. 2022).
Uncertainty- Weitthhiigiiiiggiggnnggg (UW) (Kendall et al. 2018):
CAGrad (Liu et al. 2021a and TAG (Fifty et al. 2021 (Detailed configurations for TAG are in Appendix
A.6). We also involve the random policy which samples the task uniformly at each training slot, and the
results of condensed version are presented in Table 2
Experimental settings To mitigate the impact of extraneous computations, we calculate the time necessary to complete one epoch for each task and convert the training duration into the number of training epochs for STL. Utilizing the same device, the training time for for each task with STL and MTL methods can be found in Table | and Table [ We assess three distinct training budgets: (1) Small budget: the time required to complete 500 training epochs using our method, approximately 1.59 days in GPU hours; (2)
Medium budget: 1000 training epochs, consuming 3.28 days in GPU hours; and (3) Large budget: 2000
training epochs, spanning 6.64 days in GPU hours.
All mothods were trained five times independently to ensure robust evaluation and statistical significance. Comprehensive experimental results are presented in Table Due to horizontal space constraints, we report only the average performance values for individual tasks, while for the Average Gap across all tasks, we provide both the mean and two standard deviations across the five training runs. We refer the more detailed analysis of the stability of each method on individual tasks to Figure I in Appendix A.12
Results In general, our method outperforms the MTL and STL methods in terms of averge gap across all the budgets used. When comparing with all MTT methods, our method demonstrates two su- perior advantages: (1) Better performance the solution quality and efficiency: In Table 2 typical MTL methods fail to obtain a powerful neural solver efficiently, and some of them even work worse than naive MTL and STL in limited budgets. Further, as ralustraaed in Figure 2 it can be seen that the method proposed exhibits superior efficiency regard- ing its convergence speed and overall effectiveness; (2) More resources-friendly The computation com- plexity of typical MTL methods grows linearly w.r.t. the number of tasks (a detailed analysis of the com- putational complexity of each MTL method is avail- able in Appendix (34), conducting these training methods still needs heavy training resources (High- performance GPU with quite large memories). The exact training time for one epoch w.r.t. GPU hour are listed in Table [ Under the same training setting, intermediate termination of prolonged training epoch for typical MTL methods incurs wasted computation resources. However, our method trains only one task at each time slot, resulting in rapid epoch-wise training that facilitates flexible experimentation and iteration.
Mean Objective Across Different MTL Methods During Training Serelt
Mean Oojechi TSP and CV
Mean Objectis for OP
Figure 2: Comparative analysis of MTL methods during training: The left graph shows the mean objective func- tion for TSP and CVRP (with tower-is-better criterion). and the right graph shows the same for OP and KP (with a higher-is-better criterion), demonstrating the superior performance of our proposed method under varying com- putational budgets.
As the training budgets increase, STL's 's advantages become evident in easier tasks such as TSP, CVRP-20, OP-20, and KP-50. However, our method continues to deliver robust results for more difficult tasks like CVRP-100 and OP-100. In addition to performance gains, the most notable advantage of our approach is that it does not require prior knowledge of the tasks and is capable of dynamically allocating resources for each task, which is crucial in real-world scenarios. When implementing STL, biases are inevitably introduced with equal allocation. As demonstrated in Table 2 the performance of two distinct allocation schedules can differ significantly: STT consistently outperforms STTavg. due to the introduction of appropriate priors for STL. What's more, the random policy achieves competitive results compared to MTL baselines due to its efficient training strategy (sampling one task at a time). It slightly outperforms STL as our mithod small and median budgets) because positive transfer exists between same-type COP during the early stages of model training. However, unlike our method which selects tasks that maximize positive transfers across all tasks, the random policy doesn't actively explore and exploit task relationships. This lack of strategic task selection explains why it underperforms compared to our approach.
10

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Published in Transactions on Machine Learning Research MMrchh2022)
Table 4:
The comparison results are obtained by training our model for 1000 epochs and STL models for 100
epochs each, amounting to a total of 1200 epochs.
TSP20
TSP50
TSP100
CVRP20
CVRP50
CVRP100
OP20
OP50
OP100
KP50
KP100
KP200
Avg. Cap
STL
0.013%
0.234%
1.673%
0.490%
1.613%
3.342%
-1.156%
0.812%
2.678%
0.030%
0.014%
0.012%
0.813% = 0.070
Ours
0.015%
0.192%
0.955%
0.335%
1.226%
2.260%
-1.111%
0.165%
1.127%
0.028%
0.011%
0.013%
0.434% 0.086
Comparison of Generalization Performance
our
Small
Media
Large
Optimallty Glap (S6
33
2.
3.
Epochs
Figure 3: This figure compares the generalization performance across different scales for TSP, CVRP, OP, and KP. The y-axis represents the average test optimality gap (%) on small, median and large scales for models trained on small (blue), median (orange), and large (green) scale single-task datasets. The red line denotes the results of our method after training for 1000 epochs.
4.2 Comparative Analysis and Generalization Performance
In this part, we evaluated the performance of our method against single-task training (STL) under approxi- mately equivalent total training epochs and generalization performance. Specifically, our method was trained for 1000 epochs across 12 tasks, while STL models were trained for 100 epochs per task, amounting to a total of 1200 epochs. The results are summarized in Table 4 highlighting the optimality gap for various tasks, including TSP, CVRP, OP, and KP. Compared to individual tasks, our method (trained 1000 epochs) consistently outperforms STL (trained 100 x 12 = 1200 epochs) across most tasks, with exceptions noted in TSP20, OP20, and KP50, indicating efficient utilization of the training epochs.
To assess the generalization performance, we compared our method, trained for 1000 epochs, against STL models trained individually for 1000 epochs. Figure 3 Jillustrates the average optimality gaps for TSP, CVRP, OP, and KP across different scales (small, median, and large). Our method demonstrates unparalleled superiority in three ways: (1) when considering the average performance
all problem scales for each type of COP, our method obtains the best results in CVRP, OP, and KP, and is equivalent to the results achieved by training TSP for about 500 epochs. This showcases our method's excellent generalization ability for problem scales; (2) Our method can handle various types of COPs under the same number of training epochs, which is impossible for STL due to the existence of task-specific modules; (3) Our mothod's training time is strictly shorter than the longest time-consuning task.
4.3 Study of the Influence Matrix
Our approach has an additional advantage as facilitates the identification of the task relationship through the influence matrix developed in Section 3.2 The influence matrix enables us capture the inherent relationships among tasks, providing empirical evidence based experiences and observations within the community. A detailed view of the influence matrix is presented in Figure ?7, revealing significant obscrvations: (1) Figure ?? highlights that the influence matrix computed using equation 10 possesses a diagonal-like block structure. This phenomenon suggests a strong correlation between the same type COP, which is not present within different types of COPs due to the corresponding elements being insignificant. Furthermore, within the same type of COP; we observe that the effect of training task on other tasks lessens with the increase in the difference of problem scales, Hence, training combinatorial neural solvers one problem scale leads higher benefits on similar problem scales than on those that are further
11

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Published in Transactions on Machine Learning Research (March/2025)
Influence Matrix
Influence of Training Each Task on Other Tasks
TSP-20
TSP-50
TSP-100
15P-30
0.912
0,163
.0.22
0,002 9,000 -..02 6.002 0,001 -0,022 5.003 -4.001 -..13
Siffancce
13P40
0.150
5,102
0.146
6,002 0.001
4.001 6,003 0.002 -0.01 0.000 55,00 0.000
CVRP-20
CVRP-50
CVRP-100
CVAP-30
0.003
0.001
-0.031
0.065 9.066
-0,001
6:000 0,001 -0.000 9.000 8,000 -0.000
14
0V8P50
19.00
0.001
0.003 0,054 0,873 0,031 5,000 0,001 9.000 -B 0OD 5,000 0.000
0P-20
0P-50
0P-20
08.002
0.001
-0.011 0,002 9.500 0.000 8.676 0.094 0.030 9.002 -9.000 -0.00
09:60
0.000
0.602
0,002 6,001 0,001 0.001 5,081 DATI
0269 6,000 0.000
KP-50
KP-100
KP-200
XP-50
5.003
0.000
...92 0,009 9.000 4.000 6.001
-.000
-.000
9.506 9,073 00033
KP 100
001
5,000
0,001
6.000 0,000 6.000 6.000 0.000 0.00
(6,002
5,67H 0.022
WP.200
(a) Influence matrix
(b) Influence distribution
Figure 4: This figure provides a visual representation of the mutual influence between tasks. The left-hand side displays the average influence matrix, as defined in equation ??, while the right-hand side illustrates the influence value throughout the training process.
away. Such scale generalization challenge for COPs has been noted in previous works, such as Figure in Kool al. (2019) and Figure 3 in JJohi al.] (2022), which demonstrate that neural solvers trained on specific problem sizes show degraded performance on other scales, with performance declining as the gap between training and testing sizes incrcases. This fact is in turn quantified by the inffuence matrix in our work. (2) Figure ?? presents a visualization of the influence resulting from equation f equation 7 over the course of the training process. Each point in the chart represents the influence of a particular task on another task at a specific time step. Notably, tasks belonging to the same type of COP are highly influential towards cach other due to the large variance of their influence values. This phenomenon indicates that for different scales of the same COP type, both positive and negative transfer exist simultancously. Positive transfer enables trained neural solver to maintain basic solving capabilities compared to random initialization. However, the negative transfer arising from scale differences hinders the neural solver's generalization ability across different problem sizes. In contrast, influences between different types of COPs are negligible, evident from the influence values being concentrated around O. This striking observation showcases that the employed combinatorial neural solver and algorithm, AM (Kool et al.) 2019) and POMO (Kwon et al. 2020), segregate the gradient space into distinct orthogonal subspaces, and each of these subspaces corresponds to a particular type of COP. Furthermore, this implics that the gradient of training each variant of COP is situated on a low-dimensional manifold, motivating us to develop more parameter-effiieet neural solver backbones and a algorithms.
4.4 Results on Real Datasets
In this section, we evaluate the efficacy of the proposed method using real-world datasets: TSPLib (Reinelt, 1991) and CVRPLib (Uchoa et al., 2017). The training encompasses six distinct tasks: TSP20, TSP50, TSP100, CVRP20, CVRP50, and CVRP100. The experimental configurations adhere those outlined in section 4.1 considering two budget scenarios: Small and Medium. From Table 2 we select the most effective basclines from Single-Task Learning (STL) and Multi-Task Learning (MTL), specifically STLbal. and UW [Kendall et al. 2018), respectively. The performance of STL for given instance is derived from the best outcomes across three models; for instance, the result for berlin52 in TSPLib is obtained from the optimal model among TSP20, TSP50, and TSP100. We only report the average results across the instances in Table and detailed instance- wise results can be found in Appendix A.10
12

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Published in Transactions on Machine Learning Research (March/2025)
Table 5: Comparison of different methods across TSPLib and CVRPLib w.r.t optimal- ity gap.
Our experimental results demonstrate superior performance across both TSPLib and CVRPLib datasets. Under small bud- get constraints, our method achieves optimality gaps of 4. 4.550% and 3.940% for TSPLib and CVRPLib respoctively, consis- tently outperforming the baseline methods. The improvement. is particularly significant compared to UW, which shows high variability (13. (13.145% optimality gap for TSPLib small budget). Under median budget conditions, our approach maintains its advantage with the lowest optimality gaps (3.177% for TSPL.b, 3.344% for CVRPLib), demonstrating robust performance across different problem settings and budget con- straints.
Small Budget
Median Budget
TSPLib
CVRPLib
TSPLib
CVRPLib
STL
5.953%
6.300%
3.975%
4.630%
UW Ous
13.145%
5.895%
6.277%
5.105%
4.550%
3.940%
3.177%
3.344%
5 Conclusions and Futhre Works
In the era of large models, training a unified neural solver for multiple combinatorial tasks is in increasing demand, whereas such a training process can be prohibitively expensive. In this work, we propose an efficient training framework to boost the training of multi-task combinatorial neural solvers with a multi-armed bandit sampler. With this framework, we can efficiently obtain unified neural solver capable of covering multiple types of COPs simultaneously. We believe that this framework can be powerful for multi-task learning in a broader sense, especially in scenarios where resources are limited, and generalization is crucial. It can also help analyze task relations in the absence of priors.
Furthermore, the proposed framework is model-agnostic, which makes it applicable to any existing neural solvers. We speculate that different neural solvers may produce varying results on the influence matrix, and a perfect neural solver may gain mutual improvements even from different types of COPs. Therefore, there is urgent need to study the unified backbone and representation method for solving COPs.
Acknowledgments
This work was supported by National Science and Technology Major Project under Grant 20222D011640$ and in part by the Shenzhen Agency of Innovation under Grant KJJ2D200200900395522006
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A Appendix
A.1 Problem Description
Traveling Salesman Problem (TSP) The objective is to determine the shortest possible route that visits each location once and returns to the original location. In this study, limit consideration to the two- dimensional euclidean case, where the information for each location is presented as (Zi, E R sampled from the unit square.
Vehicle Routing Problem (VRP) The Capacitated VRP (CVRP) (Toth & Vigo 2014) consists of a depot node and several demand nodes. The vehicle begins and ends at the depot, node, travels through multiple routes to satisfy all the demand nodes, and the total demand for each route must not exceed the vehicle capacily. The goal of the CVRP to minimize the total cost of the routes while adhering to all constraints.
Orientecring Problem (OP) The Orienteering Problem (OP) is variant of the Traveling Salesman Problem (TSP). Instead of visiting all the nodes, the objective is to maximize the total prize of visited nodes within a total distance constraint. Unlike the TSP and the Vehicle Routing Problem (VRP), the OP does not roquire selecting all nodes.
Knapsack Problem (KP) The Knapsack Problem strives to decide which items with various weights and values to be placed into knapsack with limited capacity fully. The objective is to attain the maximum total value of the selected items while not surpassing the knapsack's 's limit.
A.2 Loss Decomposition For Adam Optimizer
Adam optimizer Kingma & Ba 2015) is more widely used and popular in practice than standard gradient descent. Accordingly, we derive the loss decomposition for Adam optimizer in a manner consistent, with the previous method. We first summarize the update rule of Adam as follows:
85-T
Σ 31- 9(T)
@(t) O(t
= O(t - 1)- 17: Σ g g(T)
Σ T=1
81-7
D
8/- (1(())]12 +
where g(r) VL(0(T- 1)) and go 0, 1)
8,, 1,2 are exponential
8 l1. 11tc
average parameters for the first and second order gradients. Our assumption is that sharing the second moment term correction for all tasks can be easily implemented by using single optimizer during training,
Given that the update is predicated on the optimization trajectory's history, we can use comparable calcu- lations gradient descent to infer Adam's contribution breakdown. Starting at the same point:
++ The -T'"''('''
((+1-Tg(7)
if p=i
1 (aa = -T))A4((++++1
(11)
Th V gabaaro (V (1)
8 Jehare (T)
Otherwise
With a little abuse of notation, g(T) means the gradient w...t. the selection task time slot 7 and Jshare is the version of taking derivatives only one shared parameters. Then plugging equation 11 into equation
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have
- = - (()) Σ 1(a T7
Σ g -)
k=1
(a) effects of training task Ty: of (t2.)))))
V'I (t))
11 (at mg)
g(k 1)
(12)
(b) effects of training task (T 9 53: (e((((t,ta)))a4t)
K Bp
V
Johoae (F (t))
10 1(
m $
Isharo 1)),
effects of training task (T+ (1)
Three similar parts are obtained finally.
A.3 Proof and Discussion on Theorem 1
Proof 2 (Proof theorem
I
[T) 11 (2) (T
(1, 12)
12 1(a = TT²)
(O)(((()))
VIJ(0((((()))
T vLE())
12 LF 1(xr = )
[VL((
[
1(at = T
VL)(O'(t)
VL,(0((,(()))
21' I(ar = !
[V
1(0 = =Tp)
VLJ(O'()) (z.(0((T,)))))
12 PP' 1(ar = T
|VL) [O
1(a; = TT?)
(t) - VO
=i 1(up = Tz?)
G
t=]
VoLj(V") Σ
v ml) (om(t)))
l(a = =Tp)
+1
22 W''t 1(ar = T
f2
1(4 = TTp)
De
VI, m(tt) (6m(r))) n(())
Σ/ 12
-1= I(ar = Tip)
G
(xx*)
M
1(a = TT)
Σ
ne
Lv 1ar TT¹)
(0+1
where (+) holds because we suppose all gradients have the same norm G during the training which can be guaranteed by regularizing the gradients; (x4) holds due to the Taylor cxpansion for (L) @ (T))((t) and mmlt') denotes the loss for braining lask Tmi(t)) selected at time step 1'; (* * +) is oblained from the assumption on the bounded Hession of the loss function. Then let mt, 12) = max (n)) (1,, 12) =
E/e ( = TT?,, the final result is obtained:
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Error Bounds Demonstration During the Training
0.011
Performance w.r.t. Bandit Algorithm and Update Frequency
23
0.034
Fres
Prog
1.0
Upper Bound
1.6
Oyt. Gxp
0.000
Mean Error Bounds 6.73684e-05
20006
1.0
0.0004
0.6
Selectioo om Intervals
Bandit Alporithm
Figure 5: This figure showcases the fluctuation of mean error bounds during the training process. The red dashed line is the mean error bounds computed across all intervals.
Figure 6: This figure presents the comparison re- sults of various bandit algorithms and update fre- quencies in terms of optimality gap (%).
12 (tn
( (T s (2())122
I(a; = (ld==T?)(( (t- (t).
t=
(2) Eis,, 1(a = Tt) (t- (t) during the training process. Following the experimental setup in the main paper, with 1 = 10-a and a selection frequency of 12, we calculate the average of this error upper bound for all tasks within a selection cycle, i.e.,
In practice, we can easily verify the magnitude of n (L1.2))
((..2) Eijnng Ele ((o=77)(0m.,()))) The experimental results are demonstrated in Figure 5 showing (2))))
Exiig that the average
upper bound steadily maintains at the magnitude of 10-4 Regarding the upper bound of the norm of the Hessian matrix in neural networks, there have been numerous prior studies (Sagun et al.
2017
2020 Li et. al. 2020 investigating this topic. Specifically, [Sagun Ghorbani et. al.j 2019 Yao et al.
(2017) pointed out that in the late stage of neural network optimization, most of the eigenvalues of its Hessian matrix are near zero: Ghorbani et al. (2019) found that the large negative eigenvalues of the Hessian disappeared rapidly, with the overall spectrum shape stabilizing only very few training steps; Similarly, Li et al. p220)) discovered that training progresses, the larger eigenvalues of the neural network's Hessian gradually decrease and concentrate near zero. These findings have been validated various image datasets and neural network architectures. Although there are currently specific observations for COP tasks, we speculate that similar conclusions for small value of M might hold.
et al.
A.4 Discussion on The Bandit Algorithm and Update Frequency
As shown in equation 2 the effect of the training task Tςς on Tj can be computed as
1(a =
This is subject to the indicator function 1(a, TT', which determines whether the task T is selected time step We first highlight the following tips: (1) For the stability and accuracy of the gradients, it is recommended to involve more than one step in the process of collecting gradient information. However, having an overly slow update frequency may yield incorrect results due to the lazy updale of the bandit algorithm; (2) When the update frequency is larger than 1, UCB family algorithms are unsuitable as they tend groedily select the same task in the absence of updates. Therefore, the update frequency is a crucial parameerr to specify, and Thompson Sampling and adversary bandit algorithms are suitable in this framework due to their higher level of randomness.
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Based on above discussions, we present empirical evidence and elaborate on the details. We performed experiments for the 12 tasks under small budgets, with five repetitions each. Five update frequencies were considered: 6, 12, 24, 36, and 48. The performances w.r.t. optimality gap are presented in Figure G
Effects of bandit algorithms The four algorithms considered are: Exp3, Thompson Sampling (TS); Exp3R, and Discounted Thompson Sampling (DTS). They have more exploration characteristics than UCB family algorithms with update delays. Moreover, Exp3R and DTS have the capability handle changing environments. According to Figure TS performs the worst among these four algorithms, it fails to handle potential adversaries and changing environments. DIS performs more robustly than TS since it involves discounted factor. Exp3 and Exp3R provide good results because they are able to handle adversarics and detect envirommental changes. However, Exp3R does not perform siguificantly better than Exp3 due to the neural solver's gradual and slow improvement, resulting no abrupt changes for Exp3R to detect. Based on the observed performance, appears that simple procedures such as introducing a discounted factor in DTS and basic adversary bandit algorithms such as Exp3 are sufficient for handling our casc.
Effects of update frequency The update frequency affects the accuracy of influence information approx- imation and the tension in the bandit algorithm. Appropriate selections must balance these two factors. Figure 0 shows that the frequency of 12 generally yields the best results across different bandit algorithms. DTS and Exp3 exhibit deteriorating performance with higher frequencies, resulting from numerous lazy updates. By contrast, Exp3R does not have this property because increasing the frequency helps detect changing points more quickly. As a consequence, the number of tasks (12 our case) appears to be an appropriate empirical choice to balance these two factors.
A.5 Details of Multi-task Learning Methods
In this section, we provide detailed descriptions of the MTL. baselines used in our main experiments. Specif- ically, we elaborate on the algorithm-level details of Bandit-MTL (Mao et al., 2021). PCGrad (Yu et al., 2020), Nash-MTL (Navon et al. 2022). Uncertainty-Weigheeee et al. 2021a) and IMTT. (Liu et al. 202Tb). All these methods train multiple tasks simultaneously at each training step and achieve positive transfer by designing different strategics to balance task-specific losses through loss weights. Therefore, these methods share basic computational complexity of O(N(F + B)) for network forward and backward propagation in one training step, where F and B denote the compu- tational cost of forward and backward pass respectively. We also present the detailed derivation of their computational complexity as shown in Table
(UW) (Kendall et al. 2018 CAGrad {Liu
Bandit-MTL Bandit-MTL formulates the task weighting problem as an adversarial multi-armed bandit problem. At each training step, it computes the loss weights by regularizing the variance of task losses, which means it attempts to find weights that can both minimize the weighted sum of task losses and reduce the variance among different task Josses.Thess weights are optimized using mirror gradient ascent under the constraint that they sum to one. Since the computation of loss weights only depends on the task losses and the optimization process efficient in practice, Bandit-MTL introduces an additional (NN) computational complexity for updating the weights.
PCGrad PCGrad addresses the gradient conflict issue in multi-task learning through gradient surgery. At each training step, it identifies conflicting gradients between task pairs and modifies them by projecting each gradient onto the normal plane of the other, effectively removing the interfering components. Since PCGrad needs to compute pair-wise gradient projections for the gradients of model parameters, it introduces an additional computational complexity of O( at most, where D is the number of model parameters.
Nash-MTL Nash-MTL reformulates the gradient combination in multi-task learning as cooperative bargaining game, where each lask is treated as a player negotiating for an agreed update direction. It utilizes the Nash bargaining solution from game theory to find a proportionally fair update direction that benefits all tasks without being dominated by any single large gradient. To compute this solution, Nash- MTL requires calculating pairwise gradient inner products and solving a lincar system using a variation of the concave-convex procedure. Although the optimization process is efficient in practice, the computation
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of pairwise gradient operations introduces an additional computational complexity of O(N where D is the number of model parameters.
UW UW proposes a principled approach to balance multiple task losses by leveraging homoscedastie uncer- tainty. It interprets the homoscedastic uncertainty as dependenn weighting and uses this interpretation to automatically adjust the relative weights of different tasks. The method derives these weights directly from the estimated noise levels of each task, making it applicable both regression and classification tasks. Since the weights can be directly computed from given noise levels and applied to the losses, UW introduces only ((1) additional computational complexity.
CAGrad CAGrad proposes a novel approach to balance multiple tasks by considering both the average loss and the worst-case local improvement of individual tasks. At each training step, it seeks an update direction that maximizes the minimum improvement across all tasks within a neighborhood of the average gradient, thereby ensuring that no single task is disproportionately disadvantaged. Due to the complexity of solving the underlying min- max optimization problem, the explicit computational complexity analysis of CAGrad is not provided in our study.
IMTL IMTL aims to achieve fair optimization across tasks through two key components: loss balanc- ing and gradient balancing. For task-shared parameters, it computes scaling factors through a closed-form solution to ensure that the aggregated gradient has equal projections onto individual task gradients. For task- specific parameters, dynamically adjusts task loss weights to maintain comparable loss scales across tasks. The method involves two stages of computation: the loss balance stage (IMTL-L) which requires normalizod gradients computation for each task, and the gradient balance stage which involves matrix operations for computing scaling factors. Due to these computations, IMTL introduces an additional computational com- plexity of ONND + Nº), where the 22D term comes from matrix multiplications between task gradients, and the 8 term arises from solving a (N - 1) x (N- - 1) linear system.
A.6 Configurations and Detailed Results of TAG
Strictly speaking, TAG (Fffty et al. 2021) is not suitable as bascline because it involves two stages: collecting lask affinity and training. The main challenges lie in how to allocate weights to these two stages under limited resources and how to determine the optimal mumber of groups. However, to highlight the superiority of the proposed method, we disregard the time required for collecting task affinity and allocate training time proportionally to the number of tasks in each group. The number of groups is set to 2, 3, and 4, and the best- best-performing result is ultimately selected. The grouping results obtained from TAG are demonstrated in Table 6 As evident from the table, the results of TAG are comparable to those of MTL- based methods under all budget constraints and grouping strategies, both of which are significantly inferior to our proposed approach.
A.7 Experimental Settings
Solving Logic of POMO POMO (won et al. 2020) applies the construction scheme to solve combinatorial optimization problems (COPs) by generating a feasible solution for an instance incrementally. For example, in the Travelling Salesman Problem (TSP), when solving an instance with n nodes, if the partial solution currently contains nodes, one node needs be selected from the remaining n k nodes. Once a specific node is selected and added to the partial solution, the next partial solution, containing k + 1 nodes, is determined. This process continues until all nodes are included in the partial solution, which point a feasible solution is achieved.
Model structure We adopt the same model structures as in POMO (Koon et al. 2020) to build our model. To train various COPs in a unified model, we use a separate MLP on top of the model for each problem, which we call Header. This header facilitates correlation of input features with different dimensions. For TSP, we use two-dimensional coordinates, {(x, i 1.2...,, n), as input, while CVRP and OP have additional constraints on customer demand and vehicle capacity, in addition to two-dimensional coordinates. Hence, their input dimensions are 3 and 3, respectively. Moreover, in OP, the prize is assigned based on the
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Table 6: Performance of TAG under different grouping strategies. (L) in the main aspects.
The reported results depict the optimality gap
Group TSP20
TSP50
TSP100 CVRP20
CVRP50
CVRP100
OP20
OP50
OP100
KP50
KP100
KP200
Avg, Gap
0.624%
2.196%
3.965%
1.270%
0.033%
0.663%
2.857%
0.630%
2.303%
1.30%
-0.622%
1.542%
3.882%
0.050%
0.020%
0.039%
Small Budget
1
0.016%
0.237%
2
0.6188
2.317%
1.182%
1.401%
3
0.038%
0.737%
3.031%
0.660%
2.402%
4.209%
-0.553%
1.720%
4.155%
0.045%
0.015%
0.018%
1
0.021%
0.270%
2
0.050%
0.443%
2.179%
1.370%
0.678%
2.377%
4.322%
0.046%
0.792%
3.274%
0.703%
2.468%
4.344%
-0.116%
2.010%
4.649%
0.037%
0.014%
0.020%
0.379%
1.496%
2.787%
0.754%
0.025%
0.310%
1.745%
0.459%
1.820%
3.285%
-0.812%
0.896%
2.590%
0.032%
0.016%
0.023%
Median Uudget
0.008%
0.141%
0.400%
1.577%
2.915%
0.758%
0.031
0.368%
1.975%
0. 0.182%
1.883%
3.129%
-0.791%
0.957%
2.730%
0.0311%
0.015%
0.018%
0.012%
0.186%
0.028%
0.212%
1.218%
0.849%
0.447%
1.741%
3. 3.199%
0.035%
0.452%
2.378%
0.555%
2.063%
3.667%
-0.700%
1.134%
3.092%
0.035%
0.014%
0.019%
0.301%
1.174%
2,212%
0.566%
0.022%
0.210%
1.212%
0.352%
1.518%
2.797%
-0.100%
0.543%
1.910%
0.032%
0.013%
0.018%
Large Uudget
1
0.0011
0.078%
0.310%
1.207%
2. 2.260%
0.562%
0.020%
0.211%
1.251%
0.368%
1.521%
2.785%
-0.913%
0.556%
1.979%
0.031%
0.013%
0.014%
0.006%
0.093%
0.009%
0. 1.10%
0.667%
0.581%
0.328%
1.37%%
2. 5.90%%
0.024%
0.284%
1.443%
0.420%
1.670%
3. 3.040%
-0.853% 0.654%
2.163% 0.037% 0.014% 0.013%
distance between the node and the depot node, following the setting in AM (Kool al. 2019; The KP takes owo-dimensional inputs, {(w,, v n1,, with wy and os representing the weight and value of each item, respectively. As such, we introduce four kinds of Header to embed features with different dimensions to 128. The embeddings obtained from the Header are then passed through a shared Encoder, composed of six encoder layers based on the Transformer (Vaswani et al., 2017). Finally, we employ four type-specific Decoders, one for each COP, to make decisions in a sequential manner. The shared Encoder has the bulk of the model's capacity because the Header and Decoder are lightweight 1-layer MLPs. Furthermore, when solving specific COP, we only need to use the relevant Encoder, Header, and Decoder for evaluation. Since the model size is precisely the same, the inference time required is similar to that of single-task learning.
Hyperparameters In each epoch, we process total of 100x 1000 instances with batch size of 512. The POMO size is equal to the problem scale, except for KP-200, where it is 100. We optimize the model using Adam (Kingma & Ba, 2015) with a lcarning rate of 1c-4 and weight decay of 1c-6. The training of the model involves 1000 epochs in the standard setting. The learning rate is decreased by le-1 at the 900th epoch. During the first epoch, we use the bandit algorithm to explore at the beginning of the training process. We then collect gradient information by updating the bandit algorithm with every 12 batches of data. The model is trained using 8 Nvidia Tesla A100 GPUs in parallel, and the evaluations are done on single NVIDIA GeForce RTX 3090.
Bandit settings We utilized the open-source repository (Besson 2018 for implementing the bandit algo- rithms in this study with default settings.
A.8 Loss and Gradient Norm of Each Task
One intuitive method of measuring the effect of training is to calculate the ratio of losses between adjacent training sessions. These ratios can be used to calculate training rewards for each corresponding task. How-
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Training Loss For Each Task
Gradient Norm For Each Task
6250 230
Figure 7: Training loss for each task.
Figure 8: Gradient norm for each task.
ever, as shown in Figure 7 this method of calculating rewards is not effective because they are not sufficiently distinct to guide the training process properly.
Computing the inner products of corresponding gradients to analyze how training one task affects the others can lead to a mislcading calculation of rewards and training process. Figure g visualizes gradient norms for each task in the logarithmic scale. We observe that the gradient norms are not in the same scale, which becomes problematic when jointly training different COP types. In such cases, the rewards of certain COP types (such as CVRP our experiments) may dominate the rewards of other types.
A.9 Clarification on Performance Improvements and Further Ablations
We would like to clarify that pross-problem-type performance improvements do not stem from the shared encoder learning mutual improvement information. Rather, the improvements come from efficient training (see GPU hours/epoch in Table 3 and positive transfers in independent model parameter subspaces for each COP type: As observed through the influence matrix in Figure ??, different, COP types correspond to nearly orthogonal subspaces in the model parameter space. Each subspace learns solution information for its corresponding COP type, though this allocates fewer training parameters to cach COP typc. Based on this understanding, we further enhanced our method by the following setting.
Given a neural solver backbone and considering that different types of COPs exhibit minimal correlation, we investigate the effectiveness of our method by training separately each COP type. Following the experimental setup in Table 2 compared two training approaches: (1) class-specife training (Ours-4G), which groups 12 tasks into four problem classes (TSP, CVRP, OP, and KP) with the budget equally divided among these classes, and (2) collective training of all 12 tasks (Ours-12T). The comparison was conducted across three budget levels: Small, Medium, and Large.
Table 7: Comparison of performance between Ours-12T and Ours-4G under different budget sizes
Budget
Method
TSP20
TSP50
TSP100
CVRP20
CVRP50
CVRP100
OP20
OP30
OP100
KP50
KP100
KP2000
Avg. Gap
Oun-11T
0.019%
0.248
1.325%
0.373%
1.476%
2.741%
-1.107%
0.402%
1.826%
0.033%
0.014%
0.020%
0.614%
Small
Ours44G
0.013%
0.167%
0.919%
0.342%
1.32%%
2,444%
-1.025%
0.802%
2.126%
0.045%
0.017%
0.017%
0.583%
Medium
Ous-12T
0.014%
0.195%
0.911%
0.381%
1.19
2.219%
-1.138%
0.119%
1.000%
0.029%
0.012%
0.013%
0.412%
Ours-te 3G
0.008%
0.131%
0.697%
0,295%
131%
2.105%
-1.126%
0.198%
1.373%
0.017%
0.029%
0.019%
0.409%
O2rs-12T
0.013%
0.181%
0.842%
0.3155
158%
2.122%
0.041%
0.901%
0.011%
0.0115%
0.374%
Large
Ours4G
0.005%
0.077%
0.427%
0.281%
0.967%%
1.823%
-1.201%
0.03456
0.977%
0.046%
0.018%
0.015%
0.288%
Table 7 presents the performance comparison between class-specific training (Ours-4G) and collective training (Ours- 12T) under different budget sizes. Across all three budget levels, Ours-4G consistently outperforms the best results of Ours-12T among 5 repeated runs, achieving lower average optimality gaps (0.583% vs. 0.614% for small budget, 0.409% vs. 0.412% for medium budget, and 0.288% 0.374% for large budget).
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This superior performance of Ours-4G demonstrates that when task relationships are known priori (e.g.. through the influence matrix), separating tasks into groups can lead to better results. This improvement may be attributed to reduced interference from less correlated tasks, allowing the model to focus more effectively on training highly correlated tasks together. However, it's worth noting that Ours-12T, which trains all tasks collectively without requiring any prior knowledge of task relationships, exhibits more general applicability while still achieving competitive results as shown in Table 2 These results highlight the complementary advantages of our proposed methods: Ours-4G achieves optimal performance when task relationships are known, while Ours- provides robust and effective solution when no prior knowledge is available, making our framework adaptable to various practical scenarios.
A.10 Detailed Results on TSPLib and CVRPLib
In this section, we present the experimental results on TSPLib and CVRPLib that were omitted in the main text.
The results presented in Table B clearly illustrate the superiority of our proposed method across both budget scenarios on the TSPLib datasct. Notably, our method consistently achieves the lowest optimal- ity gaps, highlighting its robust performance under varying budget constraints. The average optimality gap for our method is 4.55% and 3.18% for small and median budgets respectively, which are significantly lower compared to the other approaches, including STLba. and UW. Moreover, an analysis of instances with problem scales of 100 or greater reveals that our method not only maintains but often enhances its performance advantage as the problem scale in- creases. This trend underscores our method's ex- cellent scalability and generalization across larger problem sizes, which is critical for practical applica- tions where larger datasets are common.
Table 8: Comparison results on TSPLib. The reported results depict the optimality gap (%) (1) in the main aspects.
Small Dudget UW
Median Dudget
Instances
8TT
Oun
STT4a.
UW
Ours
berlin82
0.077%
11.849%
0.988%
0.0955
3.161%
0.0%
st70
0.997%
3.125%
1.024%
0.837%
1.258%
0.33%
p=76
0.82456
2.673%
0.818%
0.032%
1.5522%
0.60%
ea176
2.61%
4.563%
2.39%
1.237'
1.978%
1.894%
rd100
1.877%
9.0145%
0.568%
0.685%
0.855%
0.673%
knoA100
1.887%
11.177%
3.229%
1.907%
4.494%6
0.772%
lroC100
1.269%
12.014%
1.50
1.308%
0.239
0.292%
kroD100
1.2235%
12:2773
2.352%
2: 2.528%
5.378%
1.699%
ell101
2.966056
7.2033%
2.3799%
2.37716
3.5/2636
1.343%
Bal1S
3.727%
15,20436
3.4199
3.873%
7.177%
3.775%
ch130
4.7%
8.7300%
2.7883%
2.80076
1.448%
ch150
4.71155
12,0645
3.295%
2. 2.53995
3.7399
2.432%
tep221
14.127%
23.313336
11.544%
10.105%
13.4522%
7.579%
a280
20.685%
32.415%%
14.464%
14.1125%
18.6339
11.46%
pcb112
21.608%
31.618%
17.406%
15.
19.507%
13.254%
Avg. Gap
5.953%
13.145%
4.53%
3.975%
0.277%
3.177%
In the results presented for CVRPLib in Table 9 our approach demonstrates clear advantage over the other methods, with an average optimality gap of 3.94% in the small budget scenario and 3.344% in the median budget scenario. These results are superior to those achieved by STLbal. (6.3% and 4.63%, respectively) and UW (5.895% and 5. 5.105%, respectively). Such findings highlight the efficacy of our method, especially under more constrained budget conditions. Furthermore, analogous to the results on TSPLib, our method exhibits superior performance particularly for instances where the problem size exceeds 100. Notably, for instances in the 'X' series, our method consistently achieves the lowest optimality gaps compared to both STL... and UW. Furthermore, we present a more comprehensive comparison of results on CVRPLib X-set instances in Table [O Our method achieves the best performance in terms of Avg. Gap under both small and median budgets, with 9.21% and 9.174% respectively. Notably, for large-scale instances (n:>536) under the small budget, our method consistently outperforms all alternatives. However, it is evident that the performance improvement from small to median budget is not significant, indicating that generalization to untrained scales remains a bottleneck.
A.11 Demonstration of the Bandit Algorithms
This section presents detailed information on various bandit algorithms, as shown in Figure 9 including the selection count and average return for each task. It is evident that TS algorithm dominates in all 12 tasks, leading to poor performance on tasks where training limited. In contrast, other bandit algorithms maintain balance across all tasks, resulting in better average results.
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Table 9: Comparison results on CVRPLib. The reported results depict the optimality gap (%) (J) in the main
aspects.
Small Budgel
Median Budget
ST
UW
Our
STLbal.
UW
Ous
Instances
UW
Ours
ST
UW
Ours
B-n35-k5
2.804%
2.7.6%
1.605%
1.915%
3.375%
1.419%
Ea76-k10
3.718%
2.96%
2.291%
2.78%
2.493%
2.49%
B-u31-k5
2.535%
2.164%
2.201%
0.887%
2.868%
1.7341%
E-n76k77
5.742%
4.817%
2.777%
3.393%
4.489%
2.7%
B-u43-66
2.07%
1.612%
1.328%
1.433%
1.162%
1.091%
E-n51-k5
3.833%
6.29638
4.167%
5.117%
6.183%
3.203%
B.n50.k8
2.653%
2.22%
0.753%
2.081%
1.537%
0.908%
E-n101-k14
5.439%
6.803%
4.439%
5.398%
5.018%
4.417%
B-utd-k7
4.102%
1.196%
2.999%
1.648%
4.500%
2.935%
E-u22-k4
0.075%
0.075%
0.073%
0.075%
0.102%
0.075%
B-u68-99
2.978%
3.814%
3.087%
2.005%
3. 3.628%
2.674%
E-u101-k8
7.085%
6.824%
5.182%
5.100%
5.855%
4.247%
X-1129.k18
4.634%
4.989%
3.272%
4.152%
4.445%
1.606%
E.u33-k k4
1.088%
2.321%
2.021%
1.234%
1.662%
0.562%
X-n157-k13
13.281%
7.763%
3.434%
8.206%
6.047%
3.243%
E.n23-k3 k3
0.9%
0.991%
0.402%
0.634%
0.634%
0.61%
X-n162-k11
9.169%
7.895%
4.925%
6.887%%
6.812%
3.77%
E-a76-18
4.534%
3. 3.425%
2.486%
2.814%
2. 802%
2.044%
X-n106-k14
6.514%
5.357%
1.021%
5.818%
4.832%
2.897%
E.u76kk14
2.931%
3.233%
2.287%
3.571%
3.945%
1.767%
X-1153-122
14.706%
12.736%%
9.842%
13.0819
12. 746%
11.582%
B-n57-ke9
2.354%
3.336%
2.666%
3.08%
2.693%
2.602%
X-1172-k51
9.696%
8.034%
5.79%
8.685%
7.601%
5.574%
B-n50-k7
4.200%
3.241%
2.666%
2.387%
2.387%
188%
X-n143-k7
13.044%
11.111%
5.389%
8.501%
9.398%
5.679%
B-nt5-k5
3.276%
3.421%
1.564%
2.706%
3. 3.11%%
2.517%
X-n139-k10
6.853%
6.412%
3.201%
4.548%
4.61%
2.667%
B-n61-k9
5.576%
7. 7.892%
6.2741%
6,225%
5.674%
5.527%
X-n167-k10
9.207%
9.677%
5.261%
6, 6.896%
7.7144%
4.51%
B-a52-k7
3.808%
5.024%
2.101%
1.8%
3.048%
1.878
X-1176-k26
9.509%
9.657%
8.049%
9.96%
9.42%
10.584%
B-338-k
2.702%
2.983%
1.901 9
1.87%
2.82%
1.727%
X-n134-k13
12.317%
11.499%
6.519%
9.%9
9.204%
5.727%
B-441-66
1.546%
1.67%
1.132%
0.719%
1. 1880%
1.413%
X-n125-k30
6.379%
5.126%
5.176%
6.327%
5.31%
4.863%
B-n39-k5
3.877%
2. 198%
1.623%
1.735%
2.463%
1.02%
X-1181-1223
7.51%
5.866%
3.14%
1.759%
1.317%
2.812%
B-n63-k10
5.197%
5.137%
3.69%
4.31%
5.008%
2.268%
X-110111:25
7.612%
6.11%
4.17%
6.356%
4.715% 1.%
B-a78-k10
7.877%
5.917%
5.123%
1.805%
1.889%
1.763%
X-11220-16
11.344%
9.381%
5.698%
6,232%
7. 7.243%
4.003%
B-n66-k9
2.698%
3.774%
2.327%
3.547%
2.084%
1.902%
X-n110-k13
3.906%
5.189%
3.022%
2.960%
3.245%
1.866%
B-n57-k7
2.612%
1.715%
0.134%
1.00%%
2.024%
0.061%
X-n115-k10
7.958%
8.94%
1.5%%
6.348%
4.232%
2.674%
B-a15-k6
6.91%
7.247%
6.786%
2.885%
6.918%
2.257%
X-n148-k46
9.003%
7.867%
5.387%
7.881%
6.486%
5.012%
B-a56-K7
6.94%
6.8233%
4.134%
4.106%
5.888
2.848
F-n15-k1
1.211%
1.158%
3.135%
3.875%
1.323%
2.141%
B-167-k10
4.882%
4.788%
4.574%
4.793%
4.582%
3.959%
F-a135-27
30.986%
28. 28.722%
16.019%
16. 8933%
24.714%
8.727%
B.344k55
2.400%
1.404%
1.278%
1 1.201%
1.911% 1.156%
F-172-k4
16.616%
14.387%
12.906%
12.541%
14.415%
11.514%
Continued in next column
Avg. Gap
6.3%
5.895%
3.94%
4.63%
5.105%
3.344%
A.12 Stability of Each Method
This section examines the stability characteristics of the compared methods presented in Figure I1 The con- fidence interval plots reveal significant differences in stability among the various approaches. For small-scale problems such as TSP20 and CVRP20, most methods demonstrate relatively narrow confidence intervals, indicating consistent performance across trials. However, approaches like CAGrad and Nash-MTL exhibit increasingly wide error bars on large-scale problems, suggesting high variance in solution quality. In contrast, our proposed method maintains remarkably tight confidence intervals across all problem scales and budget configurations. For instance, in the TSP100 scenario, while methods like TAG and UW show error bars spanning over 2% in optimality gap, our approach maintains a confidence interval below 1%, demonstrating exceptional stability. Similarly, for CVRP100, our method achieves not only the lowest mcan optimality gap but also the narrowest confidence interval among all compared approaches, particularly notable given the in- herent complexity of vehicle routing problems. This stability advantage extends to OP100 and KP200, where our method consistently delivers reliable performance regardless of the computational budget allocated.
We further demonstrate the stability of each model obtained by the specific methods on 10000 test instances from each COP and the corresponding 2-sigma error bar plot shown in Figure 10. It is generally accepted that longer training epochs lead to reduced standard variance for each method. Additionally, our method produces a model with the most stable performance in most scenarios when compared to other MTL methods across almost all cases.
A.13 Additional Experiments on Other Domains
We select the challenge domain on Time Series to evaluate the performance of our method. Following the common practice in this domain, there are multiple series in one piece of data and the prediction on each series is seen as a task. We consider Long-term Forecasting tasks comprising ETT (4 subsets), Weather,
For forecasting and imputation tasks, we ignore the Electricity and Traffic dataset because all MTL methods meet. out of memory errors because there are too many tasks.
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Table 10: Comparison results on CVRPLib large X-set instances. (%) (L) in the main aspects.
The reported results depict the optimality gap
Small Budget
Median Budget
X-1359-329
13.357%
10.196%
6.311%
8.97%
9.2588%
5.188%
Trstamors
ST.a...
UW
Ours
ST..a..
UW
Ours
X-n367-k17
26.92%
25. 243%
12.559%
22 15
17.674%
11.841% X
X-n376-k04
8.37%
5.268%
4.461%
6.62%
4.62%
19.442%
X-2253-k22
14. 71.70%%
136%
9.812%
13.081%
12.746%
11.582%
X-1384-352
12.17%
9.467%
7.249%
8.922%
8,7933%
5.539%
X-a157-k13
13.281%
7.763%
3.434%
8.206%
6.047%
3.243%
X-1399 k38
15.021%
11.46%
7.761%
13.005%
10.953%
7.308%
X-1172-k51
X-a162-k11
X-n107-k10
9.169%
9.606%
9.207%
7.895%
9.577%
8.034%
1.925%
5.261%
5.79%
0.890%
5.867%
8.683%
6.812%
7.714%
7.601%
5.574%
3.77%
4.51%
X-n-001-229
X-D411-K19
X-n420-k130
31 25.7
13.386%
162%
12 6616
9.919%
29.84%
17.378%
7.311%
6.102%
17.178%
12.665%
26.846%
12. 12.125%
22.044%
9.314%
14.923%
7.341%
5.985%
X-117661:26
X-a181-k23
X-a186-k15
9.500%
8.11%%
7.
9.657%
5.866%
6.2%
8.049%
4.538%
3.14%
5.593%
4.759%
9.96%
G. 196%
4.317%
9.42%
10.584%
2.812%
3.55%
X-n119-329
X-n439-337
X-n429-k61
17 .733%
14. 14.223%
19.9995
10.405%
12.2111%
10.818%
11.945%
8.217%
7.665%
14.041%
12. 12.2511%
10.63%
11.785%
8.6345E
9.48%
7.716%
7.431%
19.02%
X-n1900168
23.249%
22. 66%
11.448%
15.895%
19.820%
6.168%
X-n458-x26
35.717%
33. 33.2544%
15.816%
27. 27.829%
75.700%
13.068%
X-a195-k51
11.5799%
8. 8.789%
6.58%
8.67%
8.08%
7.032%
X-1200-k366
7.974%
9.087%
5.255%
6.128%
6.436%
6.136%
X-n-199-1388
839/%
10.71:1%
8.997%
11.40-1%
9.654%
13.673%
X-4204-k19
10.311%
7.49%
4.979%
7.2399
6.1%
4.065%
X-1
X-n480-X70
13.033%
12.5188
10.31%
12.1%%
7.602%
7.449%
11.76%
9.71%
8.144%
8.725%
7.053%
8.86%
X-1214kk11
X-a209-k16
9.162%
20.68%
17. 17.096%
8 6.672%
5.258%
8.92%
15.749%
6.961%
13.662%
7.312%%
7.321% %
4.02%
X-n502-139
31. 761%
11.911%
16.177%
13. 13.729%
14.196%
13.057%
X-n219-173
4.375%
3. 3.497%
4.57 19
3.615%
3. 333%
9.83%
X-1519-k21
X-n524-x153
11 971%
6662%
10.539%
411%
14.372%
10.766%
12: 80%
893%
10.372%
21.226%
12.102%
11.31%
X-a223-k34 k:44
7.853%
6.587%
4.546%
7.2766%
5.964%
4.405%
X-n538-k06
15 463%
13 18.28
9.865%
14.05%
12.029
9.003%
X-1233-k16
X-a228-k23
111.33%%
15 03.%
13.957%
13.0441%
7.996%
7.36%
11.06-1%
12.66%
121%
225%
6.318%
9.17%
X-1548-k50
13.454%%
14 10.5335%
6.862%
10.33%
8.72%%
9.076%
X-1561-k42
21 571%
133.663%
10.129%
16.558%
11.263%
8.809%
X-a237-k11
12.34%
9. 9.22%
5.873%
9.229%
8. 1296
1.985%
X-u573-330
57. 57.107%
37. 193%
21.593%
25. 765%
29. 09.072%
15.744%
X-n242-1:88
7.808%
6.473%
4.18%
5.865%
5.295%
3.768%
X-n5860-159
14.7%
12. 12.448%
9.937%
12.221%%
10.015%
12.783%
X-a251-1:88
X-u247- k50
13.286%
9.639%
11.53%
6.9%
9.837%
3.38%
12.483%
8.354%
11.252%
7.026%
4.555%
12.42%
X-n613-x62
X-n599-k92
15.26 15.264%
18. 805%
12.339%%
14.054%
8.936%
7.7
11.369%
15. 146%
10.018%
11.519%
9.232%
10.044%
X-a256-k16
X-u261-k13
15. 276%
12.550%
11.406%
12.80 90%
7.483%
8.105%
10.57%
15.6
11 87%
9.952%
6.037%
5.664%
X-16277-43
32. 8299%
29. 237%%
15.841%
21.009%
17.086%
11.597%
X-n206-1:58
11.201%
9.328%
6.785%
9.050%
7 7.605%
7.123%
X-06-11-35
X-n670-x130
18. 38.37%%
398%
14.581%
7955%
12.882%
1%
15.38-1%
17.64%
13.113%
14.725%
10.383%
14.493%
X-a270 1:35
10.776%
9.511%
6.699%
6.707%
6.614%
4.996%
X-0688-275
22 086%
14.0133%
10.585%
16.578%
12.918%
10.125%
X-a275-1:25 1:28
12.332%
9.558%
7.728%
11.486%
7.87%
11.82%
X-u280-k17
14.6277%
12.6211%
8.265%
9.79%
12.057%
7.689%
X-1701-144
X-1716-335
32.99%
035%
27.968%
13.744%
17.734%
9.216%
31, 692%
13.87%
25, 95.924%
14.000%
159%
8.404%
X-a284-k15
28. 63.6631
19. 9636
11.-225%
151151
17.287%
9.917%
X-u733-x159
18 018%
12.28-1%
9.352%
13.937%%
9.87%
9.005% %
X-n280-160
10.273%
9.266%
6.653%
8.843%
7.496%
6.114%
X-n749-x988
21.17%%
17. 183%
12.182%
19.278%
14.994%
10.94%
X-9294-150
11.923%
8.741%
6.187%
10.047%%
7.500%
6.46%
X-n766-X71
19.917%
109%
11.691%
14.925%
14.018%
10.732%
X-a298-k3 k:31
19.9%
10.425%
6. 2.40%
9.784%
9. 9.086%
5.791%
X-1783-148
17.523%% 122%
12.0833%
17.502%
15.687%
10.984%
X-a3033-k21 121
X-a308-k13
20.819%
15.677%
12. 52.%%
17 17.5866
11.602%
7.1%
12.006%
13.931%
10.066%
15.671%
6.113%
8.787%
X-n801-140
25. .3.%%
16. 16.%
13.368%
18.48%
16.591%
17.866%
X-n317-k:3
X-u313-k71
10. 10.%
8.573%
7.316%
7.698%
6.984%
6.385%
8.206%
7.461%
5.571%
7. 7.28-1%
5.151%
7.512%
X-u876-159
X-n895-337
X-n837-x142
35.00%
16.701%
37.37
22, 1.9%
26. 15.%
11.45%
14.322%
15.525%
8.473%
23. 59.%
12.523%%
16.%
10.398%
461%
286%
11.984%
16.001% 1%
10.42%
X-a322 k:28
13.077%
9.637%
7.348%
10.97%
8.948%
6.12%
X-u827-k20
16.5%
12.763%
8.583%
13.25%
11.878%
7.405%
X-n916-1207
15.837%
14.357%
11.17%
10.88%
10.047%
12.44%
X-19366k151
24.6455 24.645%
18.922%
17.4%
23.3022%
17. 17.802%
20.5233%
X-1311-k15
17.917%
14.3%
8.582%
12.255%
12.554%
7.85%
X-1957-k87
26. 339%
17.266%
13.175%
21.262%
13.76%%
16.707%
X-33511440
X-1344-1:43
X-1336-881
10.591%
23.583%
12.481%
9.243%
8.832%
19.07%
10.175%
6.135%
6.333%
19.736%
9.793%
8.698%
6.89-1%
13.74%
8.080%
8.873%
5.927%
5.15%
X-n1001-k443
X-u979-258
33.074%
21 886%
23. 709%
28. 181%
16.201%
13.832%
23.08%
14.21%
18.18%
20.06%
13.084%
14.098%
Continned in next column
Avg. Gap 17.709%
13.659%
9.21%
13.134%%
11.653%
9.174%
Table 11: For Long-term Forecasting tasks, all the results are averaged from 4 different prediction lengths, that is {24, 36,45 48, 60} for ILI and {96, 192, 336, 720] for the others. "Baseline" provides the the results in the original paper. Results bold mean achieving the best performance among all methods
ETT-hi MSE
ETT-h2
ETT-ml MSE
ETT-m2 MSE
Whether MSE
Exchange MSE
ILI MAE
Method
MAE
MSE
MAE
MAE
MAE|
MAE|
MAE
MSE
MTL
0.496
0.487
0.450
0.459
0.588
0.517
0.327
0.371
0.338
0.382
0.613
0.539
3.006
1.161
Ban
0.138
0.420
0.398
0.363
0.533
0.643
0.304
0.220
0.327
0.254
0.319
0.181
1.424
3.955
UW
0.420
0.385
0.400
0.359
0.502
0.557
0.325
0.236
0.308
0.231
0.287
0.153
1.425
3.942
CAGrad
0.468
0.466
0.390
0.351
0.477
0.488
0.304
0.220
0.312
0.241
0.310
0.174
1.411
3.856
IMTL-G
0.145
0.423
0.392
0.352
0.465
0.462
0.303
0.219
0.319
0.245
0.286
0.153
1.399
3.776
Nash-MTL
0.468
0.472
0.409
0.370
0.468
0.483
0.310
0.225
0.315
0.240
0.302
0.165
1.376
3,806
Ours
0.418
0.385
0.383
0.343
0.506
0.547
0.299
0.215
0.360
0.277
0.333
0.193
1.689
5.189
Exchange and ILI datasets, and Imputation task comprising ETT and Weather. The backbone is AutoFormer (Wu et al. 2021a) and all the experimental settings keep the same as the original paper.
28

---

Published in Transactions on Machine Learning Research March/2023
(a) TS-6
(b) DTS-6
(c) Exp3R-6
(d) Exp3R-6
24 22" 22
(e) TS-12
12
(g) Exp3-12
(h) Exp3R-12
-25
(1) TS-24
i) DTS-24
(k) Exp3-24 24
Exp3R-24 24
(m) TS-36
(n) DTS-36
(o) Exp3-36
(p) Exp3R-36
infermatiion
(q) TS-48
(r) 4TS-48
(s) Exp3-48
(t) Exp3R-48
Figure 9: Further results of the bandit information. The caption of each subfigure "A-B" means the influence matrix obtained by algorithm A with update frequency B.
Results show that there are no consisting best methods for all datasets, however, our method can achieve the best performance consistently on out of datasets.
From these results, our method performs well in some cases, but generally speaking, there no one universal approach which can handle all tasks even on all datasets in task.
29

---

Published in Transactions on Machine Learning Research (March/200 /202))
1900
CM0L
(a) Small budget
(b) Median budget
(c) Large budget
Figure 10: Stability of the model obtained by different methods on 10000 instances from each COP with different budget allocations.
75P50
CVBP5
OPSI
T8P9)
CVR19)
019)
KP10
180
12
Figure 11: Comparison of methods across different budgets (small, median, and large) on TSP, CVRP, OP, and KP with varying problem scales. The results are presented as the optimality gap (%) for each method, with error bars indicating the standard deviation over 5 runs.
Broader Impact Statement
Our contributions significantly impact machine learning and its applications by introducing novel frame- work for training combinatorial neural solvers with multi-task learning. Based this, not only enhance performance over standard paradigms but also offer scalable method for training large models under re- source constraints. This advancement has the potential to revolutionize industries that rely on complex optimization, such as logistics and manufacturing, by enabling more sophisticated AI solutions. Further- more, our thcoretical insights into loss decomposition and the introduction of an influence matrix shod light on the inner workings of neural solvers, facilitating deeper understanding and improvement of machine learning models. However, the deployment of these advanced technologies also necessitates careful exami-
30

---

Published in Transactions on Machine Learning Research (March/200)
Table 12: For Imputation tasks, time series are randomly masked (12.5% 25%, 37.5%, 50% time points length-96. The results are averaged from different mask ratios. Results with underline mean achieving the best performance among all methods and those bold mean achieving the best among all MTL methods.
ETT-h1
ETT-h2
ETT-ml
ETT-m2
Weather
Method
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Baseline
0.103
0.214
0.055
0.156
0.051
0.150
0.029
0.105
0.031
0.057
Bandit-MTL
0.324
0.201
0.437
0.414
0.579
0.576
0.603
0.792
0.255
0.154
UW
0.268
0.143
0.354
0.280
0.669
0.767
0.774
1.124
0.391
0.324
CAGrad
0.2699
0.144
0.354
0.267
0.593
0.605
0.640
0.747
0.347
0.257
IMTL
0.270
0.145
0.353
0.266
0.594
0.605
0.681
0.854
0.388
0.337
Nash-MTL
0.268
0.143
0.347
0.255
0.637
0.694
0.693
0.866
0.489
0.494
Ours
0.300
0.174
0.411
0.366
0.473
0.384
0.609
0.734
0.283
0.174
nation of ethical and societal implications, including potential impacts on employment and the importance of ensuring privacy, socurity, and fairness in AI applications.
31


Reviews:
[{"review_id": "rxdoN8o8UU", "reviewer": "Reviewer_aEVN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. Applying MAB to multi-task training for combinatorial optimization is interesting and relatively unexplored. It provides a principled mechanism for adaptively allocating training efforts among different tasks\n\n2. The loss decomposition and the introduction of an influence matrix are conceptually appealing. They give a gradient-based explanation of how training one task impacts others (both within the same problem type and across different types).\n\nWeaknesses:\n\n1. (minor) While using MAB is promising, the motivation could be clearer. It would help if the authors provided more real-world scenarios or theoretical reasoning showing why an adaptive scheduler (bandit-based) is preferable to simpler heuristic schedules (e.g., uniform or priority-based sampling).\n\n2. (major) The influence matrix shows that cross-task correlations (between different CO problem types) are fairly low, which seems to contradict the idea that training on multiple tasks can improve overall performance and generalization. This is a key question: If cross-task gradients are almost orthogonal, from where does the boost in performance come? A more thorough explanation or illustration of how partial shared knowledge (e.g., shared encoder parameters or certain patterns in solution construction) benefits all tasks would strengthen the paper.\n\n3. (moderate) While the paper includes tasks up to size 100, testing on larger-scale instances (e.g., TSP-200, TSP-500) without training on them would provide stronger evidence of how the method scales in more realistic industrial settings. A pilot experiment could demonstrate whether the approach remains effective and whether the bandit selection shifts to favor smaller tasks or invests more in the large tasks over time.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a training paradigm for multi-task neural solvers addressing combinatorial optimization problems (COPs). The authors propose a loss decomposition framework under an encoder-decoder structure, which facilitates training via a MBA based task selection algorithm. \n\nEmpirically, the approach enhances training efficiency and generalization performance compared to traditional multi-task learning (MTL) methods and single-task learning (STL).", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Address the Cross-Task vs. Intra-Task Boost. Please correct me if I were wrong, I feel a bit confused as the empirical findings of overall performance improvements with the relatively low cross-task gradient overlap shown in the influence matrix. If the shared encoder or certain universal solution heuristics are playing a role, clarifying that mechanism would remove such counter-intuitive contradictions.\n\n2. For certain tasks or scales, (see influence matrix) training them together in the same problem family (e.g., TSP-50 vs. TSP-100) may introduce conflicts. Could the authors bring some insights behind this?\n\n3. Include (at Least Preliminary) Larger-Scale Experiments (e.g. TSP200 is enough for illustration). It would be interesting to check the generalization w.r.t. scale changes (say 100 to 200 in TSP problems). This would also strengthen claims of the method’s scalability and ability to generalize across a wide range of instance sizes.\n\n4. Provide insight into why random sampling may outperform single-task schedules in some cases. If random sampling systematically explores all tasks more uniformly, it might balance knowledge accumulation in ways that a naive or a non-adaptive schedule does not. This could be an interesting angle to explore and clarify.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "2vmUhXGRMY", "reviewer": "Reviewer_W2Yt", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\nS1. In my view, the main strength of the method proposed by the authors is that it is very general and does not depend on the specifics of the combinatorial optimization problem. In fact, the \"inner loop\" algorithm does not even need to be based on supervised learning -- indeed, the authors use a reinforcement learning method, which does not require ground truth labels for training. The method is a framework in the true sense of the word, and it is not a stretch to imagine it being used for other COPs by swapping out the neural solvers. In my view, it is an appealing proposal for \"foundation models\" for COPs.\n\nS2. The analysis around the influence of tasks on each other is very interesting -- and it is a nice property of the proposed framework that it enables such an analysis in the first place. It may also give a way of \"quantifying\" how close COPs are to each other.\n\nS3. The authors consider a wide range of MTL and STL setups in their evaluation, so the evaluation is quite robust in this sense.\n\n### Weaknesses\n\nW1. A doubt that I have about the reported results is that there are no confidence intervals or error bars (on the random initialization of model weights) reported in the tables and figures. The only place that mention repetitions is a small secondary experiment in the appendix. If the main experiments were indeed repeated, the number of seeds should be specified, and CIs and error bars added. In case they were not, it is necessary to repeat them so that the results are reliable.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work considers the problem of training a neural model that can solve several combinatorial optimization problems (COPs) at once. It casts this problem in a multi-task learning (MTL) context. The model architecture contains parameters that are shared between tasks as well as some that are specialized per-task. It proposes using a bandit algorithm to select, within the training loop, what should be the next COP to train on. The reward for this bandit is designed based on task losses, which the authors show are decomposable. The authors conduct experiments where they compare their method with an extensive range of MTL methods, generally showing favorable performance over them.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "C1. Please address W1 above, which I view as a mandatory condition for securing acceptance.\n\nC2. There is little to no detail about the workings of the other MTL methods, which are essentially the baselines for this work. I believe the paper would benefit from adding a high-level discussion (perhaps to the appendix if space is limited) of how the other methods operate. It is difficult to judge the complexity analysis in Table 1 without it. It should also be clear, in my opinion, *why* we see that their performance is not so great in this setting, and the proposed method is able to perform better. \n\nC3. It does not seem that the proposed method is dependent on the tasks themselves being COPs? Is this indeed the case or not?\n\nC4. Small points: many of the equation refs are of the form \"Equation equation 1\", consider removing the second \"equation\" throughout the manuscript. Typos: \"combinarotial\" (page 2)", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "3duu1khsbd", "reviewer": "Reviewer_ZdYQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The proposed method is general and efficient.\n2. The comparison to baselines is comprehensive.\n\nWeaknesses:\nMore experimental results and additional clarifications are suggested to support the authors' claims.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a framework for efficiently training a multi-task neural solver for combinatorial optimization problems (COPs). The authors propose a multi-armed bandit (MAB) approach to dynamically select tasks during training, which avoids the need for explicit loss balancing in traditional multi-task learning (MTL). They also provide theoretical loss decomposition under an encoder-decoder framework, leading to the construction of an influence matrix that reveals task relationships. Empirical validation of the approach on sampled instances and benchmark datasets (TSPLib and CVRPLib), demonstrating superior performance compared to many single-task learning (STL) and MTL baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The influence matrix is a key component of the proposed method. While the authors argue that it captures task relationships, it would be beneficial to see a more detailed analysis of how this matrix evolves during training and how it impacts the selection of tasks. Moreover, Figure 4 shows that there are no (or little) connections between different problem types. In this case, why should we train them together with a shared encoder? Why not train one model for each problem type?\n\n2. The paper compares the proposed method with several MTL baselines and STL. However, the comparison with STL seems to be based on different training budgets. Could the authors clarify if the comparison is fair in terms of computational resources and training time? Morevoer, when the training budget of STL increases, will the final converged results outperform MTL methods including the one proposed in this paper?\n\n3. The paper focuses on combinatorial optimization problems. How well does the proposed method generalize to other domains, such as natural language processing or computer vision? Are there any limitations or specific conditions (such as strong or weak relationships) under which the method performs well? \n\n4. The performance of the proposed method may depend on the choice of hyperparameters, such as the update frequency of the bandit algorithm. Can the authors provide more analysis to show how robust the method is to different hyperparameter settings?\n\n5. While the paper demonstrates results on TSPLib and CVRPLib, the instances selected are of small sizes. It would be valuable to see how the method performs on large-scale instances, such as CVRPLib X-set with up to 1,000 nodes. Will the MTL benefit large-scale generalization when compared to STL on CVRP?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers found the proposed framework/method interesting, generalizable, and conceptually appealing, with comprehensive and robust experimental analysis. While acknowledging these strengths, they raised many concerns and requested changes in their reviews. After rebuttal, most of the concerns raised were properly addressed, and all reviewers leaned toward accepting this work. In the official recommendation, reviewer W2Yt believes two revisions are still requested for the camera-ready version of this work: (1) the learning curves should be provided, and (2) the results with 2 standard deviations (~ 95% confidence interval) should be provided instead of 1 standard deviation.   

I read the paper in detail and fully concur with the reviewers' consensus that this work clearly meets the TMLR acceptance criteria (solid and well-supported claims, potential audience). In addition, I also believe reviewer W2Yt's requested changes are reasonable. Therefore, I recommend accepting this work with minor revision. 

I expect the authors can provide both the learning curves and experimental results with 2 standard deviation in the camera-ready version of this work.
Decision: Accept (poster)