Weakness: Method:
1. Problem Formulation:
A key step in the proposed method is to convert the multi-objective optimization problem into different constrained single-objective optimization problems. However, many conversion methods have already been proposed [1-3] which can tackle the non-convex Pareto frontier. None of them are discussed in this work. Is the proposed problem formulation new or adopted from other works?
What are its advantages (and disadvantages) over other methods?
What is J(\pi_k) in (7)?
2. Finding the Whole Pareto Front:
Theorem 1 is a crucial motivation of this work to find the whole Pareto front. However, the theoretical property of Theorem 1 is relatively weak. It can only guarantee that the optimal solution of the constrained single-objective problem can dominate other solutions that satisfy the same constraint. Therefore, the obtained solution is not guaranteed to be Pareto optimal (can be dominated by other feasible solutions out of the constrained set). In addition, even assuming all the single-objective problems can be successfully optimized, there is also no guarantee that all Pareto solutions can be found by the proposed method.
It is also confusing why the obtained solutions can exactly lie on the unit vector in Figure 1. In my understanding, the proposed method can find a solution that satisfies the preference-based constraint while minimizing the L2 norm of all objectives. Since the problem has an inequality constraint, the obtained solution should be close to the unit preference vector but not guaranteed to be exactly on the unit vector.
3. The Model Structure:
To incorporate the preference into the TSP-Net model, the proposed method simply adds the preference embedding to each node (city) embedding. According to the ablation study in Appendix, the preference embedding network can be a simple single-layer FC. The rest model is the standard TSP-Net where the whole transformer-based encoder can be totally replaced by a single 1-D convolution layer. Why can this simple structure work so well for MOTSP? Will it significantly hurt the model performance of each individual preference compared to a single objective model?
More concerns on the results are listed below in the experiment part.
4. Other Problems and Related Work:
One important advantage of the learning-based method is its flexibility to solve different problems [4]. Although this work focuses on MOTSP, I believe it could have a larger impact by showing its ability to solve other problems.
Many works on deep multi-task learning and multi-objective RL have been cited and discussed multiple times in the related work section, while they are not the most related work to this paper. I think it is better to shorten this part into a single paragraph, and leave more space to discuss the related work on learning-based solvers (e.g., [4]) and other approaches for MOTSP. Experiments:
5. Competitive Baselines:
According to the experimental results, the learning-based solvers are much better than the heuristic-based solvers. However, for the single objective TSP, the SOTA heuristic-solver (e.g., Concorde) usually has the best performance. Since the obtained Pareto front is not highly non-convex (as in Figure 2), the results for linear scalarization + Concorde should be included for a better comparison.
6. Unfair Comparison:
The comparison to other learning-based solvers is questionable. First of all, one contribution of DRL-MOA is the transfer learning based training method. The required training epoch (and time) is far less than those reported in this paper. Hence the results reported in Table 1 are misleading. It seems that all the PA-Nets are trained on the 120-city instances for 2,3 and 5 objective MOTSP. However, the original DRL-MOA is only trained on the 40-city MOTSP. As reported in this work, "The trained model of bi-objective TSP for DRL-MOA (Li et al.,2020) is used.", will it lead to unfair comparison?
The TSP-Net is more powerful than the Ptr-Net used in DRL-MOA, so it is expected that it can have better performance. However, it only moderately outperforms DRL-MOA on the 2 and 5 objective problems, and has worse performance on the 3-obj problems. A more reasonable baseline is to use the TSP-Net in DRL-MOA to check whether the preference-based model could lead to worse performance.
To calculate the hypervolume, this work reports the results with {100,500,500} preferences for the 2, 3, and 5 objective MOTSP instances, while it only reports the results of {100, 91, 40} networks for DRL-MOA. Although it is understandable that one advantage of PA-Net is to generate a dense approximation, the results of {91, 40} preferences should also be reported for a clear comparison. Since PA-Net (with 500 preferences) are already outperformed by DRL-MOA (with only 91 models) for the 3-objective MOTSP, will it be significantly outperformed by DRL-MOA with the same number of solutions?
7. Missing Experiment Settings:
How many training samples and epochs are used to train the PA-Net and DRL-MOA? How many test instances are used to measure the performance for different methods? If there is only one instance for each kind of problem, the results are not convincing. What is the run time (inference) for each method to solve the MOTSP instance with different numbers of objectives and cities?
The details for Hypervolume calculation (e.g., the reference points for different instances) are missing.
8. Surprising Results on the Ablation Studies:
It is quite surprising that the transformer-based encoder is not needed for the TSP-Net, and only a simple one-layer FC is needed for preference embedding. With this setting, the whole PA-Net should have a much smaller scale. What are the total numbers of parameters for these ablation settings (1, 2, and 3)? With the 1-D Conv encoder, the ablation 3 model should be even smaller than the Prt-Net. Why it still needs ~12 hrs to train the model? Why can this model still outperform the Ptr-Net model on the 2-objective MOTSP? Reference:
[1] Das, Indraneel, and John E. Dennis. "Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems." SIAM journal on optimization 8, no. 3: 631-657, 1998.
[2] Mavrotas, George. Effective implementation of the ε-constraint method in multi-objective mathematical programming problems. Applied mathematics and computation 213, no. 2: 455-465, 2009.
[3] Miettinen, Kaisa. Nonlinear multiobjective optimization. Vol. 12. Springer Science & Business Media, 2012.
[4] Kool, Wouter, Herke van Hoof, and Max Welling. Attention, Learn to Solve Routing Problems!. ICLR 2019.
5. Competitive Baselines: According to the experimental results, the learning-based solvers are much better than the heuristic-based solvers. However, for the single objective TSP, the SOTA heuristic-solver (e.g., Concorde) usually has the best performance. Since the obtained Pareto front is not highly non-convex (as in Figure 2), the results for linear scalarization + Concorde should be included for a better comparison.