HyperLogic: Enhancing Diversity and Accuracy in Rule Learning with HyperNets

Summary

We are grateful that reviewer hMMc has a positive impression of the high-level design of our framework and the promising results of comprehensive experiments. To address your questions about the details of our method, we provide point-wise responses as follows.

Q1. The introduction of additional hyperparameters may increase sensitivity, which could impact the model's performance and generalizability.

The additional hyperparameters brought by our framework mainly include the hypernet architecture, the coefficient lamba1 of the diversity loss caused by the hypernet in the loss function, and the number of weight sampling times $k_1$ and $k_2$ during and after training. Among them, adding more layers in the architecture adjustment and the number of sampling times $k_1$ after training have little impact on the training, while the coefficient $\lambda_1$ and the number of sampling times $k_1$ during training have a more significant impact on the training results, which requires us to have a good choice of hyperparameters. Further strategies. Furthermore, we additionally consider an adaptive weight fusion mechanism to optimize model performance through weighted fusion between the weights generated by the hypernet and the original main network weights. This mechanism is controlled by a learnable parameter α, which can be automatically adjusted during the training process to deal with the instability caused by improper selection of hyperparameters. (For more details, please refer to Q1 of Reviewer Q4Et.)

Q2. The framework's applicability to a broader range of datasets and tasks needs to be investigated.

We extended the experiment and proved that our HyperLogic framework can choose other differentiable rule learning algorithms besides DRNet as the main network combination to solve more tasks, such as processing high-dimensional large-scale data with multiple tasks. Classification tasks reflect the broad application scenarios of our framework.

Specifically, the core feature of our framework, HyperLogic, is as an inclusive framework to allow enhancements to all neural methods as long as they are relevant to learning a set of interpretable weights end-to-end. This means that we can choose different differentiable rule learning networks as the main network according to needs to handle different tasks. Currently, for simplicity of presentation, our main network is derived from DRNet, which only supports binary classification tasks and is very time-consuming, preventing it from being proven on larger-scale data. To prove that our framework can handle larger data sets, we replace the rule learning model with the latest DIFFNAPS [1], which can perform multi-classification tasks in high-dimensional data.

Compared with the current data set, in our new data set, the feature dimension has been raised from a maximum of 154 dimensions to a maximum of 5k dimensions, the amount of data has been raised from a maximum of 24,000 to a maximum of 50,000, and the task has been raised from a maximum of 2 categories to a maximum of 50 categories, reflecting the characteristics of the task diversity and complexity. In this case, HyperLogic-enhanced DIFFNAPS showed a further average 6.1% F1 score improvement compared to the already good vanilla DIFFNAPS, proving the effectiveness of the framework (please refer to General Q1 of global response for task details).

[1] Walter, N. P., Fischer, J., & Vreeken, J. (2024, March). Finding interpretable class-specific patterns through efficient neural search. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 38, No. 8, pp. 9062-9070).

Q3. The computational overhead introduced by the hypernetworks should be discussed more.

Since our framework introduces an additional super network, more computing load is inevitable. We take supplementary experiments as an example (please refer to General Q1 of global response for task details) to demonstrate the original algorithm DIFFNAPS and the main network based on DIFFNAPS The average F1 score, time and number of parameters of the HyperLogic framework in different tasks are shown in the table.

It can be seen that although the parameter increase is large, considering the significant improvement in F1 score and the extremely limited increase in time consumption, especially since Hyperlogic allows us to generate diverse rule sets in one training, the increase in computational load is extremely worthwhile.

Summary: We first thank reviewer F7a8 for the insightful comments, especially for the questions about our model design and experiment details, which helped us to clarify our paper. We would like to address the concerns one by one.

Q1. The method is limited in network structure (2-layer architecture), data format(only binary datasets with labels), and rule language(only conjunctions in antecedent, single label in consequent).

Our method is a general framework that can enhance many existing neural rule-learning methods, since the idea of ​​hypernet can be applied to different types of primary network, as long as they are related to learning a set of interpretable weights in a differentiable way (for some minor restrictions on the primary network architecture caused by the introduction of hypernet, please see Q6). The HyperLogic framework has no additional restrictions on data format and rule language, and only inherits from the selected primary network.

As you mentioned, our implementation is currently limited in network structure, data format and rule language. But this is due to the limitations of DRNet, the primary network we refer to, not the limitations of our framework. We choose DRNet because the simplicity of the architecture is conducive to our proof and expression. However, considering the limitations of DR-Net, we can also consider other more flexible neural approach as the primary network to alleviate the restrictions on data format, rule language, etc. (please see Q2).

Q2. More related work, including recent neural approaches, should be considered.

Our proposed method introduces hypernetworks to generate weights for the main network, offering a novel framework. The main network is now a two-layer neural network specifically designed to retrieve rules directly from learned weights. However, other pre-existing neural methods can also be substituted, as long as they are related to end-to-end learning of a set of interpretable weights. Unlike traditional methods and recent Bayesian approaches, our method requires fewer hyperparameters and offers greater scalability. Differing from other neural methods, HyperLogic serves as a more versatile framework that can organically utilize these methods as the main network to enhance performance. Furthermore, our approach is capable of generating multiple sets of rule sets in a single training session, which is not feasible with other methods.   Theoretically, the neural methods described in the previous section can serve as the main network for our framework, just like DRNet. Since DIFFNAPS is a novel algorithm capable of differentiable pattern discovery on large-scale data, we have chosen to use DIFFNAPS as the main network in our supplemental experiments. This expansion of our experiments further demonstrates the efficacy of our framework (refer to Q3 for details).

Q3. The experiments should be extended to properly compare to existing work.

Please refer to General Q1 in the global response.

Q4. In section 4, you convert the process of drawing M samples with K components each into drawing MK single-component samples. Does the latter actually match the former?

1. The conversion mentioned is purely for theoretical proof purposes and is not implemented in the actual algorithm.
2. The preparation for the theoretical analysis (lines 175-191) remains valid regardless of the (in)dependence among the K components. Equation (9) is derived from the sample average of a distribution with equally weighted marginals.
3. To prove Theorem 1, we need to identify a set of parameters for equation (7) that satisfies the inequality stated in the theorem. One approach to achieve this is by using sampling weights transformed from Barron's theorem. It's important to note that this is just one possible set of parameters, and the actual learning algorithm does not utilize such sampling.

Q5. In Table 1, the bold numbers are confusing, as DR-Net performs similarly to the presented method on partial datasets within the standard error.

Our framework not only focuses on the accuracy of the optimal rule set, but also on the simplicity of the rule set and the richness of the learned candidate rule set. Now we simply selected the best rule set on the training set for testing, and have surpassed the results of DRNet in a more concise rule set (Figure 3), demonstrating the effectiveness of our framework. At the same time, the optimal rule set on the training set is not completely equal to the optimal rule set on the test set (Figure 2), indicating that our training strategy (such as hyperparameter selection) and the final rule set selection strategy (such as further considering the complexity of the rule set) can further obtain higher test accuracy. We leave this for future work.

Q6. The limitations of network structure.

HyperLogic has almost no special restrictions on the architecture of differentiable rule learning networks, except for two possible problems, but both can be solved:

1. If the main network has too many parameters, the hypernetwork might struggle to produce stable weights. To solve this, we tried two approaches: ① Combine the original weights with those generated by the hypernetwork to improve stability while keeping the hypernetwork's flexibility (please refer to Q1 of Reviewer Q4Et for details). ② Only generate some of the main network's weights using the hypernetwork. Our tests show this still significantly improves results, even with large datasets (please see Q3 for details).
2. Some neural methods use a non-differentiable technique called weight pruning to simplify the model. To address this, we've chosen to keep all weights and apply a sigmoid function to them after processing. This makes the entire process differentiable. While this change might affect how the data is distributed and require adjustments to the model's settings, our tests show it doesn't harm performance.

Summary

Many thanks to reviewer Q4Et for your positive comments and recognition of our contribution including an interesting framework, a robust theoretical foundation, comprehensive experiments, and a well-written paper. We would like to address your concerns one by one.

Q1. Are there any guidelines or automated processes to aid in choosing the optimal hyperparameters?

The additional hyperparameters brought by our framework mainly include the hypernet architecture, the coefficient lamba1 of the diversity loss caused by the hypernet in the loss function, and the number of weight sampling times $k_1$ and $k_2$ during and after training.

1. Regarding the hypernetwork architecture:

We followed HyperGAN, which is a simple and effective hypernetwork architecture, and has been proven in experiments that more layers will not bring significant gains; explore other more powerful hypernetworks The architecture will be interesting as our future work.

2. Regarding the number of sampling times $k_2$ after training:

$k_2$ only affects the results of our final extracted rules and does not directly affect the training of the model. Due to the simplicity constraint, the rule set we sampled is not easy to overfit. We can expand $k_2$ as much as possible to obtain a richer rule candidate set if resources and time allow.

3. Regarding the coefficient of diversity loss $\lambda_1$ and the number of sampling times $k_1$ in training:

The two have a more significant impact on the effect. In practice, we can set a smaller initial value and gradually increase it until classification error staturates. Further, inspired by your question, we considered extending our model to alleviate the challenge of hyperparameter selection. We no longer let the weight $W_{hyper}$ generated by the super network completely act on the weight of the main network, but retain the weight $W_{main}$ of the main network, and combine it with the weight generated by the super network to obtain $W_{main-final} = \alpha* W_{hyper} + (1-\alpha)*W_{main}$, where $\alpha$ is a learnable parameter restricted to 0-1. This can increase the stability of our results without losing the advantage that the hypernetwork can generate diverse weights: in extreme cases, we set very inappropriate hyperparameters about the hypernetwork and generate inappropriate $W_{hyper}$, and the algorithm will Prompt alpha to approach 0, thereby offsetting the impact of inappropriate hyperparameter selection and allowing the model to degenerate into a normal version without a hypernet. The effectiveness of such improvements is demonstrated in supplementary experiments, especially for high-dimensional data. Updated algorithms and results will be added to the final version.

Q2. The scalability of the approach for extremely large datasets or high-dimensional data remains unclear.

Our framework aims to enhance various existing neural rule-learning methods, since hypernet ideas can be applied to different types of primary networks, as long as they are related to learning a set of interpretable weights in a differentiable manner. Therefore, HyperLogic is a general framework. However, it should be noted that the framework essentially stems from the main network, that is, the framework's task scope, model capabilities, etc. are still closely related to the main network, and scalability is often limited by the size of the primary network. However, we can easily switch to a more flexible primary network according to the needs of the task. This is just a simple adaptation, while the main ideas stay the same.

Currently, for the sake of simplicity of description, our HyperLogic main network gets inspiration from DR-Net, which although has a simple architecture, only supports binary classification tasks and is very time-consuming, preventing HyperLogic from being proven on larger-scale data. To prove that our framework can handle larger data sets, we replaced HyperLogic's main network with the latest DIFFNAPS [1], which is a neural approach that can perform multi-classification tasks and pattern discovery in high-dimensional data.

Under this new configuration, we conducted experiments on a new dataset with feature dimensions up to 5,000, data size up to 50,000, and task types up to 50 categories. In this case, HyperLogic, which uses DIFFNAPS as the main network, further demonstrated an average 6.1% F1 score improvement compared to the already good vanilla DIFFNAPS, while the average time consumption only increased by 8.6% (please refer to General Q1 in global response for task details), effectively proving the practical application potential of the HyperLogic framework in processing large-scale data sets and high-dimensional data.

[1] Walter, N. P., Fischer, J., & Vreeken, J. (2024, March). Finding interpretable class-specific patterns through efficient neural search. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 38, No. 8, pp. 9062-9070).

Q3. Typo: in line 125, do all negative weights have the inputs of 0?

Thank you for your feedback. We have reviewed line 125 and acknowledge that all negative weights should have the inputs of -1 . This error will be corrected in our revised manuscript.