Paper: Beyond Differentiability: Neurosymbolic Learning with Black-Box Programs (ICLR-2024)

Reviewer comment (Reproducibility):
Poor quantitative aspects of training, including memory requirements, training time for each model, and for each dataset.

Author response:
There was a tradeoff between scalability and expressiveness in this work. ISED favors expressiveness, allowing the use of general-purpose languages such as Python, over scalability. Hence, we did not include memory requirements and training time for our benchmarks in this paper. For the sake of completeness, we will include these numbers in the final version, but we expect that the training time of ISED will generally be higher than the training time of Scallop.

Reviewer comment (Reproducibility):
What are the shortcomings related to the quantitative aspects of training, such as memory requirements, training time for each model, and for each dataset?

Author response:
There was a tradeoff between scalability and expressiveness in this work. ISED favors expressiveness, allowing the use of general-purpose languages such as Python, over scalability. Hence, we did not include memory requirements and training time for our benchmarks in this paper. For the sake of completeness, we will include these numbers in the final version, but we expect that the training time of ISED will generally be higher than the training time of Scallop.

Reviewer comment (Novelty):
In the abstract, the authors mention that ‘existing general neuro-symbolic frameworks require that programs be written in differentiable logic programming languages’. However, there already exist frameworks aiming at bridging machine learning and logic reasoning such as Semantic Loss [1], Abductive Learning [2] and NEUROLOG [3], which do not impose a requirement for differentiability and they use non-differentiable programs. The paper does not conduct comparisons with such methods.

Author response:
Thank you for pointing out these references. We will cite them in the final version of this paper. The ideas in [1] and [4] seem similar, but our main contribution is in integrating sampling-based estimation of loss in the neurosymbolic learning pipeline. We are trying to implement the exact algorithm in [4] as a baseline for experimental evaluation, and will include it in the final version.

Reviewer comment (Novelty):
The novelty of the proposed work needs to improve to meet the desired standards. The method proposed in this paper involves employing the REINFORCE algorithm to eliminate the requirement for differentiability in neural-symbolic system. However, this idea has already been introduced in the previous work [4]. Another critical component of the method, 'Estimate', fundamentally constitutes a sampling estimation of the well-known semantic loss [1], yet the paper does not provide reference to this work.

Author response:
Thank you for pointing out these references. We will cite them in the final version of this paper. The ideas in [1] and [4] seem similar, but our main contribution is in integrating sampling-based estimation of loss in the neurosymbolic learning pipeline. We are trying to implement the exact algorithm in [4] as a baseline for experimental evaluation, and will include it in the final version.

Reviewer comment (Novelty):
The introduction of the black-box programs seems to constrain the problem largely. In the writing, it isn't clear if the programs are only used as supervision or explicitly used as the symbolic aspect. If it is the latter this greatly reduces the difficulty of the problem as the symbolic aspect is largely unneeded, as you could include an expert program to solve without needing symbolic, this is especially evident in the leaf disease test as there are a number of off the shelf expert models that could be applied without needing a threshold of % diseased. The approach would have been better argued by another approach, such as logical reasoning, ideally on Knowledge Graphs, which would be a compatible setting to prior methods, increasing the number of comparisons the authors could perform.

Author response:
ISED uses the programs for both supervision and the symbolic computation. In the leaf example, we calculate ground truth severity scores using the same program that we use as the symbolic aspect during training. To calculate the ground truth severity scores, we use human-labeled bounding boxes and corresponding areas calculated by SAM and pass them to the black-box program. Since all areas are labeled 1, the program simply returns the sum of the diseased areas in the bounding boxes. During training, we try to learn which of the segments returned by SAM contain diseased areas by using the same program as the symbolic component.

We think it would be beneficial to add a benchmark where we can use the pure neural model to generate knowledge graphs, and the black box can just verify this reasoning. We will add a knowledge graph benchmark in the final version.

Reviewer comment (Experiments):
It isn't clear how this would scale to larger, more complex problems where the black-box program actually is complex. All examples are relatively trivial.

Author response:
We agree that most examples are relatively simple; we do not have benchmarks as complex as say, Sudoku, a popular task in the neurosymbolic literature. We also acknowledge that similar to prior work in neurosymbolic learning, there is a tradeoff between scalability and expressiveness in our work. ISED favors expressiveness, allowing the use of general-purpose languages such as Python, over scalability. That being said, we did include some tasks in the Leetcode benchmark suite that have large input spaces, as shown in Table 3 of the paper.

Reviewer comment (Reproducibility):
It isn't clear how they handle the gradients around the black box as they just state an optimizer solves this without any specificity.

Author response:
**Q2:** We do not compute gradients explicitly, but rather use a custom loss function that is made to reward certain inputs and penalize others.

Reviewer comment (Writing):
Greater explanation of whether the programs are learnt or if only the inputs are being estimated.

Author response:
**Q1:** The programs are not being learned, only the inputs are being estimated

Reviewer comment (Theory):
Explanation of how gradients are propagated.

Author response:
**Q2:** We do not compute gradients explicitly, but rather use a custom loss function that is made to reward certain inputs and penalize others.

Reviewer comment (Experiments):
Any prediction of how this would scale to complex problems.

Author response:
**Q3:** We still need to do more work to determine how ISED would scale to complex problems. We have found that simple tricks like increasing the sample count or moving from the add/mult to min/max computation in the loss can help solve problems that are more complex, such as HWF. However, we would need to find more complex benchmarks for testing ISED to answer this question.

Reviewer comment (Experiments):
Limited Benchmark: The evaluation compares with Scallop and CNN. However, this work did not compare with another differentiating neurosymbolic program work: DeepProbLog, which limits the significance of the performance.

Author response:
Prior work shows that Scallop achieves comparable or higher accuracy than DeepProbLog (DPL) on many simple arithmetic, comparison, and counting tasks involving MNSIT images ([1], Section 6.3, Figure 15 and Table 4). Because Scallop outperformed DPL on all these tasks, we did not include DPL as a baseline. We will make sure to add DPL experiments later for the sake of completeness.

Reviewer comment (Experiments):
Scalability. This work can only cover an input length of 7, concluding the multiplication and additional operation. However, a traditional neurosymbolic program’s statement may not limit to these two operations.

Author response:
We acknowledge that similar to prior work in neurosymbolic learning, there is a tradeoff between scalability and expressiveness in our work. ISED favors expressiveness, allowing the use of general-purpose languages such as Python, over scalability. To this end, we used a modified version of the HWF task in our benchmarks which included only multiplication and addition operations to show that hand-written formulas, with a smaller input space, can still be learned under ISED.

Reviewer comment (Theory):
How accurate the sampling-based method is? Is there any theoretical analysis about the gradient error from the estimation?

Author response:
We don’t have any theoretical analysis for the sampling-based method yet.