Neurosymbolic AI combines neural networks with symbolic reasoners in an efort to create robust and logical machine learning frameworks. In one approach, a neural component processes raw data and outputs latent concepts. A symbolic component then conducts logical reasoning with the concepts to produce the final result. A major hurdle lies in the propagation of the end label signal to the latent space when no latent labels are available. We investigate the use of active learning to alleviate this problem. In particular, we consider the neurosymbolic framework Embed2Sym. We adapt the learning framework to incorporate active learning by gaining a latent learning signal for misclassified examples. An oracle, such as a human in the loop, provides latent labels, which are used to finetune the neural component. Using the same benchmark datasets as the original paper, we empirically evaluate our method. We demonstrate that even a small amount of labelled latent data leads to a sizeable increase in accuracy.
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Neurosymbolic AI aims to combine the robustness of neural networks to real-world data with
the explainability and provable correctness of symbolic reasoners [
One such framework is Embed2Sym [
In this paper, we propose to mitigate this problem by providing latent signals using active
learning, which allows the system to ask an oracle to annotate datapoints [
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Each task contains raw and symbolic inputs and labels representing the result of a downstream operation. For example, in MNIST Addition the input consists of images of two numbers and the output equates to their sum. Crucially, no labels are provided for the intermediate, called latent, representations. In our example, there are no labels for the numbers themselves, only their sum.
The Embed2Sym framework contains a neural perception component and a symbolic reasoning
component. The system works in three stages: [
It is a two-stage architecture containing a perception and a reasoning component, both of which are neural. Each input is processed by the perception network which creates embedding vectors. The reasoning network processes the concatenated embeddings to output the predicted label. 2. Cluster discovery The k-means algorithm divides the embedding space created by the perception network into clusters. The number of clusters is predetermined. 3. Cluster labelling An ASP algorithm assigns each cluster their symbolic meaning by means of an optimisation task. It utilises a hard-coded symbolic component to compute the downstream result from the latent concepts.
Perception network
Perception
network Cluster label 1
Cluster label
3 P y
At inference time, the algorithm works in three steps, as illustrated on the left side in Figure 1. First, the neural perception network turns the inputs into embeddings ⃗ . Second, the clustering assigns each embedding a symbolic label. Third, the hard-coded symbolic reasoning component calculates the end result.
The neural component in Embed2Sym generates embeddings of the latent concepts, for which it has no labels. Instead, it uses downstream labels for training, which is a more dificult task. The core idea of this paper is to finetune the neural network with latent labels after it has been trained end-to-end. We acquire the latent labels from two sources, as shown in Figure 1: 1. For all examples with a correct end prediction, we assume that the predicted latent concepts are correct as well. Therefore, we can use the cluster labels from the trained perception networks. This represents the vast majority of examples. 2. For all examples with an incorrect end prediction, we ask an oracle for the corresponding latent labels. We refer to these as active labels. For complex tasks, the oracle is typically a human in the loop. In our case, we can use existing labels for MNIST and CIFAR10 images.
Since the last layer in the original framework uses k-means clustering, it is not diferentiable. Therefore, we replace the clustering layer with a multi-layer-perceptron, indicated by the green boxes labelled ”dense layer”.
We assess the efect of our extension by performing the tasks outlined in the original paper:
MNIST and CIFAR10 addition, as well as Member. In the first two tasks, the inputs consist of
images representing numbers and the output indicates their sum. In the member task, the input
consists of an MNIST image and a list of numbers. The output is a binary variable indicating
whether the number is a member of the list. [
We investigate two main questions: 1. What proportion of labels needs to be obtained by an oracle? 2. Does active learning improve the accuracy of the results?
We conducted each experiment for five independent runs and show the average scores and their standard deviations.
Even though the latent labels for these specific tasks are easy to come by, calls to oracles are generally very expensive. Therefore, we need to evaluate our results in light of the percentage of active labels utilised.
Table 1 presents the number of active labels used for each task, as well as their proportion of the dataset. In each case, the percentages remain below 5%. We conclude that the number of necessary labels are feasible to obtain and the added efort is justifiable to achieve better accuracies.
Figure 2 shows the learning curves while finetuning the neural component for MNIST addition with 15 digits. The baseline is the accuracy achieved from the fully trained original framework. The extended model surpasses the baseline within a few number of epochs. The test accuracies are on average significantly higher than the baseline, with only one run being slightly below. Similar results occur for MNIST additions with a lower number of digits, as well as CIFAR10 addition, albeit not as pronounced.
In Member, the accuracy already surpasses the baseline after only round of training (epoch 0), as Figure 3 illustrates on the left. Interestingly, the accuracy starts to decline again after epoch 8. The graph on the right indicates that the model is overfitting, as the train and validation curves diverge. Further investigation reveals that 5.9% of incorrect latent labels were missed during the active labelling, because the downstream label was correct for them.
The nature of the member task facilitates correct end predictions despite wrong latent
predictions in two major ways: First, the label is binary, so a random guess can achieve an
accuracy of 50%. Second, most digits in each input list are irrelevant for the task. For example,
let the list be [
Active learning shows some promising results in our experiments. Providing a stronger signal
in the latent space helped achieve a better performance, especially when the baseline had room
for improvement. This work represents a first step towards investigating active learning in
neurosymbolic AI. Future work includes extending other frameworks, such as [