A Quantitative Perspective on Values of Domain Knowledge for Machine Learning

Machine learning has achieved great success in numerous fields, such as computer vision, natural language processing, security, and robotics. The current success of machine learning, or more generally artificial intelligence (AI), owes much to the huge amount of available data. That being said, a priori domain knowledge, which comes from various sources and has multiple forms, also plays an increasingly more crucial role in the development of machine learning and may lead to new breakthroughs von Rueden et al. (2019); Wang et al. (2020b); Borghesi et al. (2020).

Domain knowledge has already been playing a part in current machine learning pipelines in many real-world applications. Specifically, machine learning with domain knowledge injection (a.k.a informed machine learning von Rueden et al. (2019)) can be accomplished in different ways. For example, in computer vision, image pre-processing such as flipping Benaim and Wolf (2018); Shyam et al. (2017), translation Benaim and Wolf (2018), cropping, Qi et al. (2018), scalingZhang et al. (2018) can help to augment data, which exploits the prior domain knowledge of image invariance. For another example, differential equations and logic rules from physical sciences and/or common knowledge provide additional constraints or new regularization terms for model training Battaglia et al. (2016); Borghesi et al. (2020); Silvestri et al. (2020); Muralidhar et al. (2018); Xu et al. (2018). Moreover, knowledge graphs can provide relational knowledge for reasoning or feature representation in deep learning Liang et al. (2018); Choi et al. (2017). In recent years, prior (domain) knowledge from one or more multiple teacher networks can also be injected into a learning task (i.e., student network) via distillation Hinton et al. (2015); You et al. (2017). As evidenced by many examples, such prior domain knowledge can effectively address the lack of labeled data commonly encountered in machine learning tasks, improving the learning performance in terms of various goals such as accuracy, generalization, and/or robustness.

In machine learning tasks, we need not only a model with good performance, but also model explainability Molnar (2020); Lipton (2018). The model explainability can help us better understand the model, use the model with more trust, or more quickly debug the deployed model. There have been many prior studies to address model explainability (especially in the context of deep neural networks or DNNs) from various perspectives, such as model features Molnar (2020), model inference Camburu (2020), and contribution of individual data samples Ghorbani and Zou (2019). Nonetheless, despite the increasingly wider usage of informed machine learning and and the clear advantages over its uninformed counterpart, a quantitative understanding of the actual contribution of domain knowledge injected into machine learning tasks has been lacking. This creates ambiguity about the role of domain knowledge in machine learning and decreases the explainability of the resulting models.

A foundation of domain knowledge explainability is to quantify the value or contribution of domain knowledge for machine learning tasks. In the first place, quantification of domain knowledge contribution gives us a better transparency of informed machine learning, which is part of the “right to explanation” legal requirement imposed in Europe for automated decision systems Goodman and Flaxman (2017). In addition, by domain knowledge quantification, we can know the actual contribution of different domain knowledge, which provides us with a principled guideline about what kind of domain knowledge is mostly needed to be collected in similar machine learning tasks. Last but not least, domain knowledge injected in machine learning may be implicit, inaccurate, and/or even corrupted. With domain knowledge quantification, we can make an informed decision as to which domain knowledge to utilize, since acquiring domain knowledge (e.g., coming from experts) may be quite expensive.

Nonetheless, it is non-trivial to quantify the value of domain knowledge in terms of its contribution of the performance of a learning task. First, domain knowledge has various forms and can be injected in machine learning in different ways von Rueden et al. (2019). Moreover, different domain knowledge is combined together and injected into a machine learning task, and the overall performance is typically not a simple linear combination of the performance corresponding to each piece of domain knowledge. Last but not least, we need to guarantee fairness for the quantification of domain knowledge in terms of its contribution to model performance.

To address these challenges, we leverage a classic tool from game theory — Shapley value — which provides a fair distribution of total payoff among several participants Shapley (1953). Shapley value has been recently used in machine learning model explainability to quantify the contribution of features Štrumbelj and Kononenko (2014), training data samples Ghorbani and Zou (2019), algorithms Yona et al. (2019), and neurons Ghorbani and Zou (2020). But, to our knowledge, quantifying the contribution of domain knowledge for machine learning is a novel perspective on model explainability that has not been considered. It will only become more important as informed machine learning becomes increasingly ubiquitous.

Our main contributions are summarized as follows.

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  We formulate the problem of quantifying the value of domain knowledge in informed machine learning, with a focus on the common approach to using regularization for domain knowledge injection into machine learning tasks.

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  We present a quantification method based on Shapley value to measure the value of domain knowledge in a fair manner. Additionally, we address the computational cost of Shapley value and give an approximated algorithm based on Monte-Carlo sampling, which achieves a polynomial time complexity.

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  As examples of quantifying the value of domain knowledge, we perform experiments on image classification tasks using semi-supervised learning over datasets of MNIST MNI and CIFAR10 CIF . Specifically, we consider symbolic domain knowledge and inject it into a learning task by adding a knowledge-related loss to the learning objective. Our experiments show the performance improvement by injecting different combinations of domain knowledge, from which we calculate the value of each domain knowledge for quantitative explainability. Our results also quantify and explain the performance degradation due to imperfect domain knowledge (on MNIST).

Explanation of DNN models. Since most of the current DNN models are black-box and hard to explain, model interpretability/explainability has attracted much attention from both academia and industry Molnar (2020); Lipton (2018). The techniques to explain DNN or AI models can be broadly categorized into intrinsic interpretation techniques and post-hoc interpretation Molnar (2020), depending on whether the adopted explainability method is integrated with the learning process and affects the model. Intrinsic explainability techniques usually rely on some explainable/interpretable models like regression models, decision tree Zhang et al. (2019), attention models Heo et al. (2018), etc. More recently, intrinsic explainability techniques are also used in deep learning to achieve self-explanatory neural networks Alvarez Melis and Jaakkola (2018); Shen et al. (2019). Intrinsic explainability can have more accurate explanations for the output of DNN models, but it may also decrease the learning performance to some extent.

Unlike intrinsic explainability, post-hoc explainability uses a new explainable model to explain another more complex model after the model is trained. Many model-based methods have been proposed for post-hoc interpretability/explainability Greenwell et al. (2018); Ribeiro et al. (2018, 2016). By contrast, explainability based on Shapley value is a model-free method and has been becoming increasingly popular for AI explanation due to its flexibility. Shapley value is proved to be the only method that satisfies a set of axioms of fairness in payoff distribution problems Shapley (1953). Among existing techniques based on Shapley value, Štrumbelj and Kononenko (2014) explains the features of model input, Ghorbani and Zou (2019) studies explanation of the value of data samples, and Ghorbani and Zou (2020) quantifies the contribution of neurons in neural networks and proposes a Bayesian optimization algorithm to reduce the evaluation complexity. The joint contribution of learning algorithms and dataset are studied in Yona et al. (2019) using Shapley value. Also, Ancona et al. (2019) propose a polynomial algorithm to compute Shapley values and explain input features for DNNs. In this paper, we leverage Shapley value for model explainability from a new perspective — quantifying the value of domain knowledge.

Informed Machine Learning Many techniques have been developed to inject domain knowledge in machine learning tasks von Rueden et al. (2019); Wang et al. (2020b). One naive approach is to generate additional synthetic training data based on domain knowledge. For example, samples generated from simulations or demonstration can help mitigate the data shortage problem in reinforcement learning Deist et al. (2019); Gao et al. (2018b); Hester et al. (2017); Karpatne et al. (2017); Rai et al. (2019). Additionally, data can be generated by knowledge graphs Jiang et al. (2018), generative models Gao et al. (2018a), etc. Domain knowledge can also benefit machine learning tasks by choosing hypothesis sets like regression model and neural architectures. A good hypothesis set can improve the learning performance as well as reduce the training complexity. Examples include domain knowledge based neural architecture search Chen et al. (2020); Towell and Shavlik (1994) and hyper-parameter tuning Mendoza et al. (2016).

Injecting domain knowledge into machine learning tasks via a loss term in the training objective function is another common method. For example, AI models applied in real world are often trained on the optimization objective with constraints originated from domain knowledge Borghesi et al. (2020); Silvestri et al. (2020); Muralidhar et al. (2018). Knowledge distillation from one or more multiple teacher networks also falls into this category of knowledge injection Hinton et al. (2015); You et al. (2017). Moreover, recent studies have shown that logical rules can also be integrated into machine learning by including a domain knowledge related loss term Xu et al. (2018); Hu et al. (2016).

To our knowledge, however, the values of domain knowledge injected into informed machine learning have not been rigorously or fairly quantified.

A key goal of machine learning is to find the intrinsic hypothesis for a learning task. Typically, a hypothesis $h$ represents a mapping from an input variable $X$ to an output variable $Y$. In some cases, a hypothesis is a distribution over the mapping. In conventional machine learning, a training dataset $\mathcal{D}$ is provided and a hypothesis set $\mathcal{H}$ is determined, usually by choosing a learning model $h(X,\theta)$. With a certain loss function and learning algorithm, the model $h(X,\theta)$ is learnt to approximate the optimal hypothesis $h^{*}=h(X,\theta^{*})$ which has the optimal performance among the chosen hypothesis set $\mathcal{H}$. For example, given a performance metric $R(h)$ (which can be risk, cost, or negative reward/utility), the optimal hypothesis in the chosen hypothesis set is $h^{*}=\arg\min_{h\in\mathcal{H}}R(h)$. By training based on the dataset $\mathcal{D}$, we get a final hypothesis $\hat{h}_{\mathcal{D}}=h(X,\hat{\theta}_{\mathcal{D}})$.

For a machine learning task, various forms of domain knowledge from multiple sources can be integrated into the machine learning pipeline for improving the model performance. For example, domain knowledge can be used for data augmentation von Rueden et al. (2019); Wang et al. (2020b) by generating synthetic data samples via simulations Deist et al. (2019); Hester et al. (2017); Karpatne et al. (2017), data transforming Qi et al. (2018); Zhang et al. (2018), and generative models Goodfellow et al. (2014); Gao et al. (2018a). With more (independent) samples in the augmented dataset $[\mathcal{D},\tilde{\mathcal{D}}_{K}]$, the generalization error can be reduced Guedj and Shawe-Taylor (2019). Also, domain knowledge is important for choosing a hypothesis set $\mathcal{H}$. This is usually done by choosing a learning model $h_{K}(X,\theta)$, such as knowledge-based neural architecture search Chen et al. (2020) and hyper-parameter tuning Mendoza et al. (2016). Moreover, domain knowledge can be used to decide whether some final hypothesis should be discarded von Rueden et al. (2019).

In this paper, we focus on another important approach to domain knowledge injection: adding knowledge-based constraints or loss terms as part of the objective function Borghesi et al. (2020); Silvestri et al. (2020); Muralidhar et al. (2018); Xu et al. (2018). Specifically, with a data-based loss $L(h,\mathcal{D})$ used in conventional machine learning and a loss $L_{K_{n}}(h,\bar{\mathcal{D}}_{n})$ based on domain knowledge $K_{n}$ applied to its corresponding (possibly unlabelled) dataset $\bar{\mathcal{D}}_{n}$, for $n=1,\cdots,N$, the training objective based on domain knowledge can be expressed as

where $\lambda_{n}\geq 0$ is used to balance the pure data-based loss and knowledge-based loss. In addition to knowledge-based constraints or losses, this formulation also applies to data augmentation based on domain knowledge: $\bar{\mathcal{D}}_{n}$ is the synthetic training dataset generated based on domain knowledge $K_{n}$ and we have $L_{K_{n}}(h,\bar{\mathcal{D}}_{n})=L(h,\bar{\mathcal{D}}_{n})$ for $n=1,\cdots,N$.

Assume that $R(h)$ is the performance metric of interest in a machine learning task given a set containing $N$ pieces of domain knowledge $\mathcal{K}=\left\{K_{1},K_{2},\cdots,K_{N}\right\}$. For conciseness, $K_{n}$ represents the $n$-th domain knowledge as well as the corresponding (possibly unlabelled) dataset $\bar{\mathcal{D}}_{n}$. With the final hypothesis $\hat{h}_{\mathcal{D},\mathcal{K}}$ learnt based on both training dataset $\mathcal{D}$ and domain knowledge in $\mathcal{K}$, the learning performance is denoted as $R(\hat{h}_{\mathcal{D},\mathcal{K}})$.

For informed machine learning with domain knowledge, the model performance highly relies on the included domain knowledge information. Naturally, different domain knowledge can have different contributions to the learning performance, and the performance metric can change a lot by adding, removing or revising a subset of the domain knowledge. Thus, to explain the learnt model and achieve better transparency, it is crucial to quantify the value of each domain knowledge for the considered learning task in terms of the metric of interest. Moreover, the quantification of domain knowledge values should be fair. In this section, we give a set of axioms for fair quantification of domain knowledge values and propose a method based on Shapley value to achieve fair quantification.

In this section, we run experiments to quantify the values of domain knowledge for two image classification tasks under a semi-supervised setting for which domain knowledge can significantly improve the learning performance.

In this paper, we study valuation of domain knowledge and provide a first step towards quantitative understanding of contribution of domain knowledge injected into informed machine learning. Specifically, we propose domain knowledge Shapley to quantify the values of domain knowledge in terms of the learning performance improvement for machine learning tasks in a fair manner. We run experiments on both MNIST and CIFAR10 datasets using semi-supervised learning and quantitatively measure the fair value of different domain knowledge. Our results also quantify the extent to which imperfect domain knowledge can affect the learning performance (for the MNIST dataset).