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Abductive Learning |
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================== |
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Traditional supervised machine learning, e.g. classification, is predominantly data-driven. Here, a set of training examples \left\{\left(x_1, y_1\right), \ldots,\left(x_m, y_m\right)\right\} is given, where x_i \in \mathcal{X} is the i-th training instance, y_i \in \mathcal{Y} is the corresponding ground-truth label. These data are then used to train a classifier model f: \mathcal{X} \mapsto \mathcal{Y} to accurately predict the unseen data. |
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Integrating the Power of Machine Learning and Logical Reasoning |
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--------------------------------------------------------------- |
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（可能加一张图，比如左边是ML，右边是ML+KB） |
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Traditional supervised machine learning, e.g. classification, is |
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predominantly data-driven, as shown in the below figure. |
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Here, a set of training examples :math:`\left\{\left(x_1, y_1\right), |
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\ldots,\left(x_m, y_m\right)\right\}` is given, |
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where :math:`x_i \in \mathcal{X}` is the :math:`i`-th training |
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instance, :math:`y_i \in \mathcal{Y}` is the corresponding ground-truth |
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label. These data are then used to train a classifier model :math:`f: |
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\mathcal{X} \mapsto \mathcal{Y}` to accurately predict the unseen data. |
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In Abductive Learning (ABL), we assume that, in addition to data as examples, there is also a knowledge base \mathcal{KB} containing domain knowledge at our disposal. We aim for the classifier f: \mathcal{X} \mapsto \mathcal{Y} to make correct predictions on unseen data, and meanwhile, the logical facts grounded by \left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\} should be compatible with \mathcal{KB}. |
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.. image:: ../img/ML.jpg |
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:width: 600px |
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The process of ABL is as follows: |
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In **Abductive Learning (ABL)**, we assume that, in addition to data as |
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examples, there is also a knowledge base :math:`\mathcal{KB}` containing |
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domain knowledge at our disposal. We aim for the classifier :math:`f: |
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\mathcal{X} \mapsto \mathcal{Y}` to make correct predictions on unseen |
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data, and meanwhile, the logical facts grounded by |
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:math:`\left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\}` |
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should be compatible with :math:`\mathcal{KB}`. |
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1. Upon receiving data inputs \left\{x_1,\dots,x_m\right\}, pseudo-labels \left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\} are obtained, predicted by a data-driven classifier model. |
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2. These pseudo-labels are then converted into logical facts \mathcal{O} that are acceptable for logical reasoning. |
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3. Conduct joint reasoning with \mathcal{KB} to find any inconsistencies. |
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4. If found, the logical facts contributing to minimal inconsistency can be identified and then modified through abductive reasoning, returning modified logical facts \Delta(\mathcal{O}) compatible with \mathcal{KB}. |
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5. These modified logical facts are converted back to the form of pseudo-labels, and used for further learning of the classifier. |
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6. As a result, the classifier is updated and replaces the previous one in the next iteration. |
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The process of ABL is as follows: |
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This process is repeated until the classifier is no longer updated, or the logical facts \mathcal{O} are compatible with the knowledge base. |
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1. Upon receiving data inputs :math:`\left\{x_1,\dots,x_m\right\}`, |
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pseudo-labels |
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:math:`\left\{f(\boldsymbol{x}_1), \ldots, f(\boldsymbol{x}_m)\right\}` |
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are predicted by a data-driven classifier model. |
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2. These pseudo-labels are then converted into logical facts |
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:math:`\mathcal{O}` that are acceptable for logical reasoning. |
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3. Conduct joint reasoning with :math:`\mathcal{KB}` to find any |
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inconsistencies. If found, the logical facts that lead to minimal |
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inconsistency can be identified. |
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4. Modify the identified facts through **abductive reasoning** (or, **abduction**), |
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returning revised logical facts :math:`\Delta(\mathcal{O})` which are |
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compatible with :math:`\mathcal{KB}`. |
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5. These revised logical facts are converted back to the form of |
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pseudo-labels, and used like ground-truth labels in conventional |
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supervised learning to train a new classifier. |
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6. The new classifier will then be adopted to replace the previous one |
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in the next iteration. |
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This above process repeats until the classifier is no longer updated, or |
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the logical facts :math:`\mathcal{O}` are compatible with the knowledge |
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base. |
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The following figure illustrates this process: |
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一张图 |
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.. image:: ../img/ABL.jpg |
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We can observe that in the above figure, the left half involves machine |
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learning, while the right half involves logical reasoning. Thus, the |
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entire abductive learning process is a continuous cycle of machine |
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learning and logical reasoning. This effectively forms a paradigm that |
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is dual-driven by both data and domain knowledge, integrating and |
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balancing the use of machine learning and logical reasoning in a unified |
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model. |
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We can observe that in the above figure, the left half involves machine learning, while the right half involves logical reasoning. Thus, the entire abductive learning process is a continuous cycle of machine learning and logical reasoning. This effectively form a dual-driven (data & knowledge driven) learning system, integrating and balancing the use of machine learning and logical reasoning in a unified model. |
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What is Abductive Reasoning? |
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
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Abductive reasoning, also known as abduction, refers to the process of |
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selectively inferring certain facts and hypotheses that explain |
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phenomena and observations based on background knowledge. Unlike |
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deductive reasoning, which leads to definitive conclusions, abductive |
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reasoning may arrive at conclusions that are plausible but not conclusively |
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proven. It is often described as an ‘inference to the best explanation.’ |
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In Abductive Learning, given :math:`\mathcal{KB}` (typically expressed |
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in first-order logic clauses), one can perform deductive reasoning as |
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well as abductive reasoning. Deductive reasoning allows deriving |
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:math:`b` from :math:`a` only where :math:`b` is a formal logical |
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consequence of :math:`a`, while abductive reasoning allows inferring |
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:math:`a` as an explanation of :math:`b` (as a result of this inference, |
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abduction allows the precondition :math:`a` to be abducted from the |
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consequence :math:`b`). Put simply, deductive reasoning and abductive |
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reasoning differ in which end, left or right, of the proposition |
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“:math:`a\models b`” serves as conclusion. |
Gao Enhao
2 years ago
