Nearest Neighbor Classifiers over Incomplete Information: From Certain Answers to Certain Predictions (2005.05117v2)

Abstract: Machine learning (ML) applications have been thriving recently, largely attributed to the increasing availability of data. However, inconsistency and incomplete information are ubiquitous in real-world datasets, and their impact on ML applications remains elusive. In this paper, we present a formal study of this impact by extending the notion of Certain Answers for Codd tables, which has been explored by the database research community for decades, into the field of machine learning. Specifically, we focus on classification problems and propose the notion of "Certain Predictions" (CP) -- a test data example can be certainly predicted (CP'ed) if all possible classifiers trained on top of all possible worlds induced by the incompleteness of data would yield the same prediction. We study two fundamental CP queries: (Q1) checking query that determines whether a data example can be CP'ed; and (Q2) counting query that computes the number of classifiers that support a particular prediction (i.e., label). Given that general solutions to CP queries are, not surprisingly, hard without assumption over the type of classifier, we further present a case study in the context of nearest neighbor (NN) classifiers, where efficient solutions to CP queries can be developed -- we show that it is possible to answer both queries in linear or polynomial time over exponentially many possible worlds. We demonstrate one example use case of CP in the important application of "data cleaning for machine learning (DC for ML)." We show that our proposed CPClean approach built based on CP can often significantly outperform existing techniques in terms of classification accuracy with mild manual cleaning effort.

• The paper introduces Certain Predictions (CP) and develops efficient polynomial-time algorithms for K-Nearest Neighbor classifiers operating on incomplete data.
• It provides polynomial-time algorithms for KNN to efficiently answer Certain Prediction queries on incomplete datasets, overcoming exponential complexity.
• The framework has practical uses in guiding data cleaning priorities and improving accuracy, introducing CPClean which outperforms existing data cleaning solutions.

Nearest Neighbor Classifiers over Incomplete Information: A Study on Certain Predictions

In the landscape of ML, the quality and completeness of data have long been recognized as critical factors in determining the performance and effectiveness of models. Real-world datasets often suffer from various forms of data incompleteness or inconsistency, which can adversely impact the models trained on them. This paper introduces the concept of Certain Predictions (CP) and rigorously explores its significance by extending the established database notion of Certain Answers to the domain of ML.

Overview of Certain Predictions

The paper defines Certain Predictions (CP) as a framework to analyze how incomplete information in training data can affect the outcomes of ML models. CP is stipulated as a property of a test example: it can be "certainly predicted" if all possible classifiers trained on all possible world representations of the incomplete dataset yield the same class label for it. Two fundamental queries related to CP are proposed:

1. Checking Query (Q1): Determines whether a test data example can be CP'ed.
2. Counting Query (Q2): Computes the number of classifiers that support a particular prediction for non-CP'ed data examples.

These queries are inherently challenging, given their reliance on reasoning across potential combinatorial versions of training data. However, the authors hone in on Nearest Neighbor (NN) classifiers to establish efficient methods for answering CP queries.

Efficient CP Algorithms for Nearest Neighbor Classifiers

To address the computational challenges associated with CP queries, the paper devises algorithms specific to K-Nearest Neighbor (KNN) classifiers. Surprising results are achieved wherein both CP queries can be executed in polynomial time, even over exponentially large sets of possible worlds. Notably:

• The paper introduces algorithms with complexity scaling polynomially in scenarios where traditional computations would demand exponential time. For instance, the general solution for Q1 and Q2 queries over KNN classifiers scales as $\mathcal{O}(N \cdot M \cdot \log(N \cdot M))$, significantly reducing computation time while handling large datasets.
• Additional optimizations such as sorting-based incremental computation further enhance the efficiency, allowing for real-time, scalable ML operations on incomplete datasets.

Practical and Theoretical Implications

The implications of the CP concept are multifaceted. Practically, CP provides insights into the prioritization of data cleaning efforts and enhances the accuracy of models trained over noisy data. The paper introduces CPClean, a strategic approach leveraging CP's framework to simplify manual data cleaning workload, thereby outperforming existing solutions such as ActiveClean and BoostClean.

Theoretically, the CP framework offers a structured method to propagate uncertainty and validation within ML models. It aligns with consistent query answering paradigms from database theory, suggesting broader applicability to diverse models beyond KNN, including models not reliant on gradient-based learning.

Future Directions

Future work should explore the extension of CP methods across a wider array of ML classifiers, examining both exact and approximate algorithmic approaches. Moreover, there’s potential in merging CP with sensitivity-analysis strategies from existing frameworks like ActiveClean—potentially yielding novel insights and improvements in data-driven machine learning applications.

In conclusion, the paper lays a substantial foundation for handling incomplete information with indexical precision in ML tasks, offering efficient, theoretically backed algorithms customizable to real-world data challenges. The findings represent a significant stride towards competence in data incompleteness, fostering a more robust integration of database theoretical methods in machine learning contexts.