A Link between Coding Theory and Cross-Validation with Applications
This work provides theoretical insights linking coding theory to cross-validation, with applications in statistical testing for machine learning algorithms, but it is incremental as it builds on existing frameworks.
The paper tackles the problem of determining how many binary classification problems a learning algorithm can solve on fixed data with zero or limited cross-validation errors, showing that the exact answers are given by error detecting codes theory, with specific results for AUC and leave-pair-out cross-validation, including bounds on code words in light constant weight codes and development of new randomization tests.
How many different binary classification problems a single learning algorithm can solve on a fixed data with exactly zero or at most a given number of cross-validation errors? While the number in the former case is known to be limited by the no-free-lunch theorem, we show that the exact answers are given by the theory of error detecting codes. As a case study, we focus on the AUC performance measure and leave-pair-out cross-validation (LPOCV), in which every possible pair of data with different class labels is held out at a time. We show that the maximal number of classification problems with fixed class proportion, for which a learning algorithm can achieve zero LPOCV error, equals the maximal number of code words in a constant weight code (CWC), with certain technical properties. We then generalize CWCs by introducing light CWCs, and prove an analogous result for nonzero LPOCV errors and light CWCs. Moreover, we prove both upper and lower bounds on the maximal numbers of code words in light CWCs. Finally, as an immediate practical application, we develop new LPOCV based randomization tests for learning algorithms that generalize the classical Wilcoxon-Mann-Whitney U test.