Learning from Imprecise Observations: An Estimation Error Bound based on Fuzzy Random Variables

In the problem of multi-class classification, researchers have proved that we can train a classifier that has good performance on the test set, as long as the training and test sets are precisely drawn from the same distribution and the size of the training set approaches infinity. However, in a realworld situation, such precise observations are often unavailable in some cases. For example, readings on analogue measurement equipment are not precise numbers but intervals since there is only a finite number of decimals available. Hence, in this paper, we propose a more realistic problem called learning from imprecise observations (LIMO), where we train a classifier with fuzzy observations (i.e., fuzzy vectors). We prove the estimation error bound of this novel problem based on the distribution of fuzzy random variables. This bound demonstrates that we can always learn the best classifier when we have infinite fuzzy observations. We also develop a practical algorithm to train a classifier using fuzzy observations. The experiment results verify the efficacy of our theory and algorithm.