Naive Feature Selection: a Nearly Tight Convex Relaxation for Sparse Naive Bayes
This work addresses feature selection for naive Bayes classification, offering a faster alternative to existing methods while maintaining statistical effectiveness, though it appears incremental as it builds on established sparse modeling techniques.
The authors tackled the problem of feature selection for naive Bayes classification by proposing a sparse version that leads to a combinatorial maximum-likelihood problem, providing an exact solution for binary data and a bound for multinomial cases with a convex relaxation that becomes tight as marginal contributions decrease. Numerical experiments on text data showed their method is statistically as effective as state-of-the-art methods like LASSO while being orders of magnitude faster.
Due to its linear complexity, naive Bayes classification remains an attractive supervised learning method, especially in very large-scale settings. We propose a sparse version of naive Bayes, which can be used for feature selection. This leads to a combinatorial maximum-likelihood problem, for which we provide an exact solution in the case of binary data, or a bound in the multinomial case. We prove that our convex relaxation bounds becomes tight as the marginal contribution of additional features decreases, using a priori duality gap bounds dervied from the Shapley-Folkman theorem. We show how to produce primal solutions satisfying these bounds. Both binary and multinomial sparse models are solvable in time almost linear in problem size, representing a very small extra relative cost compared to the classical naive Bayes. Numerical experiments on text data show that the naive Bayes feature selection method is as statistically effective as state-of-the-art feature selection methods such as recursive feature elimination, $l_1$-penalized logistic regression and LASSO, while being orders of magnitude faster.