GBHT: Gradient Boosting Histogram Transform for Density Estimation
This work provides a theoretical explanation for why boosting improves density estimation, which is incremental for researchers in machine learning and statistics.
The authors tackled the problem of density estimation by proposing GBHT, a gradient boosting method using histogram transforms, and proved it achieves faster convergence rates than its base learner under certain smoothness assumptions.
In this paper, we propose a density estimation algorithm called \textit{Gradient Boosting Histogram Transform} (GBHT), where we adopt the \textit{Negative Log Likelihood} as the loss function to make the boosting procedure available for the unsupervised tasks. From a learning theory viewpoint, we first prove fast convergence rates for GBHT with the smoothness assumption that the underlying density function lies in the space $C^{0,α}$. Then when the target density function lies in spaces $C^{1,α}$, we present an upper bound for GBHT which is smaller than the lower bound of its corresponding base learner, in the sense of convergence rates. To the best of our knowledge, we make the first attempt to theoretically explain why boosting can enhance the performance of its base learners for density estimation problems. In experiments, we not only conduct performance comparisons with the widely used KDE, but also apply GBHT to anomaly detection to showcase a further application of GBHT.