Fully-Dynamic Decision Trees

We develop the first fully dynamic algorithm that maintains a decision tree over an arbitrary sequence of insertions and deletions of labeled examples. Given $ε> 0$ our algorithm guarantees that, at every point in time, every node of the decision tree uses a split with Gini gain within an additive $ε$ of the optimum. For real-valued features the algorithm has an amortized running time per insertion/deletion of $O\big(\frac{d \log^3 n}{ε^2}\big)$, which improves to $O\big(\frac{d \log^2 n}ε\big)$ for binary or categorical features, while it uses space $O(n d)$, where $n$ is the maximum number of examples at any point in time and $d$ is the number of features. Our algorithm is nearly optimal, as we show that any algorithm with similar guarantees uses amortized running time $Ω(d)$ and space $\tildeΩ (n d)$. We complement our theoretical results with an extensive experimental evaluation on real-world data, showing the effectiveness of our algorithm.