On Imbalanced Regression with Hoeffding Trees

Learning high-quality oblique decision trees remains a significant challenge due to the discrete and non-convex nature of split optimization. We present the Hinge Regression Tree (HRT) framework, which reframes each oblique split as a nonlinear least-squares problem over two linear predictors whose max/min envelope induces ReLU-like representation capacity. We show that the resulting node-level optimization can be interpreted as a damped Newton method, and we establish the monotonic decrease of the node objective for its backtracking line-search variant. We establish, theoretically, that HRT is a universal approximator with an explicit $O(δ^2)$ approximation rate. Building upon this base learner, we propose HRT-Boost, a mathematically synergistic ensemble extension that couples node-level Newton updates with stage-wise functional gradient descent. We show that this ensemble construction admits a stage-wise empirical risk reduction guarantee under the squared loss. Empirical evaluations on synthetic and real-world benchmarks show that HRT is highly competitive with established single-tree baselines, and HRT-Boost compares favorably with strong ensemble baselines and often yields substantially more compact models. The code is publicly available at https://github.com/Hongyi-Li-sz/HRT-Boost.

Bagging-based ensembles, most notably Adaptive Random Forests, are among the strongest performers for learning from data streams. A common denominator across these methods is their reliance on Hoeffding Trees as base learners, which grow decision trees incrementally by testing whether a candidate split is significantly better than its alternatives using concentration inequalities. Despite their empirical success, existing variants lack valid statistical guarantees. Current analyses rely on fixed-sample concentration bounds, while split decisions are made using data-dependent stopping rules, which invalidates their guarantees and can drive the probabilty of incorrect splits to one. We introduce a principled alternative based on anytime-valid inference. Our method provides: (i) anytime-valid control of false splits under arbitrary data streams, including non-stationary settings; (ii) finite commitment time under a predictive advantage; and (iii) under stationary i.i.d. data, risk is monotone decreasing and strictly improves at every split. Empirically, we evaluate both standalone trees and their use within Adaptive Random Forests on non-stationary streams. Our method improves performance while producing substantially smaller trees.

Streaming decision trees are natural candidates for open-world continual learning, as they perform local updates, enjoy bounded memory, and static decision boundaries. Despite these, they still fail in online class-incremental learning due to two coupled miscalibrations: (i) their split criterion grows unreliable as the class count K expands, and (ii) the absence of knowledge transfer at split time. Both failures share a common root: the range of Information Gain intrinsically scales with log2 K. Consequently, any Hoeffding-style confidence radius derived from it must inevitably grow with the class count, making a K-independent split criterion structurally impossible, taking away the potential benefits of applying streaming decision trees to continual learning. To fix this issue, we present MIST (McDiarmid Incremental Streaming Tree), which resolves both failures through three integrated components: (i) a tight, K-independent McDiarmid confidence radius for Gini splitting that acts as a structural regulariser; (ii) a Bayesian inheritance protocol that projects parent statistics to child nodes via truncated-Gaussian moments, with variance reduction guarantees strongest precisely when splitting is most conservative; and (iii) per-leaf KLL quantile sketches that support both continuous threshold evaluation and geometry-adaptive leaf prediction from a single data structure. On standard and stress-test tabular streams, MIST is competitive with global parametric methods on near-Gaussian benchmarks and uniquely robust on non-Gaussian geometry where SOTA benchmarks collapse.