Chien-Ming Chi

Chien-Ming Chi, Yingying Fan, Jinchi Lv

Quantifying the usefulness of individual features in random forests learning can greatly enhance its interpretability. Existing studies have shown that some popularly used feature importance measures for random forests suffer from the bias issue. In addition, there lack comprehensive size and power analyses for most of these existing methods. In this paper, we approach the problem via hypothesis testing, and suggest a framework of the self-normalized feature-residual correlation test (FACT) for evaluating the significance of a given feature in the random forests model with bias-resistance property, where our null hypothesis concerns whether the feature is conditionally independent of the response given all other features. Such an endeavor on random forests inference is empowered by some recent developments on high-dimensional random forests consistency. Under a fairly general high-dimensional nonparametric model setting with dependent features, we formally establish that FACT can provide theoretically justified feature importance test with controlled type I error and enjoy appealing power property. The theoretical results and finite-sample advantages of the newly suggested method are illustrated with several simulation examples and an economic forecasting application.

Chien-Ming Chi

We propose o1Neuro, a new neural network model built on sparse indicator activation neurons, with two key statistical properties. (1) Constructive universal approximation: At the population level, a deep o1Neuro can approximate any measurable function of $\boldsymbol{X}$, while a shallow o1Neuro suffices for additive models with two-way interaction components, including XOR and univariate terms, assuming $\boldsymbol{X} \in [0,1]^p$ has bounded density. Combined with prior work showing that a single-hidden-layer non-sparse network is a universal approximator, this highlights a trade-off between activation sparsity and network depth in approximation capability. (2) Sure convergence: At the sample level, the optimization of o1Neuro reaches an optimal model with probability approaching one after sufficiently many update rounds, and we provide an example showing that the required number of updates is well bounded under linear data-generating models. Empirically, o1Neuro is compared with XGBoost, Random Forests, and TabNet for learning complex regression functions with interactions, demonstrating superior predictive performance on several benchmark datasets from OpenML and the UCI Machine Learning Repository with $n = 10000$, as well as on synthetic datasets with $100 \le n \le 20000$.

Chien-Ming Chi

Orthogonal-split trees perform well, but evidence suggests oblique splits can enhance their performance. This paper explores optimizing high-dimensional $s$-sparse oblique splits from $\{(\vec{w}, \vec{w}^{\top}\boldsymbol{X}_{i}) : i\in \{1,\dots, n\}, \vec{w} \in \mathbb{R}^p, \| \vec{w} \|_{2} = 1, \| \vec{w} \|_{0} \leq s \}$ for growing oblique trees, where $ s $ is a user-defined sparsity parameter. We establish a connection between SID convergence and $s_0$-sparse oblique splits with $s_0\ge 1$, showing that the SID function class expands as $s_0$ increases, enabling the capture of more complex data-generating functions such as the $s_0$-dimensional XOR function. Thus, $s_0$ represents the unknown potential complexity of the underlying data-generating function. Learning these complex functions requires an $s$-sparse oblique tree with $s \geq s_0$ and greater computational resources. This highlights a trade-off between statistical accuracy, governed by the SID function class size depending on $s_0$, and computational cost. In contrast, previous studies have explored the problem of SID convergence using orthogonal splits with $ s_0 = s = 1 $, where runtime was less critical. Additionally, we introduce a practical framework for oblique trees that integrates optimized oblique splits alongside orthogonal splits into random forests. The proposed approach is assessed through simulations and real-data experiments, comparing its performance against various oblique tree models.