Tan Yan

Xin Chen, Jason M. Klusowski, Yan Shuo Tan

Stacking regressions is an ensemble technique that forms linear combinations of different regression estimators to enhance predictive accuracy. The conventional approach uses cross-validation data to generate predictions from the constituent estimators, and least-squares with nonnegativity constraints to learn the combination weights. In this paper, we learn these weights analogously by minimizing a regularized version of the empirical risk subject to a nonnegativity constraint. When the constituent estimators are linear least-squares projections onto nested subspaces separated by at least three dimensions, we show that thanks to an adaptive shrinkage effect, the resulting stacked estimator has strictly smaller population risk than best single estimator among them, with more significant gains when the signal-to-noise ratio is small. Here "best" refers to an estimator that minimizes a model selection criterion such as AIC or BIC. In other words, in this setting, the best single estimator is inadmissible. Because the optimization problem can be reformulated as isotonic regression, the stacked estimator requires the same order of computation as the best single estimator, making it an attractive alternative in terms of both performance and implementation.

Yan Shuo Tan, Jason M. Klusowski, Krishnakumar Balasubramanian

Models based on recursive adaptive partitioning such as decision trees and their ensembles are popular for high-dimensional regression as they can potentially avoid the curse of dimensionality. Because empirical risk minimization (ERM) is computationally infeasible, these models are typically trained using greedy algorithms. Although effective in many cases, these algorithms have been empirically observed to get stuck at local optima. We explore this phenomenon in the context of learning sparse regression functions over $d$ binary features, showing that when the true regression function $f^*$ does not satisfy Abbe et al. (2022)'s Merged Staircase Property (MSP), greedy training requires $\exp(Ω(d))$ to achieve low estimation error. Conversely, when $f^*$ does satisfy MSP, greedy training can attain small estimation error with only $O(\log d)$ samples. This dichotomy mirrors that of two-layer neural networks trained with stochastic gradient descent (SGD) in the mean-field regime, thereby establishing a head-to-head comparison between SGD-trained neural networks and greedy recursive partitioning estimators. Furthermore, ERM-trained recursive partitioning estimators achieve low estimation error with $O(\log d)$ samples irrespective of whether $f^*$ satisfies MSP, thereby demonstrating a statistical-computational trade-off for greedy training. Our proofs are based on a novel interpretation of greedy recursive partitioning using stochastic process theory and a coupling technique that may be of independent interest.

Xin Chen, Jason M. Klusowski, Yan Shuo Tan et al.

Combining randomized estimators in an ensemble, such as via random forests, has become a fundamental technique in modern data science, but can be computationally expensive. Furthermore, the mechanism by which this improves predictive performance is poorly understood. We address these issues in the context of sparse linear regression by proposing and analyzing an ensemble of greedy forward selection estimators that are randomized by feature subsampling -- at each iteration, the best feature is selected from within a random subset. We design a novel implementation based on dynamic programming that greatly improves its computational efficiency. Furthermore, we show via careful numerical experiments that our method can outperform popular methods such as lasso and elastic net across a wide range of settings. Next, contrary to prevailing belief that randomized ensembling is analogous to shrinkage, we show via numerical experiments that it can simultaneously reduce training error and degrees of freedom, thereby shifting the entire bias-variance trade-off curve of the base estimator. We prove this fact rigorously in the setting of orthogonal features, in which case, the ensemble estimator rescales the ordinary least squares coefficients with a two-parameter family of logistic weights, thereby enlarging the model search space. These results enhance our understanding of random forests and suggest that implicit regularization in general may have more complicated effects than explicit regularization.

Yan Shuo Tan, Abhineet Agarwal, Bin Yu

Decision trees are important both as interpretable models amenable to high-stakes decision-making, and as building blocks of ensemble methods such as random forests and gradient boosting. Their statistical properties, however, are not well understood. The most cited prior works have focused on deriving pointwise consistency guarantees for CART in a classical nonparametric regression setting. We take a different approach, and advocate studying the generalization performance of decision trees with respect to different generative regression models. This allows us to elicit their inductive bias, that is, the assumptions the algorithms make (or do not make) to generalize to new data, thereby guiding practitioners on when and how to apply these methods. In this paper, we focus on sparse additive generative models, which have both low statistical complexity and some nonparametric flexibility. We prove a sharp squared error generalization lower bound for a large class of decision tree algorithms fitted to sparse additive models with $C^1$ component functions. This bound is surprisingly much worse than the minimax rate for estimating such sparse additive models. The inefficiency is due not to greediness, but to the loss in power for detecting global structure when we average responses solely over each leaf, an observation that suggests opportunities to improve tree-based algorithms, for example, by hierarchical shrinkage. To prove these bounds, we develop new technical machinery, establishing a novel connection between decision tree estimation and rate-distortion theory, a sub-field of information theory.

Raaz Dwivedi, Yan Shuo Tan, Briton Park et al.

Building on Yu and Kumbier's PCS framework and for randomized experiments, we introduce a novel methodology for Stable Discovery of Interpretable Subgroups via Calibration (StaDISC), with large heterogeneous treatment effects. StaDISC was developed during our re-analysis of the 1999-2000 VIGOR study, an 8076 patient randomized controlled trial (RCT), that compared the risk of adverse events from a then newly approved drug, Rofecoxib (Vioxx), to that from an older drug Naproxen. Vioxx was found to, on average and in comparison to Naproxen, reduce the risk of gastrointestinal (GI) events but increase the risk of thrombotic cardiovascular (CVT) events. Applying StaDISC, we fit 18 popular conditional average treatment effect (CATE) estimators for both outcomes and use calibration to demonstrate their poor global performance. However, they are locally well-calibrated and stable, enabling the identification of patient groups with larger than (estimated) average treatment effects. In fact, StaDISC discovers three clinically interpretable subgroups each for the GI outcome (totaling 29.4% of the study size) and the CVT outcome (totaling 11.0%). Complementary analyses of the found subgroups using the 2001-2004 APPROVe study, a separate independently conducted RCT with 2587 patients, provides further supporting evidence for the promise of StaDISC.