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Dynamic Trees for Learning and Design

Matthew A. Taddy, Robert B. Gramacy, Nicholas G. Polson

arXiv:0912.1586v4stat.MEstat.CO

TL;DR

For complicated response surfaces in online settings, tree inference can require O(Bn^3) computation and batch MCMC must be rerun as data arrive. The paper reformulates regression trees as a dynamic model fit online and reports superior performance at lower cost than common alternatives.

  • Problem

    Sequential design with tree models requires O(Bn^3) computation for a posterior sample of size B, while batch MCMC must be rerun as new data arrive.

    Source: §1 · p. 3, §1.2 · p. 7

  • Method

    The paper reformulates regression trees as a dynamic model and uses sequential particle-based inference to fit them online.

    Source: §4 · p. 35

  • Results

    Dynamic regression trees provide superior performance and are less expensive than common alternatives.

    Source: §1 · p. 2, §4 · p. 35

  • Takeaways & Limitations

    The sequential characterization yields models that can be fit online, with scale-free automatic prediction.

    Source: §4 · p. 35

    Batch MCMC must be rerun for each new observation, constraining its use for sequential updates.

    Source: §1.2 · p. 7

Abstract

from arXiv · show

Dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in on-line application settings. We create a sequential tree model whose state changes in time with the accumulation of new data, and provide particle learning algorithms that allow for the efficient on-line posterior filtering of tree-states. A major advantage of tree regression is that it allows for the use of very simple models within each partition. The model also facilitates a natural division of labor in our sequential particle-based inference: tree dynamics are defined through a few potential changes that are local to each newly arrived observation, while global uncertainty is captured by the ensemble of particles. We consider both constant and linear mean functions at the tree leaves, along with multinomial leaves for classification problems, and propose default prior specifications that allow for prediction to be integrated over all model parameters conditional on a given tree. Inference is illustrated in some standard nonparametric regression examples, as well as in the setting of sequential experiment design, including both active learning and optimization applications, and in on-line classification. We detail implementation guidelines and problem specific methodology for each of these motivating applications. Throughout, it is demonstrated that our practical approach is able to provide better results compared to commonly used methods at a fraction of the cost.

1 Introduction

The paper develops dynamic regression trees and sequential particle inference for online regression, classification, and sequential design. By treating tree structure as a changing model state and marginalizing leaf-model parameters, the approach supports online prediction while addressing challenges of exploring tree space and updating models as observations arrive.

1 Introduction

  • Treating the partition tree as a dynamic state lets each new observation trigger small changes, reducing tree-space dimension without sacrificing flexibility.Particle learning provides online filtering of tree states.Source: §1 · p. 2
  • Examples show the proposed approach can outperform commonly used methods at a fraction of their cost.The introduction reports this broad comparison without specifying metrics or values.Source: §1 · p. 2
  • Given a tree, prediction uncertainty is available in closed form after marginalizing leaf-model parameters, leaving posterior sampling only for the partition state.The proposed prior specification is scale-free and automatic.Source: §1 · pp. 2, 4
  • Trees support flexible, nonstationary means and nonconstant variance, but trade off smoothness and explicit correlation structure for fast, interpretable online inference.Prediction requires searching for the rectangle containing a new input, and individual realizations yield interpretable decision trees.Source: §1 · p. 3

1.1 Partition Trees

  • Binary recursive partition trees split input space into axis-aligned rectangular regions, with predictions assigned at terminal nodes.The introduction motivates these partitions as a simple modeling structure with a model fitted within each region.Source: §1 · p. 2, §1.1 · p. 5
  • A partition tree assigns a simple regression or classification decision rule to each leaf, with the data likelihood factored across leaves.Each leaf receives the observations whose covariates fall within its partition.Source: §1.1 · p. 6

1.2 Inference for Tree Models

  • Tree MCMC can mix poorly because local moves make it difficult to reach distant tree structures, and batch inference must be rerun for each new observation.Moving between similarly probable trees with drastically different hierarchical splits can require an improbable sequence of prunes and grows.Source: §1.2 · p. 7
  • The authors introduce dynamic regression trees and sequential particle inference to address difficulties with existing tree MCMC methods.The passages characterize the approach as a new class of models with a sequential framework intended to alleviate MCMC difficulties.Source: §1.2 · p. 7

2 Dynamic Regression Trees

This section develops dynamic regression trees for sequential regression and classification, specifying constant, linear, and multinomial leaf models alongside particle learning for tree inference. It also describes approximate marginal-likelihood estimation and Bayes-factor comparisons of leaf models, including how these comparisons depend on the initial training sample and data order.

2 Dynamic Regression Trees

  • Dynamic regression trees update partition states as covariates arrive, combining local tree changes around each observation with particle resampling to capture global uncertainty.Each particle contains a tree and sufficient statistics for its leaf models.Source: §2 · pp. 7–8

2.1 Partition Evolution

  • The covariate-dependent transition prior permits stay, prune, or grow moves near the leaf containing the new covariate, with move types equally likely when available.Restricting changes to this neighborhood is intended to limit posterior simulation cost.Source: §2.1 · pp. 8–9

2.2 Prediction and Leaf Regression Models

  • Given a tree, posterior likelihood factors into leaf marginals, while prediction uses the containing leaf integrated over its parameters for constant, linear, and multinomial models.The predictive distribution conditions on the current tree rather than averaging over potential next trees.Source: §2.2 · pp. 10–11
  • Simple leaf models avoid the inference costs of integrating more complex regression models, and extra modeling does not necessarily improve performance.Integrating over regression parameters can require reversible-jump MCMC or higher-dimensional particle representations.Source: §2.2 · p. 12

2.2.1 Constant Mean Leaves

  • Constant-mean leaves use a minimum-data condition, and their sufficient statistics are easy to update when leaf membership changes.The stated condition is |η| > 2.Source: §2.2.1 · p. 12

2.2.2 Linear Mean Leaves

  • Linear leaves use centered covariates and updated regression statistics, with a minimum leaf size of |η| ≥ d+2.The matrices can be adapted through partitioned inverse equations.Source: §2.2.2 · p. 13

2.2.3 Multinomial Leaves

  • Multinomial leaves summarize class responses with counts and use a default Dirichlet(1_C/C) prior to obtain posterior class probabilities.The posterior probability vector is summarized by (z_η + 1/C)/(|η| + 1).Source: §2.2.3 · p. 14

2.3 Particle Learning for Posterior Simulation

  • Particle learning resamples globally and propagates local tree changes, integrating over leaf parameters so particles retain split rules and sufficient statistics.Tree-change probabilities can be calculated using only the neighborhood of the observation’s leaf; the resulting updates are low-dimensional.Source: §2.3 · pp. 15–17

2.4 Marginal Likelihood Estimation

  • Simple leaf models make reliable posterior marginal-likelihood estimates available through sequential estimation, while minimum-data restrictions provide training samples for proper leaf-parameter priors.Source: §2.4 · p. 17
  • Approximate marginal likelihood conditions on an initial sample large enough for a proper root-tree posterior predictive distribution, with predictive factors approximated via particle learning.The marginal likelihood increment is the normalizing constant for particle-learning resampling probabilities.Source: §2.4 · p. 18
  • Bayes factors compare leaf-model specifications, including linear versus constant leaves, and can assess which covariates should serve as leaf regressors.The linear-to-constant marginal likelihood ratio measures relative evidence for linear leaves.Source: §2.4 · p. 18
  • The inference is data-order dependent, and Bayes factors are conditional on the initial training sample, so comparisons require the same order and sample across models.Source: §2.4 · p. 18
  • Bayes factors yield consistent model choices when evidence is strong, while averages across repeated data reorderings and training samples provide an effective model-selection criterion.Source: §2.4 · p. 18

3 Application and Illustration

This section illustrates dynamic trees through regression-model comparisons, sequential optimization and active learning, and online multinomial classification. The examples examine tree-based criteria for choosing evaluation locations and search strategies.

3.1 Regression Model Comparison

  • Linear leaves adapt better to parabola data, while constant leaves have lower average tree height on motorcycle data (10.5 vs 12.8 for linear leaves).For parabola data, average tree height is 9.4 for constant leaves versus 5.4 for linear leaves.Source: §3.1 · p. 19
  • Linear leaves are preferred in every parabola-data run, whereas motorcycle-data Bayes factors lack consistent support and their mean shows fairly strong evidence against linear leaves.The Bayes factors are conditional on the first five observations and are not directly comparable across runs because orderings and training samples differ.Source: §3.1 · p. 20, Figure 3 · p. 21
  • On motorcycle data, TGP and BART have mean fits similar to dynamic trees, but their 90% predictive intervals vary in estimating uncertainty, and BART’s global variance is misspecified for heteroskedastic data.For parabola data, the models find practically identical fits except for the poorly performing constant-leaf dynamic tree.Source: §3.1 · p. 20, Figure 2 · p. 21
  • On Friedman-data predictive RMSE, dynamic tree models clearly outperform static counterparts; DTL performs nearly as well as or better than GP and BART; Figure 4 reports a sample of 100 for each estimator.Source: §3.1 · p. 22, Figure 4 · p. 23

3.2 Search Optimization

  • Dynamic regression trees are proposed as an efficient global model for hybrid local-global stochastic optimization, but their lack of interpolation limits their usefulness in deterministic optimization.The regression model is used to locate promising input areas and guide local search toward robust optimality.Source: §3.2 · pp. 22–23
  • With the 1-D sin/Cauchy function, higher precision quickly locates a robust minimum but does not continue wider exploration, and linear leaves fit the test function efficiently.The example uses linear leaves, 500 particles, and 100 candidate locations; φ controls the global scope of the search.Source: §3.2 · pp. 24–25
  • Constant-leaf trees yield better results than linear-leaf trees and the TGP routine on the 2-D exponential search, while TGP requires nearly 10 times the computation.The reported comparison is based on 50 runs; the table caption describes the mean solution and standard deviation but supplies no values in the passage.Source: §3.2 · p. 25, Table 1 · p. 26

3.3 Active Learning

  • Particle-learning active learning evaluates candidate heuristics across particles and updates the trees after each observation; constant and linear leaves permit closed-form heuristic calculations conditional on a tree.The procedures implement ALM and ALC, which select locations by predictive variance and expected reduction in predictive variance, respectively.Source: §3.3 · pp. 27–28
  • On the parabola, ALM and ALC plots are roughly similar and linear-model statistics are flatter than constant-model statistics; on heteroskedastic motorcycle data, the plots differ substantially.Figure 6 presents these comparisons for both constant- and linear-leaf dynamic trees.Source: Figure 6 · p. 29, §3.3 · p. 29
  • On 1-d sin/Cauchy, TGP ALC RMSE was 0.0641 versus 0.0660 for TGP ALM and 0.0755 for DTL ALC; TGP methods dominated.ALC edged out ALM, while sequential tree alternatives ranked among the better performers despite weaker results for static treed linear models.Source: §3.3 · pp. 30–31, Table 2 · p. 30
  • At t = 10, when data are insufficient to support a tree split, ALC is high where the response changes direction, especially where it changes most rapidly.Figure 7 shows snapshots of the linear-leaf dynamic tree’s ALC progression on the sin/Cauchy data.Source: Figure 7 · p. 30, §3.3 · p. 31
  • On 2-d exponential data, dynamic trees shared the top six spots only with TGP and clearly dominated their static analogues; DTC was the second-best performer.The passage notes that this fits the function well with constant leaves, while offline LHS/ME and GP methods performed poorly because the region of interest was confined to the bottom-left quadrant.Source: Table 2 · p. 30, §3.3 · p. 31

3.4 Classification

  • The multinomial-leaf tree assigns each new input to the class with the highest mean conditional posterior probability.Source: §3.4 · p. 32
  • For Cushing’s data, tree boundaries were axis-symmetric rather than GP’s radial-basis surface, and tree entropy concentrated near the top-left three-class border instead of being high almost everywhere.The plotted surfaces did not establish which classifier performed better.Source: §3.4 · pp. 32–33
  • On credit approval, the dynamic tree was the clear performance winner and the only classifier to beat 14% average misclassification; it was orders of magnitude faster than GP and TGP classifiers.The comparison used 1,000 particles for the dynamic tree versus 15,000 MCMC iterations for the GP and TGP classifiers.Source: §3.4 · p. 34, Table 3 · p. 34

4 Discussion

The paper reformulates regression trees as dynamic models that can be fit online, with scale-free prior specification and no need for parameter sampling. It reports superior performance at lower cost than common alternatives.

4 Discussion

  • The dynamic formulation creates a class of models that can be fit online, with scale-free automatic priors, no parameter sampling, and robust, flexible prediction.Source: §4 · p. 35
  • Dynamic regression trees provide superior performance and are less expensive than common alternatives.Source: §4 · p. 35
  • Filtering particles contain only split rules and sufficient statistics, allowing sequential filtering to discard partition models that predict poorly.Narrowing the posterior search is essential given the size and complexity of the tree space.Source: §4 · p. 35
  • Restrictions on partition size permit improper priors for constant and linear leaves, which is not usually possible in particle inference.Source: §4 · p. 35
  • The authors hope sequential model characterizations and covariate-dependent update mechanisms will inform dynamic versions of other graphical structures.Source: §4 · p. 35

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