ROOT: Weighted Trees/Forests for ROOT-style Functional Optimization

Rashomon-set Optimal Trees for interpretable functional optimization and treatment effect generalizability

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Overview

ROOT (Rashomon set of Optimal Trees) is a general functional optimization framework for learning interpretable binary weight functions, represented as sparse decision trees.

At its core, ROOT searches over functions $w(X) \in {0,1}$ that assign inclusion or exclusion weights to units based on their covariates $X$. The optimization objective (or loss function) can be chosen to reflect different scientific goals.

• Framework flexibility: By substituting different loss functions, ROOT can be applied to diverse settings.
• Interpretability: Solutions are restricted to sparse decision trees, ensuring outputs are human-readable and suitable for communication with collaborators.
• Rashomon principle: Rather than return a single solution, ROOT identifies a Rashomon set of near-optimal trees, highlighting multiple plausible explanations. A characteristic tree can then be extracted to summarize common patterns.

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ROOT for generalizability

The original ROOT paper demonstrates this framework in the context of generalizing causal effects from randomized trials to a target population.

• Problem: Randomized trials are internally valid but often differ from the target population. If trial participants underrepresent certain effect-modifying subgroups, estimates of the Target Average Treatment Effect (TATE) can be imprecise or misleading.
• Solution: ROOT introduces a refined estimand, the Weighted Target Average Treatment Effect (WTATE), defined over the subset of the population that is sufficiently represented.
  • Units are either included ($w(X) = 1$) or excluded ($w(X) = 0$) according to learned tree-structured rules.
  • The loss function is designed to minimize the variance of the estimator while retaining as much of the target population as possible.
• Interpretation: ROOT’s decision trees explicitly describe who is underrepresented and why they are excluded. The Rashomon set provides multiple, equally valid ways of describing these subgroups.

For a detailed worked example of ROOT in generalizability mode, see the generalizability_path_example vignette.

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Installation

You can install the development version of ROOT from GitHub with:

# install.packages("devtools")
devtools::install_github("peterliu599/ROOT")

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Example: Portfolio selection via variance minimization

ROOT can be applied to any optimization problem that can be expressed as a binary inclusion/exclusion decision. In this example, we use ROOT to select a minimum-variance portfolio from a universe of 100 simulated assets, each characterized by its market beta and annualized volatility. ROOT learns an interpretable tree-structured rule describing which assets to include ($w = 1$) to minimize portfolio return variance.

library(ROOT)
set.seed(123)

n_assets <- 100

# Asset features
volatility <- runif(n_assets, 0.05, 0.40)  # annualised volatility
beta       <- runif(n_assets, 0.5,  1.8)   # market beta
sector     <- sample(c("Tech", "Finance", "Energy", "Health"),
                     n_assets, replace = TRUE)

# Simulate returns correlated with beta and volatility
market      <- rnorm(1000, 0.0005, 0.01)
returns_mat <- sapply(seq_len(n_assets), function(i)
  beta[i] * market + rnorm(1000, 0, volatility[i] / sqrt(252))
)
vsq <- apply(returns_mat, 2, var)  # per-asset return variance (objective)

dat_portfolio <- data.frame(
  vsq    = vsq,
  vol    = volatility,
  beta   = beta,
  sector = as.integer(factor(sector))
)

portfolio_fit <- ROOT(
  data        = dat_portfolio,
  num_trees   = 20,
  top_k_trees = TRUE,
  k           = 10,
  seed        = 42
)

summary(portfolio_fit) # for a full summary output

## ROOT object
##   Generalizability mode: FALSE 
## 
## Summary classifier (f):
## n= 100 
## 
## node), split, n, loss, yval, (yprob)
##       * denotes terminal node
## 
## 1) root 100 4 1 (0.0400000 0.9600000)  
##   2) beta>=1.658134 12 4 1 (0.3333333 0.6666667)  
##     4) vol>=0.3285209 4 0 0 (1.0000000 0.0000000) *
##     5) vol< 0.3285209 8 0 1 (0.0000000 1.0000000) *
##   3) beta< 1.658134 88 0 1 (0.0000000 1.0000000) *
## 
## Global objective function:
##   User-supplied: No (default objective used)
## 
## Diagnostics:
##   Number of trees grown: 20
##   Rashomon set size: 10
##   % observations with w_opt == 1: 96.0%

print(portfolio_fit) # for a brief summary print

## ROOT object
##   Generalizability mode: FALSE 
## 
## Summary classifier (f):
## n= 100 
## 
## node), split, n, loss, yval, (yprob)
##       * denotes terminal node
## 
## 1) root 100 4 1 (0.0400000 0.9600000)  
##   2) beta>=1.658134 12 4 1 (0.3333333 0.6666667)  
##     4) vol>=0.3285209 4 0 0 (1.0000000 0.0000000) *
##     5) vol< 0.3285209 8 0 1 (0.0000000 1.0000000) *
##   3) beta< 1.658134 88 0 1 (0.0000000 1.0000000) *

plot(portfolio_fit)

The characterized tree recovers an intuitive and interpretable portfolio construction rule. Assets with beta < 1.7 are included (88% of the universe). Among the remaining high-beta beta $\geq$ 1.7 assets, those with volatility $\geq$ 0.33 are excluded ($w = 0$, 4%), while high-beta assets with volatility < 0.33 are retained (8%). Overall, the final majority vote by the Rashomon set of 10 near-optimal trees includes 96% of assets in the final portfolio and screens out only the most risk-concentrated subset.

For a detailed worked example of ROOT in optimization mode, see the optimization_path_example vignette.

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Issues

If you encounter any bugs or have any specific feature requests, please file an issue.

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License

The contents of this repository are distributed under the MIT license.

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Reference

Parikh, H., Ross, R. K., Stuart, E., & Rudolph, K. E. (2025). Who Are We Missing?: A Principled Approach to Characterizing the Underrepresented Population. Journal of the American Statistical Association, 120(551), 1414–1423. https://doi.org/10.1080/01621459.2025.2495319