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

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

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.

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.

Installation

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

# install.packages("devtools")
# devtools::install_github("peterliu599/ROOT")
install.packages("ROOT", repos = c('https://peterliu599.r-universe.dev', 'https://cloud.r-project.org'))

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 choose which of 100 simulated assets to include in an equally weighted, low-variance portfolio. Each asset is characterized by its market beta and annualized volatility, and ROOT learns an interpretable tree-structured rule describing which assets to include ($w = 1$).

The objective is the return variance of the equally weighted portfolio of the included assets, $L(w) = w^\top \Sigma w / (\mathbf{1}^\top w)^2$, where $\Sigma$ is the return covariance matrix. Because every pair of included assets contributes a covariance term, the objective cannot be split into independent per-asset contributions; it is supplied to ROOT as a custom global_objective_fn.

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 from a single-factor market model
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))
)
Sigma_hat <- cov(returns_mat)  # estimated return covariance matrix

# Variance of the equally weighted portfolio of included assets
portfolio_variance <- function(D) {
  w <- D$w
  if (sum(w) == 0) return(Inf)
  sum(Sigma_hat[w == 1, w == 1]) / sum(w)^2  # = w' Sigma w / (1'w)^2
}

dat_portfolio <- data.frame(
  vsq    = diag(Sigma_hat),  # marginal variances; guide split sampling only
  vol    = volatility,
  beta   = beta,
  sector = as.integer(factor(sector))
)

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

summary(portfolio_fit)  # full summary
print(portfolio_fit)    # brief summary
plot(portfolio_fit)     # characterized tree

The characterized tree gives a plain-language rule for which asset types belong in the portfolio, and portfolio_fit$D_rash$w_opt holds the final inclusion decision for each asset. In this example the rule is driven mainly by market beta: ROOT keeps 33 mostly low-beta assets (beta below about 0.98), and the equally weighted portfolio of those assets has about 47% lower return variance than the equally weighted portfolio of all 100 assets.

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

Issues

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

License

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

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